SHSAT Sunset

Practice Test

Order of operations

Change this testNew TestYour practice tests

Question 1 of 23

2026 Practice Test A Math, question 10

Question

Three expressions are given:

−2(2.5n+4)-2(2.5n + 4)

5−n5 - n

3(2n+7)3(2n + 7)

Which of these are equivalent to the sum of these three expressions?

Select the two correct answers.

Choose every answer that is correct, then check. The DOE has not published how answers with several parts are scored; this site counts one right only when every part is right.

  • A5−n+6n+21−5n−85 - n + 6n + 21 - 5n - 8
  • B5−n+6n+21−5n+85 - n + 6n + 21 - 5n + 8
  • C5−n+5n+21−5n−85 - n + 5n + 21 - 5n - 8
  • D18−n18 - n
  • E18
  • F34
Show the answer

Correct answer: A and E

Add the three expressions: 5−n+3(2n+7)+(−2)(2.5n+4)5 - n + 3(2n + 7) + (-2)(2.5n + 4).

Distribute 3 and −2: 5−n+6n+21−5n−85 - n + 6n + 21 - 5n - 8.

Combine the variable terms and combine the constants: (−n+6n−5n)+(5+21−8)=18(-n + 6n - 5n) + (5 + 21 - 8) = 18.

Question analysis

This algebra question gives three expressions, one with a decimal coefficient and a negative factor, and asks which two of six choices are equivalent to their sum. It tests distributing, combining like terms and recognizing equivalent expressions. Signs from the negative factor are easy to lose, and several choices are built from exactly those slips.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
  • 6.EE.4 identifying equivalent expressions
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Expanding and combining linear expressions with a negative factor and a decimal coefficient is standard grade 7 work. Choosing the two equivalent ones out of six, where each common slip appears as a choice, takes more care than one simplification.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
4 steps
Reading level
Question and choices: about grade 8, 28 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The six choices form three consistent pairs, each an expansion with its simplified total: the right pair, a pair built on a sign slip with −2 × 4 that totals 34, and a pair built on 3 × 2n read as 5n that totals 18 − n. A student who makes either slip finds two choices that agree with each other and seem to confirm the work. The question rewards one careful simplification and the confidence to accept a sum with no n left.

  1. Error-prone arithmetic moderate Working it out

    The first expression puts a negative factor in front of a decimal term: −2 times 2.5n is −5n, and the −2 must reach the 4 as well, giving −8, not +8. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    The first two expansions are identical except for their last sign, − 8 against + 8, and the third differs from the first only in 5n for 6n, so only a term-by-term check separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Many steps minor Working it out

    Two distributions come first, then the n terms and the constants are collected separately, and six choices are compared with the result; both right ones must be picked to score. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  4. An answer in an unexpected form minor Choosing the answer

    One right answer is the expansion before any terms are combined and the other is the plain number 18, which has to be accepted as equal to a sum in which every expression held n. (Enright, Morley & Sheehan, 2002)

Strategy: Distribute each factor across its parentheses, keeping the sign with every product, then combine the n terms and the constants separately and test each of the six choices against the result.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
  • Answer in another form: The right answer is written differently from the way you worked it out.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

The distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number in front of the parentheses, and b and c are the terms inside. You multiply a by every term inside, not only the first one. If a is negative, its minus sign goes into every product. It works because a groups of b + c is a groups of b plus a groups of c.
Multiplying signed numbers
(−a)×b=−(a×b)(-a) \times b = -(a \times b)
a and b are positive numbers, so −a is negative. A negative number times a positive number is negative. Use it whenever a minus sign sits in front of the parentheses: −2 times 4 is −8, not 8.
Combining like terms
an+bn=(a+b)nan + bn = (a + b)n
Like terms have the same letter part, such as 6n and −5n. a and b are their coefficients, the numbers in front of n. Add the coefficients and keep the n. Constants, the plain numbers, combine only with other constants.

  1. Sort out the question

    An expression is a math phrase made of numbers, letters and signs, with no equal sign. The sum of the three expressions means you add all of them:

    −2(2.5n+4)+(5−n)+3(2n+7)-2(2.5n + 4) + (5 - n) + 3(2n + 7)

    You need the two choices that are equivalent to this sum. Equivalent means equal for every value of n. Some choices show the sum before it is simplified. Others show the result.

  2. Distribute the 3

    To distribute means to multiply the number outside the parentheses by each term inside.

    3(2n+7)=3×2n+3×7=6n+213(2n + 7) = 3 \times 2n + 3 \times 7 = 6n + 21

    Watch out: 3 times 2n is 6n. Adding 3 and 2 instead gives 5n. That slip leads to choice C, and then to 18 − n, choice D.

  3. Distribute the −2

    Now multiply −2 by both terms. A negative times a positive is negative, so both products are negative.

    −2(2.5n+4)=−2×2.5n+(−2)×4=−5n−8-2(2.5n + 4) = -2 \times 2.5n + (-2) \times 4 = -5n - 8

    Watch out: the −2 multiplies the 4 too. Writing +8 instead of −8 leads to choice B. That sum works out to 34, choice F.

  4. Write the whole sum

    Put the three pieces together, starting with 5 − n:

    5−n+6n+21−5n−85 - n + 6n + 21 - 5n - 8

    This matches choice A exactly, term by term. So A is one of the answers.

  5. Combine like terms

    Gather the n terms first. Remember that −n means −1 times n.

    −n+6n−5n=(−1+6−5)n=0n=0-n + 6n - 5n = (-1 + 6 - 5)n = 0n = 0

    Then gather the constants:

    5+21−8=185 + 21 - 8 = 18

    The n terms cancel out, so the whole sum is just 18.

    Watch out: a sum with no n left is fine. Keeping an extra −n leads to choice D.

  6. Pick the two answers

    Choice A is the sum before you combine like terms. Choice E, 18, is the sum after you combine them. Both equal the sum for every value of n. So the answers are A and E.

Check your answer

Try a value, such as n = 2, in each expression. −2(2.5×2+4)=−2×9=−18-2(2.5 \times 2 + 4) = -2 \times 9 = -18. 5−2=35 - 2 = 3. 3(2×2+7)=3×11=333(2 \times 2 + 7) = 3 \times 11 = 33. The sum is −18+3+33=18-18 + 3 + 33 = 18, which matches E. Choice D would give 18 − 2 = 16 instead.

More questions like this

The original DOE question
Question 10 as shown on the DOE’s digital test, page 81 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test A Math, page 81, question 10.

Question 2 of 23

2026 Practice Test A Math, question 27

Question

Complete the steps to show one way to solve the equation 24=−0.5(x+8)24 = -0.5(x + 8) for x.

Move the correct answer to each box. Each answer may be used more than once. Not all answers will be used.

24=−0.5(x+8)24 = -0.5(x + 8)

24=−0.5x+24 = -0.5x +

=−0.5x= -0.5x

=x= x

On the test you drag tiles into the boxes; here, choose each box’s tile from its menu. A tile can be used more than once. The DOE has not published how answers with several parts are scored; this site counts one right only when every part is right.

Show the answer

Correct answer: blank 1: −4; blank 2: 28; blank 3: −56

Solution:

Answer for second equation: −4

Answer for third equation: 28

Answer for fourth equation: −56

24=−0.5(x+8)24 = -0.5(x + 8)

Distribute the −0.5 on the right side of the equation: 24=−0.5x−424 = -0.5x - 4.

Add 4 to both sides of the equation to eliminate the −4 on the right side: 28=−0.5x28 = -0.5x.

Divide both sides by −0.5: −56=x-56 = x.

Question analysis

Students complete a step-by-step solution of the equation 24 = −0.5(x + 8) by dragging number tiles into three boxes: one after distributing, one after collecting the constants, and one for x. It tests solving a two-step linear equation with a negative decimal. Tiles may be reused, and many come in positive and negative pairs, so every sign counts.

NYS standards
  • 7.EE.4a two-step equations
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Solving equations of the form p(x + q) = r with rational numbers is named in the grade 7 standards. Here the steps are laid out, but the negative decimal factor forces a sign decision at every one of the three boxes.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
3 steps
Reading level
Question and choices: about grade 3, 39 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The layout settles the method, so the question comes down to one sign at each box. The printed plus before the first box nudges students toward the positive 4, and the pool then supplies every number that follows from it, down to −40. Students who distribute correctly can still lose the item at the last box by dividing 28 by 0.5. Substituting −56 back, −0.5 × (−48) = 24, confirms the final box.

  1. Error-prone arithmetic moderate Working it out

    Two of the three boxes turn on a negative decimal: −0.5 times 8 must give −4, and dividing 28 by −0.5 means multiplying by −2, which doubles the size and makes it negative. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer minor Choosing the answer

    The pool pairs −4 with 4, −40 with 40 and −56 with 56, so a box filled with the right size and the wrong sign looks just as finished; estimating cannot separate −56 from 56. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Many steps minor Working it out

    The three boxes form a chain and all three must be right, so a slip in the first carries through; the pool even holds the whole wrong chain 4, 20 and −40. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  4. An answer in an unexpected form minor Choosing the answer

    The second line prints a plus sign before its box, so the distributed −0.5x − 4 must be written as −0.5x + (−4), with the negative tile in the box. (Enright, Morley & Sheehan, 2002)

Strategy: Multiply −0.5 by both x and 8, undo the constant term by doing the opposite operation to both sides, then divide by −0.5, which is the same as multiplying by −2.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

The distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number in front of the parentheses, and b and c are the terms inside. Multiply a by each term inside. If a is negative, the minus sign goes into every product. It works because a groups of b + c is the same as a groups of b plus a groups of c.
Signs when you multiply or divide
(−)×(+)=(−)(−)×(−)=(+)(-) \times (+) = (-) \qquad (-) \times (-) = (+)
A negative number times a positive number gives a negative result. A negative times a negative gives a positive result. The same rules hold when you divide. Use them at every step where a negative number appears.
Doing the same to both sides
if a=b, then a+c=b+c and ac=bc\text{if } a = b \text{, then } a + c = b + c \text{ and } \frac{a}{c} = \frac{b}{c}
a and b are the two sides of an equation, and c is any number (not zero when you divide). An equation is like a balanced scale. Doing the same thing to both sides keeps it level. That is how you get x by itself.

  1. Sort out the question

    You solve 24=−0.5(x+8)24 = -0.5(x + 8) by filling three boxes:

    • Box 1 comes after −0.5x+-0.5x +, once the parentheses are gone.
    • Box 2 is the number equal to −0.5x-0.5x.
    • Box 3 is the value of x.

    A tile can be used more than once, and some tiles are not used.

  2. Distribute the −0.5

    Multiply −0.5 by each term inside the parentheses. A negative times a positive is negative:

    −0.5×x=−0.5x-0.5 \times x = -0.5x

    −0.5×8=−4-0.5 \times 8 = -4

    So 24=−0.5x+(−4)24 = -0.5x + (-4). The line already shows a plus sign, so box 1 gets −4.

    Watch out: the −0.5 multiplies the 8 too. Writing 4 instead starts a wrong chain: 4, then 20, then −40. Every number in that chain is a tile.

  3. Add 4 to both sides

    You want −0.5x alone on one side, and the −4 is in the way. Adding 4 undoes subtracting 4, so add 4 to both sides:

    24+4=−0.5x−4+424 + 4 = -0.5x - 4 + 4

    28=−0.5x28 = -0.5x

    Box 2 gets 28.

    Watch out: taking 4 away from 24 instead gives 20, another tile.

  4. Divide both sides by −0.5

    Now x is multiplied by −0.5. Dividing undoes multiplying, so divide both sides by −0.5:

    28÷(−0.5)=−5628 \div (-0.5) = -56

    Dividing by 0.5 is the same as multiplying by 2, since every whole holds two halves. A positive divided by a negative is negative. So x=−56x = -56.

    Watch out: dropping the minus sign gives 56, which is also a tile.

  5. Name the three tiles

    From top to bottom, the finished steps read:

    24=−0.5x+(−4)24 = -0.5x + (-4)

    28=−0.5x28 = -0.5x

    −56=x-56 = x

    So the boxes get −4, 28 and −56, in that order. On the test, you drag each tile into its box. On this site, you choose it from the box’s menu.

Check your answer

Put x=−56x = -56 back into the first equation. −56+8=−48-56 + 8 = -48. Then −0.5×(−48)=24-0.5 \times (-48) = 24, since a negative times a negative is positive. That gives 24, the left side, so −56 is right.

More questions like this

The original DOE question
Question 27 as shown on the DOE’s digital test, page 98 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test A Math, page 98, question 27.

Question 3 of 23

2026 Practice Test C Math, question 28

Question

If z+3w4=5w\frac{z+3w}{4} = 5w, what is the value of z in terms of w?

Enter your answer in the space provided. Enter only your answer.

On the test you build the answer on an on-screen keypad; here, type it. Use ^ for a power, sqrt( ) or √ for a root, pi for π, and <= or >= for ≤ and ≥.

Show the answer

Correct answer: z = 17w

The goal is to isolate the variable z by using inverse operations.

First, multiply both sides by 4:

z+3w4=5w\frac{z+3w}{4} = 5w

z+3w=20wz + 3w = 20w

Then subtract 3w3w from both sides:

z=17wz = 17w

Question analysis

This algebra question gives an equation in which z plus 3w, all divided by 4, equals 5w, and students type z in terms of w after a printed "z =". It tests solving a literal equation with inverse operations. The 4 divides the whole sum, so undoing the steps in the right order, with the right signs, is the tricky part.

NYS standards
  • 8.EE.7b linear equations with rational coefficients
  • 7.EE.4a two-step equations
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students solve two-step equations with numbers, such as (x + 3)/4 = 5. Solving for one letter in terms of another is grade 8 work, and the typed answer gives no choices to check against.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
2 steps
Reading level
Question and choices: about grade 4, 23 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The equation takes only two inverse operations, but their order is the trap: the 4 divides the whole sum z + 3w, so it has to be cleared first. Students who subtract 3w from 5w straight away, or who move 3w to the right without changing its sign, reach a tidy multiple of w that looks as finished as the right one. Substituting w = 1, which needs z = 17 to make (z + 3)/4 equal 5, checks any entry.

  1. Variables instead of numbers moderate Understanding the question

    With w on both sides and no numbers to substitute, the result has to stay a multiple of w, and a slip such as 23w gives no sign that it is wrong. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

  2. An answer in an unexpected form minor Choosing the answer

    The answer is typed after a printed “z =”, with no choices to compare, so the entry must be a single term in w, not an equation or a number. (Enright, Morley & Sheehan, 2002)

Strategy: Treat w like a known number: undo the division by 4 first, multiplying the whole of each side, then undo the added 3w, keeping the terms in w together.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Doing the same thing to both sides
if A=B, then 4A=4B and A−C=B−C\text{if } A = B, \text{ then } 4A = 4B \text{ and } A - C = B - C
A and B are the two sides of an equation, so they are equal. C is any amount. An equation is like a balanced scale. If you multiply both sides by the same number, or take the same amount from both, it stays balanced. That is how you get one letter alone.

  1. Sort out the question

    The equation is z+3w4=5w\frac{z + 3w}{4} = 5w. You need z in terms of w. That means getting z alone on one side. The other side will use only w and numbers.

    Treat w like a number you do not know yet. The answer box already shows z =, so you type only what comes after it.

  2. Undo the division first

    The fraction bar means the whole top, z + 3w, is divided by 4. So undo that first. Multiply both sides by 4, because multiplying undoes dividing:

    z+3w4×4=5w×4\frac{z + 3w}{4} \times 4 = 5w \times 4

    z+3w=20wz + 3w = 20w

    Watch out: do not subtract 3w first. The 3w sits inside the fraction, so it is divided by 4 too. Subtracting it from 5w first leads to the wrong answer 8w.

  3. Undo the adding

    Now 3w is added to z. Subtract 3w from both sides:

    z+3w−3w=20w−3wz + 3w - 3w = 20w - 3w

    z=17wz = 17w

    Think of w as a thing, like apples. 20 apples take away 3 apples leaves 17 apples.

    Watch out: moving 3w to the other side without changing its sign gives 20w+3w=23w20w + 3w = 23w. That is wrong.

  4. Enter it after the printed z =

    On the test, the box already shows z =. Press 1, then 7, then the w key. You do not need a times sign between 17 and w. Do not type z = again. On this site, type 17w in the box.

    So you would enter 17w, which makes z = 17w.

Check your answer

Pick a number for w, such as w = 1. Then z = 17. Put both into the first equation. The left side is 17+34=204=5\frac{17 + 3}{4} = \frac{20}{4} = 5. The right side is 5×1=55 \times 1 = 5. The two sides match, so 17w works.

More questions like this

The original DOE question
Question 28 as shown on the DOE’s digital test, page 101 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test C Math, page 101, question 28.

Question 4 of 23

2026 Practice Test D Math, question 4

Question

If x=14x = \frac{1}{4}, what is the value of 22−x\frac{2}{2 - x}?

  • A34\frac{3}{4}
  • B89\frac{8}{9}
  • C1171\frac{1}{7}
  • D3123\frac{1}{2}
Show the answer

Correct answer: C

The first step is to substitute the value for x into the expression to get 22−14\frac{2}{2 - \frac{1}{4}}.

Then, evaluate the denominator and finally divide the numerator by the denominator.

2(74)=2÷(74)=2(47)=87=117\frac{2}{\left(\frac{7}{4}\right)} = 2 \div \left(\frac{7}{4}\right) = 2\left(\frac{4}{7}\right) = \frac{8}{7} = 1\frac{1}{7}

Question analysis

This substitution question sets x equal to one fourth and asks for the value of 2 divided by (2 − x). Students substitute, subtract a fraction from a whole number, and divide by the result. The expression is a stacked fraction, so treating the whole denominator as one quantity and dividing rather than multiplying are where students slip.

NYS standards
  • 6.EE.2c evaluating expressions and order of operations
  • 6.NS.1 dividing fractions by fractions
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Substituting a value and dividing by a fraction are grade 6 skills, and here they take three short steps with friendly numbers. The stacked fraction asks for care, but the demand is below typical grade 7 work with signed rational numbers.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 10, 12 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Substitution is one step, so the question is about handling a fraction inside a fraction. The bar over 2 − x groups the whole denominator, and the division by 7/4 must flip it; both errors lead to listed choices. A size check helps: 2 − 1/4 is a little less than 2, so the value must be a little more than 1, which rules out every wrong choice.

  1. Error-prone arithmetic moderate Working it out

    The denominator 2 − 1/4 has to become 7/4 before anything is divided, and dividing 2 by 7/4 means multiplying by 4/7; multiplying by 7/4 instead is the usual slip. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. An answer in an unexpected form minor Choosing the answer

    The work ends at the improper fraction 8/7, but the choices show it only as a mixed number, listed among 3/4, 8/9 and 3 1/2. (Enright, Morley & Sheehan, 2002)

Strategy: Simplify the denominator to a single fraction first, then divide 2 by that fraction by multiplying by its reciprocal; treating the fraction bar as a grouping symbol avoids the common slip.

Traps

  • Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Dividing by a fraction
a÷bc=a×cba \div \frac{b}{c} = a \times \frac{c}{b}
a is any number, and b/c is a fraction. To divide by a fraction, multiply by its reciprocal (the fraction flipped upside down). It works because dividing asks how many copies of b/c fit into a.
Changing an improper fraction to a mixed number
ab=qrb\frac{a}{b} = q\frac{r}{b}
Use it when the top, a, is bigger than the bottom, b. q is how many whole times b fits into a. r is the remainder, what is left over. For 8/7, the 7 fits once with 1 left, so 8/7 = 1 1/7.

  1. Sort out the question

    You know x=14x = \frac{1}{4}. You need the value of 2 divided by (2 − x). A fraction bar means divide. It also groups, so you work out the whole bottom, 2 − x, before you divide.

  2. Work out the bottom

    Put in 1/4 for x. To subtract, write 2 as fourths, since 2 = 8/4:

    2−14=84−14=742 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}

    The bottom is 7/4.

    Watch out: the sign is minus. Adding 1/4 gives 9/4 and then 8/9, choice B.

  3. Divide 2 by 7/4

    Now the problem is 2 ÷ 7/4. Use the rule for dividing by a fraction. Flip 7/4 to get 4/7, then multiply:

    2÷74=2×47=872 \div \frac{7}{4} = 2 \times \frac{4}{7} = \frac{8}{7}

    Watch out: multiplying by 7/4 instead of 4/7 gives 14/4 = 3 1/2, choice D.

  4. Write it as a mixed number

    The choices use mixed numbers. 7 fits into 8 once, with 1 left over:

    87=117\frac{8}{7} = 1\frac{1}{7}

    Watch out: the fraction bar covers all of 2 − 1/4. Working 2/2 = 1 first and then taking away 1/4 gives 3/4, choice A.

    So the answer is C, 1171\frac{1}{7}.

Check your answer

Estimate. The bottom, 1 3/4, is a bit less than 2. So 2 divided by it should be a bit more than 1, and 1 1/7 fits. Also check by multiplying: 7/4 × 8/7 = 56/28 = 2. That is the top, so the division is right.

More questions like this

The original DOE question
Question 4 as shown on the DOE’s digital test, page 77 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D Math, page 77, question 4.

Question 5 of 23

2026 Practice Test D Math, question 20

Question

What is the value of −9(−4)2+36(−4)+304−4\frac{-9(-4)^{2} + 36(-4) + 304}{-4}?

  • A−76
  • B−4
  • C4
  • D148
Show the answer

Correct answer: B

Use the order of operations to evaluate the expression. It is important to recognize that the fraction bar is a grouping symbol and so the entire numerator should be evaluated first. Exponents are evaluated first.

−9(−4)2+36(−4)+304−4\frac{-9(-4)^{2} + 36(-4) + 304}{-4}

−9(16)+36(−4)+304−4\frac{-9(16) + 36(-4) + 304}{-4}

Then multiplication, recognizing that the product of a negative factor and a positive factor is negative.

−9(16)+36(−4)+304−4\frac{-9(16) + 36(-4) + 304}{-4}

−144−144+304−4\frac{-144 - 144 + 304}{-4}

Then addition and subtraction in order from left to right, recognizing the rules for adding and subtracting signed numbers.

−144−144+304−4\frac{-144 - 144 + 304}{-4}

−288+304−4=16−4\frac{-288 + 304}{-4} = \frac{16}{-4}

And finally the fraction can be evaluated by division, recognizing that the quotient of a positive and a negative number is negative.

16−4=−4\frac{16}{-4} = -4

Question analysis

This arithmetic question asks for the value of a fraction whose numerator, −9(−4)² + 36(−4) + 304, is divided by −4. Students follow the order of operations with negative numbers: a square, two products, a sum and a division. Every step carries a sign, and one dropped negative changes the result completely.

NYS standards
  • 7.NS.2a rational number multiplication
  • 7.NS.2b rational number division
  • 6.EE.2c evaluating expressions and order of operations
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Order of operations with negative numbers is standard grade 7 practice. This expression stacks a squared negative, two products and a division by a negative into one fraction, so it takes more steps than usual, each with a sign to track.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
What makes it hard
Solution
4 steps
Reading level
Question and choices: about grade 5, 6 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Nothing here is conceptually new; the question measures care with signs under the order of operations. The parentheses in (−4)² decide whether −9 multiplies 16 or −16, and the two products of −144 then nearly cancel the 304, leaving a small numerator whose sign is easy to lose. Writing each intermediate value with its sign, line by line, prevents most errors.

  1. Error-prone arithmetic moderate Working it out

    Five sign decisions sit in one expression: squaring −4, two products with a negative factor, a sum that climbs from −288 past zero, and a division by −4. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    The numerator alone takes an exponent, two multiplications and two additions before the single division, and all of it must be finished before dividing. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  3. A very close wrong answer minor Choosing the answer

    The right value and the choice 4 differ only in sign, so a student with the right size must still check the sign of a positive numerator over −4. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Treat the fraction bar as grouping: square −4 first, do the two multiplications, combine the numerator from left to right, and only then divide by −4, writing each sign as you go.

Traps

  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Order of operations
grouping→exponents→×,÷→+,−\text{grouping} \rightarrow \text{exponents} \rightarrow \times, \div \rightarrow +, -
This is the order for working out an expression. Grouping means parentheses and the fraction bar. Exponents are small raised numbers, like the 2 in (−4)². Next, multiply and divide from left to right. Last, add and subtract from left to right.
Squaring a negative number
(−a)2=(−a)×(−a)=a2(-a)^{2} = (-a) \times (-a) = a^{2}
a is a positive number, so −a is negative. Squaring means multiplying a number by itself. A negative times a negative is positive, so a negative number squared is positive. For example, (−4)² = 16.
Signs when you multiply or divide
positivenegative=negative\frac{\text{positive}}{\text{negative}} = \text{negative}
When two numbers have different signs, their product or quotient is negative. When they have the same sign, it is positive. A quotient is the answer to a division.

  1. Sort out the question

    You need the value of one big fraction. The fraction bar is a grouping symbol, so finish the whole top first. Then divide by −4. Follow the order of operations, and track every minus sign.

  2. Do the exponent first

    The only exponent is (−4)². Squaring means multiplying a number by itself:

    (−4)2=(−4)×(−4)=16(-4)^{2} = (-4) \times (-4) = 16

    Watch out: (−4)² is 16, not −16. Using −16 makes the top 304, which gives −76, choice A.

  3. Do the multiplication

    Now multiply. A negative times a positive is negative:

    −9×16=−144-9 \times 16 = -144

    36×(−4)=−14436 \times (-4) = -144

    So the top becomes:

    −144+(−144)+304-144 + (-144) + 304

  4. Add from left to right

    Add the two negatives first:

    −144+(−144)=−288-144 + (-144) = -288

    Then add 304. Since 304 is more than 288, the result is positive:

    −288+304=16-288 + 304 = 16

    The whole top is 16.

  5. Divide by −4

    Now divide the top by the bottom. A positive divided by a negative is negative:

    16−4=−4\frac{16}{-4} = -4

    Watch out: positive 4, choice C, ignores the negative bottom. Dropping every minus sign gives 592 ÷ 4 = 148, choice D.

    So the answer is B, −4.

Check your answer

Multiply back. If the answer is −4, then −4 times the bottom, −4, should give the top. It does: (−4) × (−4) = 16. Also, the top is positive and the bottom is negative, so the answer must be negative.

More questions like this

The original DOE question
Question 20 as shown on the DOE’s digital test, page 93 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D Math, page 93, question 20.

Question 6 of 23

2024 Test A Math, question 10

Question

Simplify:

(39(x−3)3)+3913\frac{\left(\frac{39(x - 3)}{3}\right) + 39}{13}

  • Axx
  • Bx−2x - 2
  • C13x−3613x - 36
  • Dx+3613x + \frac{36}{13}
Show the answer

Correct answer: A

(39(x−3)3)+3913\frac{\left(\frac{39(x - 3)}{3}\right) + 39}{13}

Divide the first 39 by 3.

13(x−3)+3913\frac{13(x - 3) + 39}{13}

Distribute 13 through (x−3)(x - 3).

13x−39+3913\frac{13x - 39 + 39}{13}

In the numerator, the terms −39 and 39 cancel out.

13x13\frac{13x}{13}

Finally, divide the numerator and denominator by 13 to get x.

Question analysis

An algebra question that asks students to simplify a stacked fraction: 39 times (x − 3) over 3, plus 39, all divided by 13. It tests dividing, distributing and combining like terms in the right order. The nested fraction bars make it easy to divide only some of the terms or to distribute to only one.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
  • 6.EE.3 generating equivalent expressions
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students expand and simplify linear expressions such as 13(x − 3) + 39. This wraps that work inside two fraction bars, which adds reading load and chances to misstep but no new mathematics.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 16+, 6 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The expression looks heavier than it is: every number is a multiple of 13, and the constants cancel to leave a single term. What makes it risky is scope, knowing which terms each fraction bar and each multiplier covers. Each wrong choice comes from stretching or shrinking that scope by one term, so working outward from the innermost parentheses, one line per step, is the safe way.

  1. Error-prone arithmetic moderate Working it out

    Two fraction bars are nested: 39 ÷ 3 applies only to the first term, the 13 must reach both x and −3, and only then does the whole numerator divide by 13. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    Four moves come in a fixed order, divide 39 by 3, distribute 13, cancel −39 with +39, divide by 13, and each is a chance to apply a step to the wrong terms. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Work from the inside out: divide 39 by 3, distribute through the parentheses, combine the constants in the numerator, and only then divide everything by 13.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distributive property
a(b−c)=a×b−a×ca(b - c) = a \times b - a \times c
a is the number outside the parentheses. b and c are the terms inside. Multiply a by each term inside, and keep the minus sign between them. It works because a groups of (b − c) are a groups of b, take away a groups of c.

  1. Sort out the expression

    You need to simplify the big fraction, which means write it in its shortest form. It has a small fraction inside its top. The top is 39(x−3)3+39\frac{39(x - 3)}{3} + 39, and the bottom is 13. Work from the inside out, one step at a time.

  2. Divide the first 39 by 3

    The small fraction bar is only under 39(x−3)39(x - 3). So divide just that 39 by 3.

    39(x−3)3=13(x−3)\frac{39(x - 3)}{3} = 13(x - 3)

    Now the top is 13(x−3)+3913(x - 3) + 39. Watch out: the small bar does not reach the added 39. Dividing that 39 by 3 as well leads to x−2x - 2, choice B.

  3. Multiply out the parentheses

    a(b−c)=a×b−a×ca(b - c) = a \times b - a \times c

    13(x−3)=13×x−13×3=13x−3913(x - 3) = 13 \times x - 13 \times 3 = 13x - 39

    Watch out: multiply the 3 by 13 too, not just the x. Leaving −3 as it is leads to choice D.

  4. Combine the plain numbers on top

    13x−39+39=13x13x - 39 + 39 = 13x

    −39 and 39 add up to 0, so they cancel. The top is now just 13x, which means 13 times x.

  5. Divide by the bottom

    Now divide the whole top by 13. Dividing by 13 undoes times 13.

    13x13=x\frac{13x}{13} = x

    Watch out: the long bar is under the whole top. Dividing only the last 39 by 13 gives 13x−3613x - 36, choice C. So the answer is A, xx.

Check your answer

Try x = 5. Then x−3=2x - 3 = 2 and 39×2=7839 \times 2 = 78. Next, 78÷3=2678 \div 3 = 26 and 26+39=6526 + 39 = 65. Last, 65÷13=565 \div 13 = 5. You get back 5, the same as x. So x is right.

More questions like this

The original DOE question
Question 67 as printed on page 78 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 78, question 67. Open page 78 of the handbook (PDF)

Question 7 of 23

2024 Test B Math, question 13

Question

p+2r=r(p+1)+1p+2r=r(p+1)+1

In the equation above, if r=2r=2, what is the value of p?

  • A0
  • B1
  • C2
  • D3
Show the answer

Correct answer: B

First substitute 2 for r in the equation.

p+2r=r(p+1)+1p+2r=r(p+1)+1

p+2(2)=2(p+1)+1p+2(2)=2(p+1)+1

Then solve for p.

p+2(2)=2(p+1)+1p+2(2)=2(p+1)+1

p+4=2p+2+1p+4=2p+2+1

p+4=2p+3p+4=2p+3

p+(4−3)=2p+(3−3)p+(4-3)=2p+(3-3)

p+1=2pp+1=2p

(p−p)+1=2p−p(p-p)+1=2p-p

1=p1=p

The value of p is 1.

Question analysis

An equation question: p + 2r equals r times (p + 1), plus 1, and r is given as 2. Students substitute for r, expand the parentheses and solve the resulting linear equation for p. With p on both sides and a product to expand, distributing is where a slip is most likely.

NYS standards
  • 8.EE.7b linear equations with rational coefficients
  • 6.EE.2c evaluating expressions and order of operations
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
At grade 7 Equations with the variable on both sides are formally grade 8, but after substituting, this is a short solve with small whole numbers. It asks about as much as a typical grade 7 two-step equation.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 6, 13 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Substituting r = 2 is immediate; the points are lost in expanding 2(p + 1) + 1. Each wrong choice solves a slightly wrong expansion: forgetting to multiply the 1 inside, folding the outer 1 into the parentheses, or losing a term. Putting the answer back into the original equation, with both r and p, is a quick and complete check.

  1. Error-prone arithmetic moderate Working it out

    After substituting, 2(p + 1) + 1 must become 2p + 3, multiplying both terms in the parentheses before adding the 1 outside, with p left on both sides. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Replace every r with 2, multiply 2 by both terms in the parentheses, gather the p terms on one side, and check by putting the value back into the original equation.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number outside the parentheses. b and c are the terms inside. A term is a number, a letter, or a number times a letter. To multiply out the parentheses, multiply each term inside by a. It works because a groups of b + c is a groups of b plus a groups of c.
Keeping an equation balanced
if a=b, then a−c=b−c\text{if } a = b \text{, then } a - c = b - c
a and b are the two sides of an equation, and c is any number or term. An equation is like a balanced scale. If you subtract the same thing from both sides, it stays balanced. Adding the same thing, or dividing both sides by the same number that is not zero, works too.

  1. Sort out the question

    The equation has two letters, p and r. You are told that r = 2. You need the value of p. Here 2r means 2 times r, and r(p + 1) means r times the group p + 1.

  2. Put 2 in for r

    Replace every r with 2. Keep the parentheses so you can see what gets multiplied.

    p+2(2)=2(p+1)+1p + 2(2) = 2(p + 1) + 1

    On the left, 2(2)=42(2) = 4, so the left side is p+4p + 4.

  3. Multiply out the parentheses

    The 2 multiplies both the p and the 1 inside the parentheses. Then add the 1 that is outside.

    a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c

    2(p+1)+1=2p+2+1=2p+32(p + 1) + 1 = 2p + 2 + 1 = 2p + 3

    Watch out: multiply the 1 inside by 2 as well. Writing 2p + 1 + 1 gives p = 2, choice C. Adding the outside 1 inside first gives p = 0, choice A.

  4. Get p by itself

    Now the equation is p+4=2p+3p + 4 = 2p + 3. Subtract p from both sides. Then subtract 3 from both sides.

    if a=b, then a−c=b−c\text{if } a = b \text{, then } a - c = b - c

    p+4−p=2p+3−pp + 4 - p = 2p + 3 - p

    4=p+34 = p + 3

    4−3=p+3−34 - 3 = p + 3 - 3

    1=p1 = p

    So p = 1, and the answer is B, 1.

Check your answer

Put p = 1 and r = 2 back into the first equation. Left side: 1+2×2=51 + 2 \times 2 = 5. Right side: 2×(1+1)+1=52 \times (1 + 1) + 1 = 5. Both sides are 5, so p = 1 works.

More questions like this

The original DOE question
Question 70 as printed on page 202 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 202, question 70. Open page 202 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 8 of 23

2024 Test B Math, question 27

Question

Simplify:

8x−(7+2.5x)+28x-(7+2.5x)+2

  • A5.5x−95.5x-9
  • B5.5x−55.5x-5
  • C10.5x−910.5x-9
  • D10.5x−510.5x-5
Show the answer

Correct answer: B

8x−(7+2.5x)+28x-(7+2.5x)+2

First, distribute −1 over 7+2.5x7+2.5x:

8x−7−2.5x+28x-7-2.5x+2

Then combine the like terms:

5.5x−55.5x-5

The simplified form of the expression is 5.5x−55.5x-5.

Question analysis

An algebra question asks students to simplify 8x minus the quantity 7 + 2.5x, plus 2. They remove the parentheses and combine the x terms and the constants. Decimal coefficients and a subtraction in front of a group make sign errors likely.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Simplifying a linear expression with a subtracted group and decimal coefficients is standard grade 7 work. This is a typical two-step textbook exercise, with the usual sign trap and nothing extra.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 16+, 6 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The question checks one skill, removing parentheses that follow a minus sign, and its choices are built from the two ways to get it wrong. A student who applies the minus to too few terms gets 10.5x, and one who extends it past the parentheses gets −9. Handling the x terms and the constants as two separate small checks makes each slip easy to catch.

  1. Error-prone arithmetic moderate Working it out

    The minus sign before the parentheses must turn + 2.5x into − 2.5x and 7 into − 7, while the + 2 written after the parentheses keeps its sign. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    The choices pair two x terms, 5.5x and 10.5x, with two constants, −9 and −5, so each wrong one differs from the right one in a single part. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Change the minus in front of the parentheses into a minus on each term inside, then combine like terms, keeping the + 2 outside the group as it is.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Subtracting a group in parentheses
−(a+b)=−a−b-(a + b) = -a - b
a and b are the terms inside the parentheses. A minus sign in front means take away the whole group. So each term inside is taken away, and its sign flips. Terms outside the parentheses keep their own signs.
Combining like terms
ax+bx=(a+b)xax + bx = (a + b)x
a and b are numbers, and x is the letter. Like terms have the same letter part. You add or subtract their numbers and keep the letter. Plain numbers combine only with plain numbers.

  1. Sort out the problem

    You need to simplify 8x−(7+2.5x)+28x - (7 + 2.5x) + 2. To simplify means to write it in its shortest form. The tricky part is the minus sign in front of the parentheses. It takes away everything inside.

  2. Remove the parentheses

    −(a+b)=−a−b-(a + b) = -a - b

    8x−(7+2.5x)+2=8x−7−2.5x+28x - (7 + 2.5x) + 2 = 8x - 7 - 2.5x + 2

    Both 7 and 2.5x get a minus sign. The + 2 is outside the parentheses, so it stays + 2. Watch out: 2.5x must become −2.5x. Leaving it as + 2.5x gives 10.5x, as in choices C and D.

  3. Combine the x terms

    Like terms have the same letter part. Put the two x terms together.

    ax+bx=(a+b)xax + bx = (a + b)x

    8x−2.5x=(8−2.5)x=5.5x8x - 2.5x = (8 - 2.5)x = 5.5x

  4. Combine the plain numbers

    −7+2=−5-7 + 2 = -5

    Watch out: the + 2 is not inside the parentheses. Making it −2 gives −9, as in choice A.

    Put the parts together: 5.5x−55.5x - 5. So the answer is B, 5.5x−55.5x - 5.

Check your answer

Pick x = 2 and try both forms. The first form: 16−(7+5)+2=616 - (7 + 5) + 2 = 6. Choice B: 5.5×2−5=11−5=65.5 \times 2 - 5 = 11 - 5 = 6. They match. Choice D gives 10.5×2−5=1610.5 \times 2 - 5 = 16, so D is wrong.

More questions like this

The original DOE question
Question 84 as printed on page 206 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 206, question 84. Open page 206 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 9 of 23

2024 Test B Math, question 44

Question

Which inequality is equivalent to 6x−20<5x−2(5−3x)6x-20<5x-2(5-3x)?

  • Ax<16x<16
  • Bx<52x<\frac{5}{2}
  • Cx<−2x<-2
  • Dx>−2x>-2
Show the answer

Correct answer: D

6x−20<5x−2(5−3x)6x-20<5x-2(5-3x)

Distribute the coefficient −2 on the right-hand side:

6x−20<5x+(−2)(5)+(−2)(−3x)6x-20<5x+(-2)(5)+(-2)(-3x)

6x−20<5x−10+6x6x-20<5x-10+6x

Subtract 6x6x from both sides:

−20<5x−10-20<5x-10

Add 10 to both sides:

−10<5x-10<5x

Divide both sides by 5:

−2<x-2<x

This is equivalent to

x>−2x>-2

Question analysis

Students choose the simplest equivalent form of a linear inequality that has x on both sides and a negative number multiplied into parentheses. It tests distributing, combining like terms and solving inequalities. Sign errors in the distribution, and reading the direction of the final inequality, are where most mistakes happen.

NYS standards
  • 7.EE.4b two-step inequalities
  • 8.EE.7b linear equations with rational coefficients
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 inequalities have the form px + q < r. This one has x on both sides and a negative factor to distribute across a subtraction, steps that belong to grade 8 equation work, so it is harder than typical grade 7 practice.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Solution
4 steps
Reading level
Question and choices: about grade 16+, 10 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every move is standard algebra, but the inequality is built so that small slips change the answer completely. The −2 has to reach both terms in the parentheses, and once the boundary −2 is found its direction still has to be read correctly, since the variable ends up on the right as −2 < x. Rewriting that as x > −2, or substituting x = 0 into the original, catches the trap.

  1. Error-prone arithmetic moderate Working it out

    Distributing −2 over (5 − 3x) gives −10 + 6x, two sign decisions in one move, and the x terms then sit on both sides of the inequality. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    x < −2 and x > −2 share the boundary and differ only in direction, which depends on reading −2 < x correctly or flipping the sign when dividing by a negative. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Many steps minor Working it out

    Four moves follow one another, distributing, collecting 5x + 6x, moving the x terms and dividing by 5, and a slip at any of them changes the result. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Distribute the −2 to both terms in the parentheses, combine the x terms on the right, then gather the x terms on the side that keeps their coefficient positive.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number outside the parentheses. b and c are the terms inside. Multiply a by each one. A minus sign inside counts as a negative term, so 5 − 3x is 5 + (−3x). A negative times a negative is positive.
Keeping an inequality true
if a<b, then a+c<b+c\text{if } a < b \text{, then } a + c < b + c
a and b are the two sides, and c is any number. Adding or subtracting the same number on both sides keeps the inequality true. Dividing both sides by a positive number also keeps it true. Only multiplying or dividing by a negative number flips the sign.

  1. Sort out the question

    An inequality compares two sides with a sign like <, which means is less than. To solve it, you get x alone on one side.

    You need the short inequality that has the same answers as the long one.

  2. Distribute the −2

    Multiply −2 by both terms inside the parentheses, 5 and −3x.

    −2(5−3x)=(−2)×5+(−2)×(−3x)=−10+6x-2(5 - 3x) = (-2) \times 5 + (-2) \times (-3x) = -10 + 6x

    A negative times a negative is positive, so the second part is +6x. Watch out: If the −2 multiplies only the 5, you end up with x<52x < \frac{5}{2}, choice B.

  3. Combine the x terms

    The right side is now 5x − 10 + 6x. Put the x terms together: 5x + 6x = 11x.

    6x−20<11x−106x - 20 < 11x - 10

  4. Gather the x terms on one side

    Subtract 6x from both sides. That leaves the x terms on the right, where they stay positive.

    6x−20−6x<11x−10−6x6x - 20 - 6x < 11x - 10 - 6x

    −20<5x−10-20 < 5x - 10

  5. Get x by itself

    Add 10 to both sides. Then divide both sides by 5.

    −10<5x-10 < 5x

    −2<x-2 < x

    You divided by a positive 5, so the sign does not flip.

  6. Read it with x first

    −2<x-2 < x says −2 is less than x. That is the same as saying x is greater than −2, just as 3 < 5 means 5 > 3.

    x>−2x > -2

    Watch out: x<−2x < -2, choice C, has the same number but points the wrong way.

    So the answer is D, x>−2x > -2.

Check your answer

Test x = 0, which is greater than −2. The left side is 6(0)−20=−206(0) - 20 = -20. The right side is 5(0)−2(5−0)=−105(0) - 2(5 - 0) = -10. And −20 < −10 is true. Now test x = −3. The left side is −38 and the right side is −43. That is false, as it should be.

More questions like this

The original DOE question
Question 101 as printed on page 211 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 211, question 101. Open page 211 of the handbook (PDF)

Question 10 of 23

2023 Test A Math, question 2

Question

6−9÷|−3|+(−2)3×1126 - 9 \div \left|-3\right| + (-2)^{3} \times 1\frac{1}{2}

What is the value of the expression shown above?

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: -9

6−9÷|−3|+(−2)3×1126 - 9 \div \left|-3\right| + (-2)^{3} \times 1\frac{1}{2} Calculate the exponent and convert the mixed number to an improper fraction.
6−9÷|−3|+(−8)×326 - 9 \div \left|-3\right| + (-8) \times \frac{3}{2} Next, calculate the absolute value.
6−9÷3+(−8)×326 - 9 \div 3 + (-8) \times \frac{3}{2} Perform the indicated multiplication and division.
6−3+(−12)6 - 3 + (-12) Perform the addition and subtraction.
3+(−12)=−93 + (-12) = -9

Question analysis

This grid-in asks for the value of an arithmetic expression that mixes an absolute value, a negative number raised to the third power, a division and a mixed number. Students apply the order of operations with signed numbers. Cubing a negative base, and doing the division and multiplication before any adding, are where most slips happen.

NYS standards
  • 7.NS.2a rational number multiplication
  • 6.EE.2c evaluating expressions and order of operations
  • 6.NS.7c absolute value as distance
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students routinely evaluate expressions with negative numbers in the right order. This one combines an absolute value, a cubed negative and a mixed number in a single line, so it takes more steps than usual, each with a sign to watch.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
4 steps
Reading level
Question and choices: about grade 5, 10 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The expression packs four kinds of notation into one line: a subtraction, a division by an absolute value, a negative base raised to the third power and a mixed number. None of the operations is hard alone, but each has a rule that is easy to break under time pressure, and the slips lead to very different totals. Rewriting the power and the mixed number first, then dividing and multiplying before any addition, avoids them.

  1. Error-prone arithmetic moderate Working it out

    (−2)³ must stay negative, −8, and the mixed number 1½ must become 3/2 before multiplying, so the last product is −12; a sign or fraction slip here moves the total far from −9. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    Six numbers and six operations, including a power and an absolute value, have to be worked in a planned order, and every addition waits until the division and multiplication are done. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  3. An answer in an unexpected form minor Choosing the answer

    The value is negative and the grid-in offers no choices, so the minus sign has to be gridded and nothing on the page confirms that the total falls below zero. (Enright, Morley & Sheehan, 2002)

Strategy: Rewrite the absolute value and the power first, change the mixed number to an improper fraction, do the division and multiplication, and only then add and subtract from left to right.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Order of operations
grouping→powers→× and ÷→+ and −\text{grouping} \rightarrow \text{powers} \rightarrow \times \text{ and } \div \rightarrow + \text{ and } -
This is the order for working out an expression. First work inside grouping symbols, such as parentheses and absolute value bars. Next do powers. Then multiply and divide from left to right. Last, add and subtract from left to right. Using one order means everyone gets the same answer.
A power
a3=a×a×aa^{3} = a \times a \times a
a is the base, the number being multiplied. The small 3 is the exponent. It says how many times the base is used as a factor, a number being multiplied. It does not mean a times 3.
Absolute value
|a|=the distance from a to 0\left|a\right| = \text{the distance from } a \text{ to } 0
The bars around a number mean absolute value. It is how far the number is from 0 on a number line. A distance is never negative, so the absolute value of −3 is 3.
Mixed number to fraction
abc=a×c+bca\frac{b}{c} = \frac{a \times c + b}{c}
a is the whole number part, and b over c is the fraction part. Each whole is c pieces, so a wholes are a × c pieces. Add the b extra pieces. The bottom number stays c.

  1. Sort out the expression

    You need the value of 6−9÷|−3|+(−2)3×1126 - 9 \div \left|-3\right| + (-2)^{3} \times 1\frac{1}{2}. It has many operations, so the order matters. It has absolute value bars, a power of a negative number and a mixed number. The answer may be negative, so keep track of every sign.

  2. Do the absolute value and the power

    Grouping symbols and powers come first.

    |−3|=3\left|-3\right| = 3

    (−2)3=(−2)×(−2)×(−2)=4×(−2)=−8(-2)^{3} = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8

    Two negatives multiply to a positive, 4. One more negative makes it negative again. Watch out: (−2)³ is −8, not +8. Using +8 leads to 15. It is also not −2 × 3. That mistake leads to a total of −6.

  3. Change the mixed number to a fraction

    A fraction is easier to multiply than a mixed number.

    112=1×2+12=321\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}

    Now the expression looks like this:

    6−9÷3+(−8)×326 - 9 \div 3 + (-8) \times \frac{3}{2}

  4. Multiply and divide next

    Multiplying and dividing come before adding and subtracting.

    9÷3=39 \div 3 = 3

    (−8)×32=−242=−12(-8) \times \frac{3}{2} = \frac{-24}{2} = -12

    A negative times a positive is negative. Watch out: don’t just work from left to right. Doing 6 − 9 first, then dividing by 3, leads to −13.5.

  5. Add and subtract last

    Now only adding and subtracting are left. Work from left to right.

    6−3+(−12)=3+(−12)=−96 - 3 + (-12) = 3 + (-12) = -9

    Adding −12 is the same as taking away 12. So you would grid −9, with the minus sign in its own column.

Check your answer

Look at the size of each part. The part 6 − 3 is a small positive number, 3. The last part, −8 times 1½, is a bigger negative number, −12. A big negative plus a small positive is negative. So an answer below zero, −9, makes sense.

More questions like this

The original DOE question
Question 59 as printed on page 75 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 75, question 59. Open page 75 of the handbook (PDF)

Question 11 of 23

2023 Test A Math, question 3

Question

Solve for x:

7x+3−2(2x+1)=137x + 3 - 2(2x + 1) = 13

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: 4

7x+3−2(2x+1)=137x + 3 - 2(2x + 1) = 13 Apply the distributive property; multiply the −2 by 2x2x and 1.
7x+3−4x−2=137x + 3 - 4x - 2 = 13 Combine like terms.
3x+1=133x + 1 = 13 Apply the additive inverse property; subtract 1 from both sides of the equation.
3x=123x = 12 Apply the multiplicative inverse property; divide both sides of the equation by 3.
x=4x = 4

Question analysis

This grid-in gives the equation 7x + 3 − 2(2x + 1) = 13 and asks students to solve for x. It tests the distributive property with a negative factor, combining like terms and solving a two-step equation. The minus sign in front of the parentheses has to reach both terms inside, a step many students rush.

NYS standards
  • 7.EE.4a two-step equations
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students expand expressions with negative factors and solve equations such as p(x + q) = r. This equation needs both skills plus combining like terms, which is routine work with one negative sign that is easy to lose.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
4 steps
Reading level
Question and choices: about grade 3, 4 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

This is a standard multi-step linear equation, and the work is short once the parentheses are cleared. Everything rests on distributing −2 correctly; the later steps, combining 7x with −4x and 3 with −2, then undoing the + 1 and the × 3, are routine. Substituting 4 back into the left side, 28 + 3 − 18 = 13, is a fast check that catches a sign slip.

  1. Error-prone arithmetic moderate Working it out

    The −2 in front of (2x + 1) must reach both terms, giving −4x − 2; the second product, −2 × 1, is the one most often written as +2 or left as +1. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. An answer in an unexpected form minor Choosing the answer

    As a grid-in with no choices, a value such as 3 or 8/3 from a distributing slip looks just as finished as the right one; only substituting back exposes it. (Enright, Morley & Sheehan, 2002)

Strategy: Distribute −2 to both terms inside the parentheses, combine the x terms and the constants, then undo the addition and the multiplication in turn.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number outside the parentheses. b and c are the terms inside. A term is a number, a letter, or a number times a letter. Multiply each term inside by a. If a is negative, every product gets its sign too.
Keeping an equation balanced
if a=b, then a−c=b−c and a÷c=b÷c\text{if } a = b \text{, then } a - c = b - c \text{ and } a \div c = b \div c
a and b are the two equal sides. c is any number, but not 0 when you divide. Doing the same thing to both sides keeps them equal, like a balanced scale. Use it to get the unknown by itself.

  1. Sort out the equation

    You need the value of x that makes 7x+3−2(2x+1)=137x + 3 - 2(2x + 1) = 13 true. x is the unknown number. The tricky part is the −2 in front of the parentheses. It multiplies everything inside them.

  2. Multiply out the parentheses

    Use the distributive property with a = −2, b = 2x and c = 1.

    −2(2x+1)=(−2)×2x+(−2)×1=−4x−2-2(2x + 1) = (-2) \times 2x + (-2) \times 1 = -4x - 2

    So the equation becomes this:

    7x+3−4x−2=137x + 3 - 4x - 2 = 13

    Watch out: the −2 must multiply the 1 too. Leaving + 1 gives x = 3. Writing + 2 instead of − 2 gives x=83x = \frac{8}{3}.

  3. Combine like terms

    Like terms are terms of the same kind. Put the x terms together. Put the plain numbers together.

    7x−4x=3x7x - 4x = 3x

    3−2=13 - 2 = 1

    3x+1=133x + 1 = 13

  4. Get x by itself

    Undo the + 1 first. Subtract 1 from both sides, so the equation stays balanced.

    3x+1−1=13−13x + 1 - 1 = 13 - 1

    3x=123x = 12

    3x means 3 times x. So divide both sides by 3.

    x=12÷3=4x = 12 \div 3 = 4

    Watch out: don’t stop at 3x = 12. 12 is three times the answer. So you would grid 4.

Check your answer

Put x = 4 back into the first equation: 7×4+3−2(2×4+1)=28+3−2×9=31−18=137 \times 4 + 3 - 2(2 \times 4 + 1) = 28 + 3 - 2 \times 9 = 31 - 18 = 13. That matches the 13 on the right side, so x = 4 is right.

More questions like this

The original DOE question
Question 60 as printed on page 75 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 75, question 60. Open page 75 of the handbook (PDF)

Question 12 of 23

2023 Test A Math, question 6

Question

34+74=3^{4} + 7^{4} =

  • A40
  • B370
  • C2,482
  • D10,000
Show the answer

Correct answer: C

34+74=81+2,401=2,4823^{4} + 7^{4} = 81 + 2{,}401 = 2{,}482

Question analysis

This exponents question asks for the sum of 3 to the fourth power and 7 to the fourth power. Students evaluate each power separately and then add. The traps are treating an exponent as a multiplier and adding the bases before raising them to a power, and the larger power takes careful multiplication.

NYS standards
  • 6.EE.1 whole-number exponent expressions
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Evaluating whole-number powers is grade 6 work, and this question asks only for two of them and a sum. The one demand beyond routine practice is computing a fourth power of 7 accurately without a calculator.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 16+, 1 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The sum is short, but it tests exactly what an exponent means and when to apply it. The wrong choices come from three misreadings of the notation: treating the power as a multiplier, using one factor too few, and adding the bases before raising them. A size check settles it: 7⁴ alone is 49 × 49, a little under 2,500, so 40 and 370 are far too small and 10,000 far too large.

  1. Error-prone arithmetic moderate Working it out

    7⁴ means 7 × 7 × 7 × 7 = 2,401, a four-digit product built by hand, most easily as 49 × 49; 3⁴ = 81 is quick by comparison. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Something you must already know minor Working it out

    An exponent counts repeated factors rather than acting as a multiplier, so 3⁴ is 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12; the item assumes that meaning. (Anderson & Krathwohl, 2001)

Strategy: Compute each power on its own, squaring and then squaring again, and add the two results only at the end.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

A power
a4=a×a×a×aa^{4} = a \times a \times a \times a
a is the base, the number being multiplied. The small 4 is the exponent. It tells how many times the base is used as a factor, a number being multiplied. So the exponent counts factors. It does not mean multiply by 4.
Powers come before adding
a4+b4≠(a+b)4a^{4} + b^{4} \neq (a + b)^{4}
a and b are any two positive numbers. The sign ≠ means is not equal to. Powers come before adding. So raise each number to its power first, then add. Adding the bases first gives a much bigger number.

  1. Sort out the question

    You need the value of 34+743^{4} + 7^{4}. Each number has a small 4 above it. This 4 is an exponent. Work out each power on its own. Then add the two results.

  2. Find 3 to the fourth power

    3⁴ means four 3s multiplied together. Pair them up to make it easier.

    34=3×3×3×3=9×9=813^{4} = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81

    Watch out: 3⁴ is not 3 × 4 = 12. Doing that to both powers gives 12 + 28 = 40, choice A.

  3. Find 7 to the fourth power

    7⁴ means four 7s multiplied together. Pair them up the same way.

    74=7×7×7×7=49×49=2,4017^{4} = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2{,}401

    To find 49 × 49, do 49 × 50 = 2,450. Then take away one 49. That gives 2,401. Watch out: count four 7s, not three. Three 7s make 343. Using three factors in both powers gives 370, choice B.

  4. Add the two powers

    81+2,401=2,48281 + 2{,}401 = 2{,}482

    Watch out: don’t add 3 and 7 first. That gives (3+7)4=104=10,000(3 + 7)^{4} = 10^{4} = 10{,}000, choice D. Powers come before adding. So the answer is C, 2,482.

Check your answer

Estimate. 7⁴ is 49 × 49, a little less than 50 × 50 = 2,500. Adding 81 keeps it near 2,500. Only 2,482 is close. 40 and 370 are far too small, and 10,000 is far too big.

More questions like this

The original DOE question
Question 63 as printed on page 77 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 77, question 63. Open page 77 of the handbook (PDF)

Question 13 of 23

2023 Test A Math, question 20

Question

32+(−8)2+22(3−8+2)2=\frac{3^{2} + (-8)^{2} + 2^{2}}{(3 - 8 + 2)^{2}} =

  • A−60
  • B−173-\frac{17}{3}
  • C779\frac{77}{9}
  • D68
Show the answer

Correct answer: C

32+(−8)2+22(3−8+2)2=\frac{3^{2} + (-8)^{2} + 2^{2}}{(3 - 8 + 2)^{2}} =

9+64+4(−3)2=779\frac{9 + 64 + 4}{(-3)^{2}} = \frac{77}{9}

Question analysis

This arithmetic question asks for the value of a fraction whose numerator is the sum of the squares of 3, −8 and 2, and whose denominator is the square of 3 − 8 + 2. Students apply the order of operations with negative numbers. Squaring a negative number and adding inside parentheses before squaring are the points to watch.

NYS standards
  • 7.NS.2a rational number multiplication
  • 6.EE.2c evaluating expressions and order of operations
  • 7.NS.1c rational number subtraction
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Squaring negative numbers and simplifying with the order of operations is standard grade 7 practice. Here the fraction bar groups a numerator and a denominator that each involve a negative sign, but the work stays short.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 16+, 3 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The expression is built so that signs decide everything: the top's 3², (−8)² and 2² are all positive, and so is the square of the bottom's −3. Students who keep a negative after squaring, or who cancel the 9 on top against the 9 below, reach the other choices. The result, 77/9, does not reduce, which can make careful students doubt a correct answer.

  1. Error-prone arithmetic moderate Working it out

    Two negatives sit inside squares: (−8)² must be +64, and the denominator's 3 − 8 + 2 = −3 must be squared to +9, so every term ends up positive. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    The numerator needs three squares and a sum, the denominator a sum and then a square, and the fraction bar groups each side, so neither part can be split or cancelled. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Simplify the top and the bottom separately: square each number in the numerator, add inside the parentheses before squaring in the denominator, then write the fraction in lowest terms.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Square of a negative
(−a)2=(−a)×(−a)=a2(-a)^{2} = (-a) \times (-a) = a^{2}
a is any number. To square means to multiply a number by itself. A negative times a negative is positive. So the square of a negative number is always positive.
The fraction bar groups
topbottom=(top)÷(bottom)\frac{\text{top}}{\text{bottom}} = (\text{top}) \div (\text{bottom})
A fraction bar means divide. It also groups: work out the whole top and the whole bottom first, then divide. You can only cancel a number that multiplies the whole top, not one part of a sum.

  1. Sort out the expression

    The top, the numerator, is three squares added together. The bottom, the denominator, is a sum in parentheses that is then squared. To square means to multiply a number by itself. Work out the top and the bottom on their own. Then divide.

  2. Work out the top

    Square each number first. Then add.

    32=93^{2} = 9

    (−8)2=(−8)×(−8)=64(-8)^{2} = (-8) \times (-8) = 64

    22=42^{2} = 4

    9+64+4=779 + 64 + 4 = 77

    Watch out: (−8)² is +64, not −64. The minus sign is inside the parentheses, so it is squared too. Using −64 makes the top −51 and gives −173-\frac{17}{3}, choice B.

  3. Work out the bottom

    Work inside the parentheses first. Add and subtract before you square.

    3−8+2=−5+2=−33 - 8 + 2 = -5 + 2 = -3

    (−3)2=(−3)×(−3)=9(-3)^{2} = (-3) \times (-3) = 9

    The bottom is 9.

  4. Divide and simplify

    779\frac{77}{9}

    77 is 7 × 11 and 9 is 3 × 3. They share no factor, so the fraction is in lowest terms. Watch out: don’t cancel the 9 on top with the 9 below. That 9 is only one part of a sum. Canceling it leaves 64 + 4 = 68, choice D. So the answer is C, 779\frac{77}{9}.

Check your answer

Every square is positive or zero. So the top and the bottom are both positive, and the answer must be positive. That rules out −60 and −173-\frac{17}{3}. Also, 779\frac{77}{9} is about 8.6, which is far from 68.

More questions like this

The original DOE question
Question 77 as printed on page 80 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 80, question 77. Open page 80 of the handbook (PDF)

Question 14 of 23

2023 Test A Math, question 40

Question

Ken has k video games, and Jeff has j video games. If Ken gives 6 video games to Jeff, Ken will have twice as many video games as Jeff. Which equation shows the relationship between k and j?

  • Ak−6=2(j+6)k - 6 = 2(j + 6)
  • Bk−6=2j+6k - 6 = 2j + 6
  • C2(k−6)=j2(k - 6) = j
  • D2(k−6)=j+62(k - 6) = j + 6
Show the answer

Correct answer: A

If Ken gives 6 video games to Jeff, the number Ken has is k−6k - 6, and the number Jeff has is j+6j + 6. After the exchange, Ken has twice as many video games as Jeff. The equation is:

k−6=2(j+6)k - 6 = 2(j + 6)

Question analysis

This algebra word problem says Ken has k video games and Jeff has j, and if Ken gives Jeff 6 games, Ken will have twice as many as Jeff. Students choose the equation that models the situation. Getting the comparison the right way round, and grouping each new total correctly, is where most errors happen.

NYS standards
  • 7.EE.4 equations and inequalities from problems
  • 6.EE.2a writing algebraic expressions
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students write equations from word problems. This one tracks two people's counts after a transfer and states a times-as-many comparison, so both counts change and the factor of 2 must go on the right side: typical but error-prone.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
2 steps
Reading level
Question and choices: about grade 4, 41 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The question is all setup: no solving is needed, just an equation that says what the story says. The trade moves games out of one count and into the other, and the comparison has to be written with the 2 on the smaller side. The wrong equations reverse the comparison, forget Jeff's gain, or drop the parentheses, and all four look alike. Plugging in sample counts is the surest check.

  1. Turning words into math moderate Understanding the question

    After the gift both counts change, Ken's to k − 6 and Jeff's to j + 6, and “twice as many video games as Jeff” must double Jeff's whole new count, putting the 2 on his side. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  2. A very close wrong answer moderate Choosing the answer

    All four equations share the pieces k − 6, j + 6 and a 2; the right one differs from 2(k − 6) = j + 6 only in where the 2 sits, and from k − 6 = 2j + 6 by parentheses. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Variables instead of numbers minor Understanding the question

    There are no numbers to check against, only k and j, so testing an equation means inventing counts that fit the story, such as Jeff with 4 games and Ken with 26. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

Strategy: Write each boy's number of games after the exchange as an expression, then translate "Ken will have twice as many as Jeff" into an equation and test it with sample numbers.

Traps

  • Equation written backwards: Writes a comparison the wrong way round, word for word: "six times as many students as professors" as 6S = P. The bigger quantity gets the bigger number.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Twice as many
A=2×BA = 2 \times B
A and B are two amounts. “A has twice as many as B” means A equals 2 times B. The 2 goes with the smaller amount, because that is the one you double to match the bigger one.
The distributive property
a(b+c)=ab+aca(b + c) = ab + ac
a, b and c are numbers. Multiplying a by a sum means multiplying a by each part, then adding. For example, 2(4 + 6) = 8 + 12 = 20.

  1. Sort out the question

    Ken has k video games, and Jeff has j. Then Ken gives 6 of his games to Jeff. After that, Ken has twice as many games as Jeff. You need the equation that links k and j.

  2. Write each new amount

    Ken gives away 6 games, so he has k − 6 left. Jeff gets those 6 games, so he has j + 6. Both amounts change, so use both new amounts.

  3. Write “twice as many”

    After the gift, Ken has more games. So Jeff’s new amount is the one that gets doubled.

    A=2×BA = 2 \times B

    k−6=2(j+6)k - 6 = 2(j + 6)

    Watch out: putting the 2 on Ken’s side, as in 2(k−6)=j+62(k - 6) = j + 6, would mean Jeff has twice as many. That is choice D.

  4. Keep the parentheses

    a(b+c)=ab+aca(b + c) = ab + ac

    2(j+6)=2j+122(j + 6) = 2j + 12

    Twice Jeff’s new amount is 2j + 12, not 2j + 6. Watch out: dropping the parentheses gives choice B. So the answer is A, k−6=2(j+6)k - 6 = 2(j + 6).

Check your answer

Make up numbers that fit the story. Say Jeff has 4 games and Ken has 26. After the gift, Ken has 20 and Jeff has 10, and 20 is twice 10. Choice A gives 26−6=2(4+6)26 - 6 = 2(4 + 6), or 20 = 20. Choice B gives 20 = 14, which is false.

More questions like this

The original DOE question
Question 97 as printed on page 85 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 85, question 97. Open page 85 of the handbook (PDF)

Question 15 of 23

2023 Test A Math, question 44

Question

For what value of x is the equation x5−4=3(4−2x)−1\frac{x}{5} - 4 = 3(4 - 2x) - 1 true?

  • A7511\frac{75}{11}
  • B7531\frac{75}{31}
  • C157\frac{15}{7}
  • D6531\frac{65}{31}
Show the answer

Correct answer: B

x5−4=3(4−2x)−1\frac{x}{5} - 4 = 3(4 - 2x) - 1 Apply the distributive property; multiply the 3 by 4 and −2x-2x.
x5−4=12−6x−1\frac{x}{5} - 4 = 12 - 6x - 1 Combine like terms.
x5−4=11−6x\frac{x}{5} - 4 = 11 - 6x Apply the additive inverse property; add 6x6x and 4 to both sides of the equation.
x5+6x=15\frac{x}{5} + 6x = 15 Get a common denominator.
x+30x5=15\frac{x + 30x}{5} = 15 Apply the multiplicative inverse property; multiply both sides of the equation by 5.
31x=7531x = 75 Apply the multiplicative inverse property; divide both sides of the equation by 31.
x=7531x = \frac{75}{31}

Question analysis

This equation question asks for the value of x that makes x/5 − 4 = 3(4 − 2x) − 1 true. Students use the distributive property, collect variable terms on one side, and clear a fraction before solving. Each wrong choice matches a common slip with the parentheses or with the fraction, so careful bookkeeping matters more than speed.

NYS standards
  • 8.EE.7b linear equations with rational coefficients
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 equations have the form px + q = r. This one has x on both sides, a fractional coefficient and a product to expand, which are grade 8 equation skills, and every wrong choice matches a plausible slip along the way.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Solution
5 steps
Reading level
Question and choices: about grade 8, 14 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every step is a routine move in solving a linear equation, but there are five of them and the fraction raises the cost of each. The listed wrong answers come from distributing the 3 to only one term, clearing the fraction from one term only, and stretching the parentheses over the −1. Substituting the result back is slow with a denominator of 31, so careful step-by-step work matters more than checking.

  1. Error-prone arithmetic moderate Working it out

    The term x/5 sits beside whole-number terms and a negative −6x, so clearing the fraction means multiplying every term by 5, and the answer is the unfriendly 75/31. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps moderate Working it out

    Five moves in a row, distributing the 3, combining 12 and −1, collecting the x terms, clearing the fifth and dividing by 31, each give a slip a place to enter. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  3. A very close wrong answer minor Choosing the answer

    The fraction 75/11 shares its numerator with 75/31 and differs by one digit of the denominator, so only the coefficient of x, 6 1/5 rather than 2 1/5, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Distribute the 3 to both terms in the parentheses, combine constants, collect the x terms on one side, then multiply every term by 5 to clear the fraction; check by substituting.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

The distributive property
a(b−c)=ab−aca(b - c) = ab - ac
a, b and c are numbers or terms. Multiplying a by a difference means multiplying a by each part. The number outside must reach every term inside the parentheses.
Doing the same to both sides
if a=b, then a+c=b+c and a×c=b×c\text{if } a = b\text{, then } a + c = b + c \text{ and } a \times c = b \times c
a and b are the two sides of an equation, and c is any number. An equation is like a balanced scale. Adding the same amount to both sides, or multiplying both sides by the same number, keeps it balanced.

  1. Sort out the question

    You need the value of x that makes both sides of the equation equal. The plan: clear the parentheses, gather the x terms on one side, and get rid of the fraction. Then divide.

  2. Clear the parentheses

    a(b−c)=ab−aca(b - c) = ab - ac

    3(4−2x)=12−6x3(4 - 2x) = 12 - 6x

    x5−4=12−6x−1=11−6x\frac{x}{5} - 4 = 12 - 6x - 1 = 11 - 6x

    Watch out: the 3 multiplies both 4 and 2x. Writing 12 − 2x leads to 7511\frac{75}{11}, choice A. The −1 is outside the parentheses, so the 3 does not touch it.

  3. Gather the x terms

    Add 6x to both sides to move the x terms to the left. Add 4 to both sides to move the plain numbers to the right.

    if a=b, then a+c=b+c and a×c=b×c\text{if } a = b\text{, then } a + c = b + c \text{ and } a \times c = b \times c

    x5+6x=11+4=15\frac{x}{5} + 6x = 11 + 4 = 15

  4. Clear the fraction

    Multiply both sides by 5. Every term gets multiplied, not only the fraction.

    5×x5+5×6x=5×155 \times \frac{x}{5} + 5 \times 6x = 5 \times 15

    x+30x=75x + 30x = 75

    31x=7531x = 75

    Watch out: multiplying only the fraction by 5 gives x + 6x = 15. That leads to 157\frac{15}{7}, choice C.

  5. Divide by 31

    x=7531x = \frac{75}{31}

    75 and 31 share no factor except 1, so the fraction is already simplified. So the answer is B, 7531\frac{75}{31}.

Check your answer

Put x=7531x = \frac{75}{31} back in. Then x5=1531\frac{x}{5} = \frac{15}{31} and 2x=150312x = \frac{150}{31}. Write 4 as 12431\frac{124}{31}. The left side is 1531−12431=−10931\frac{15}{31} - \frac{124}{31} = -\frac{109}{31}. On the right, 4−15031=−26314 - \frac{150}{31} = -\frac{26}{31}, and 3×(−2631)−1=−7831−3131=−109313 \times \left(-\frac{26}{31}\right) - 1 = -\frac{78}{31} - \frac{31}{31} = -\frac{109}{31}. Both sides match.

More questions like this

The original DOE question
Question 101 as printed on page 86 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 86, question 101. Open page 86 of the handbook (PDF)

Question 16 of 23

2023 Test B Math, question 37

Question

Which expression is equivalent to 12x+7−(x−92)\frac{1}{2}x+7-\left(x-\frac{9}{2}\right)?

  • A12(x−5)\frac{1}{2}(x-5)
  • B12(x+5)\frac{1}{2}(x+5)
  • C−12(x+23)-\frac{1}{2}(x+23)
  • D−12(x−23)-\frac{1}{2}(x-23)
Show the answer

Correct answer: D

First, distribute the negative sign, which is the same as multiplying both terms in the parentheses by −1.

Next, combine like terms:

12x+7−(x−92)\frac{1}{2}x+7-\left(x-\frac{9}{2}\right)

12x+7−x+92\frac{1}{2}x+7-x+\frac{9}{2}

−12x+232-\frac{1}{2}x+\frac{23}{2}

Then, factor the common factor of −12-\frac{1}{2} from both terms, paying careful attention to the signs:

−12x+232=−12(x−23)-\frac{1}{2}x+\frac{23}{2}=-\frac{1}{2}(x-23)

Question analysis

An algebra question asking which factored expression equals one half x plus 7 minus the quantity x minus nine halves. Students distribute the subtraction, combine like terms with fractions, and match a factored form. Signs are the danger at every step, and the choices differ only in the signs and numbers inside the parentheses.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students expand and simplify linear expressions with fractions. Here a subtracted group must be distributed and the result factored by a negative fraction, two places where a sign is easily lost, a longer chain than typical grade 7 work.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
3 steps
Reading level
Question and choices: about grade 15, 10 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Two separate sign decisions decide this question: distributing the subtraction over x − 9/2, and factoring a negative half out of +23/2. Students who handle the first correctly often stumble on the second and choose the look-alike with +23. Expanding each choice, rather than factoring by hand, is quicker and safer, since it turns the comparison into matching −1/2x + 23/2 term by term.

  1. Error-prone arithmetic moderate Working it out

    The minus before (x − 9/2) must change two signs, and the result mixes halves with whole numbers: 1/2x − x and 7 + 9/2 both need care. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. An answer in an unexpected form moderate Choosing the answer

    The simplified result, −1/2x + 23/2, appears nowhere; every choice is factored by ±1/2, so students must factor out −1/2 themselves or expand each choice to compare. (Enright, Morley & Sheehan, 2002)

  3. A very close wrong answer moderate Choosing the answer

    The answer, −1/2(x − 23), and −1/2(x + 23) differ only in the sign inside the parentheses, and the simplified form’s +23/2 makes the plus sign look right. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Change the sign of both terms inside the parentheses, combine the x terms and the constants, then expand each choice to see which one matches rather than factoring yourself.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
  • Answer in another form: The right answer is written differently from the way you worked it out.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Subtracting a group
−(b−c)=−b+c-(b - c) = -b + c
b and c are the two terms inside the parentheses. Terms are the parts joined by plus or minus signs. A minus sign in front of parentheses takes away the whole group. So it changes the sign of every term inside, not only the first one.
Distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number outside the parentheses. b and c are the terms inside. Multiply a by each term inside. It works because a groups of (b + c) hold a of each. Use it to multiply out the choices and compare them.

  1. Sort out the question

    Equivalent means equal for every value of x. The choices are written in a different form from the problem. So simplify the problem first. Then multiply out the choices and compare.

  2. Remove the parentheses

    The minus sign in front of the parentheses takes away both terms inside. So x becomes −x, and −92-\frac{9}{2} becomes +92+\frac{9}{2}.

    12x+7−(x−92)=12x+7−x+92\frac{1}{2}x + 7 - \left(x - \frac{9}{2}\right) = \frac{1}{2}x + 7 - x + \frac{9}{2}

    Watch out: If you change only the sign of x, the numbers become 7−92=527 - \frac{9}{2} = \frac{5}{2}. That mistake leads to choice A or B.

  3. Combine the x terms

    Like terms have the same letter part. The x terms are 12x\frac{1}{2}x and −x. Write x as 22x\frac{2}{2}x so the fractions match.

    12x−22x=−12x\frac{1}{2}x - \frac{2}{2}x = -\frac{1}{2}x

    The result is negative, because you take away more x than you have.

  4. Combine the numbers

    Now combine the plain numbers, 7 and 92\frac{9}{2}. Write 7 as fourteen halves so the bottoms match. Then add.

    7+92=142+92=2327 + \frac{9}{2} = \frac{14}{2} + \frac{9}{2} = \frac{23}{2}

    The expression simplifies to −12x+232-\frac{1}{2}x + \frac{23}{2}.

  5. Multiply out choices C and D

    Only C and D start with −12-\frac{1}{2}, so test those. Use the distributive property. A negative times a negative is positive.

    −12(x+23)=−12x−232-\frac{1}{2}(x + 23) = -\frac{1}{2}x - \frac{23}{2}

    −12(x−23)=−12x+232-\frac{1}{2}(x - 23) = -\frac{1}{2}x + \frac{23}{2}

    The first line is choice C, and the second is choice D. Only D matches.

    Watch out: Choice C looks almost the same. But it gives −232-\frac{23}{2}, not +232+\frac{23}{2}.

    So the answer is D, −12(x−23)-\frac{1}{2}(x - 23).

Check your answer

Try x = 1 in the problem: 12+7−(1−92)=712+312=11\frac{1}{2} + 7 - \left(1 - \frac{9}{2}\right) = 7\frac{1}{2} + 3\frac{1}{2} = 11. Now put x = 1 into choice D: −12(1−23)=−12×(−22)=11-\frac{1}{2}(1 - 23) = -\frac{1}{2} \times (-22) = 11. Both give 11, so they match.

More questions like this

The original DOE question
Question 94 as printed on page 202 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 202, question 94. Open page 202 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 17 of 23

2023 Test B Math, question 46

Question

03+40+2−1+(−1)20^{3}+4^{0}+2^{-1}+(-1)^{2}

What is the value of the expression above, in decimal form?

  • A0
  • B0.5
  • C1.5
  • D2.5
Show the answer

Correct answer: D

03+40+2−1+(−1)20^{3}+4^{0}+2^{-1}+(-1)^{2}

=0+1+12+1=0+1+\frac{1}{2}+1

=212=2.5=2\frac{1}{2}=2.5

Question analysis

An exponents question: students find the decimal value of a sum of four powers, zero cubed, four to the zero power, two to the negative first power and negative one squared. Each of the four terms follows a different rule, and a slip on any one of them changes the total.

NYS standards
  • 8.EE.1 integer exponent properties
  • 7.NS.2a rational number multiplication
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 work uses whole-number exponents and signed numbers. Zero and negative exponents are grade 8 content, and this expression tests both alongside a negative base, each with its own common misconception.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
4 steps
Reading level
Question and choices: about grade 6, 12 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every term tests a different exponent rule, and each wrong choice comes from breaking exactly one of them. The zero exponent is the most common casualty, since 4⁰ looks as if it should be 0 like 0³; a negative exponent read as a negative number is close behind. Writing each term’s value above it before adding, 0, 1, 1/2 and 1, keeps the four rules separate.

  1. Something you must already know major Working it out

    Three rules the test does not give decide the sum: any nonzero number to the 0 power is 1, 2 to the −1 power is 1/2, and (−1)² is positive 1. (Anderson & Krathwohl, 2001)

  2. Error-prone arithmetic minor Working it out

    The four terms carry a zero base, a zero exponent, a negative exponent and a negative base, so a sign or a stray 0 can slip in at each one. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Evaluate each power on its own, using the rules for a zero exponent and a negative exponent and the sign of a squared negative number, then add the four values.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Zero exponent
a0=1a^0 = 1
a is any number except 0. An exponent of 0 always gives 1. It works because each step down in the exponent divides by a. For example, 4 to the first power is 4, and 4 ÷ 4 = 1.
Negative exponent
a−n=1ana^{-n} = \frac{1}{a^n}
a is the base, and n is a positive whole number. A negative exponent does not make the number negative. It means 1 divided by the base to the positive power. For example, 2 to the −1 power is one half.

  1. Sort out the question

    The expression adds four powers. A power is a base with an exponent, a small raised number. The exponent tells how many times to use the base as a factor, a number you multiply. Find each power, then add.

  2. Find the first two powers

    Zero times itself is still zero.

    03=0×0×0=00^3 = 0 \times 0 \times 0 = 0

    Next, use the zero exponent rule.

    40=14^0 = 1

    Watch out: 404^0 is 1, not 0. Taking it as 0 gives choice C.

  3. Find the negative power

    Use the negative exponent rule.

    2−1=121=12=0.52^{-1} = \frac{1}{2^1} = \frac{1}{2} = 0.5

    Watch out: 2−12^{-1} is not −2. That mistake gives choice A.

  4. Find the last power

    The base is −1, inside the parentheses. Multiply it by itself. A negative times a negative is positive.

    (−1)2=(−1)×(−1)=1(-1)^2 = (-1) \times (-1) = 1

    Watch out: The answer is not −1. That mistake gives choice B.

  5. Add the four values

    Now add them. The question wants a decimal, so write one half as 0.5.

    0+1+0.5+1=2.50 + 1 + 0.5 + 1 = 2.5

    So the answer is D, 2.5.

Check your answer

See why 40=14^0 = 1 with a pattern. 42=164^2 = 16 and 41=44^1 = 4. Each step down divides by 4, so 40=4÷4=14^0 = 4 \div 4 = 1. In the same way, 20=12^0 = 1 and 2−1=1÷2=122^{-1} = 1 \div 2 = \frac{1}{2}.

More questions like this

The original DOE question
Question 103 as printed on page 204 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 204, question 103. Open page 204 of the handbook (PDF)

Question 18 of 23

2022 Test A Math, question 7

Question

x2y6xy2\frac{x^{2}y^{6}}{xy^{2}}

What is the value of the expression above if x=−2x = -2 and y=2y = 2?

  • A−64
  • B−32
  • C16
  • D32
Show the answer

Correct answer: B

First, substitute −2 for x and 2 for y.

(−2)2(2)6(−2)(2)2\frac{(-2)^{2}(2)^{6}}{(-2)(2)^{2}}

Then, simplify.

(−2)(−2)(2)2(2)4(−2)(2)2\frac{(-2)(-2)(2)^{2}(2)^{4}}{(-2)(2)^{2}}

(−2)(2)41\frac{(-2)(2)^{4}}{1}

−2×161\frac{-2 \times 16}{1}

−32-32

Question analysis

This question asks for the value of a fraction built from powers of x and y when x is −2 and y is 2. Students evaluate an expression with exponents and a negative number, or simplify it first by cancelling common factors. Keeping track of the negative sign through the squaring and the division is the tricky part.

NYS standards
  • 6.EE.2c evaluating expressions and order of operations
  • 7.NS.2a rational number multiplication
  • 7.NS.2b rational number division
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students evaluate expressions with negative values and whole-number exponents. This one is a single fraction of powers, routine in method, but raising a negative base to powers and then dividing leaves several chances to lose the sign.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 8, 13 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Two routes work: substitute and compute 256 ÷ (−8), or simplify to xy⁴ first and evaluate (−2)(16). Simplifying first is safer, because it leaves a single negative factor to track. Substituting first brings in (−2)², where treating the minus sign as outside the parentheses flips the final sign, and each listed wrong value comes from an exponent or a sign handled wrongly.

  1. Error-prone arithmetic moderate Working it out

    Substituting first gives (−2)²(2)⁶ over (−2)(2)², so the work runs through a squared negative, 2⁶ = 64 and a negative divisor, and one factor of −2 decides the sign. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Something you must already know minor Working it out

    The quotient rule for powers is not stated; students who know that x²/x = x and y⁶/y² = y⁴ can reduce the expression to xy⁴ before any number goes in. (Anderson & Krathwohl, 2001)

Strategy: Cancel the factors of x and y that the top and bottom share before substituting, so there is less to compute, and track the sign of every factor of −2 through to the end.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

A power
a4=a×a×a×aa^{4} = a \times a \times a \times a
a is the base, the number being multiplied. The small number is the exponent. It says how many times the base is used as a factor, a number being multiplied. So y⁶ means six y’s multiplied together. It does not mean y times 6.
Dividing powers with the same base
aman=am−n\frac{a^{m}}{a^{n}} = a^{m - n}
a is the base. m and n are exponents. The top has m factors of a and the bottom has n. Each matching pair cancels, because a ÷ a = 1. So m − n factors are left. A letter with no exponent, like x, has exponent 1.

  1. Sort out the question

    You need the value of x2y6xy2\frac{x^{2}y^{6}}{xy^{2}} when x = −2 and y = 2. The top means x2×y6x^{2} \times y^{6}. The bottom means x×y2x \times y^{2}. You could put the numbers in right away. But simplifying first is safer. It leaves only one negative number to track.

  2. Simplify the x part

    Use the rule for dividing powers. The x on the bottom has exponent 1.

    x2x=x2−1=x1=x\frac{x^{2}}{x} = x^{2 - 1} = x^{1} = x

    You can see it by writing it out: x×xx\frac{x \times x}{x} leaves one x. Watch out: x² ÷ x is x, not 1. If you cancel every x, you lose the −2 and get 16, choice C.

  3. Simplify the y part

    y6y2=y6−2=y4\frac{y^{6}}{y^{2}} = y^{6 - 2} = y^{4}

    So the whole fraction becomes a simpler expression:

    x2y6xy2=x×y4\frac{x^{2}y^{6}}{xy^{2}} = x \times y^{4}

    Watch out: the bottom has y², not y. Using y leaves y⁵ and leads to −64, choice A.

  4. Put in the numbers

    Now substitute x = −2 and y = 2.

    x×y4=(−2)×24x \times y^{4} = (-2) \times 2^{4}

    24=2×2×2×2=162^{4} = 2 \times 2 \times 2 \times 2 = 16

    (−2)×16=−32(-2) \times 16 = -32

    A negative times a positive is negative. So the answer is B, −32.

Check your answer

Substitute first, the long way. Top: (−2)2×26=4×64=256(-2)^{2} \times 2^{6} = 4 \times 64 = 256. Bottom: (−2)×22=(−2)×4=−8(-2) \times 2^{2} = (-2) \times 4 = -8. Then 256÷(−8)=−32256 \div (-8) = -32. Watch out: (−2)² is +4, since two negatives multiply to a positive. Writing −4 leads to 32, choice D.

More questions like this

The original DOE question
Question 64 as printed on page 81 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 81, question 64. Open page 81 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 19 of 23

2022 Test B Math, question 26

Question

(23−45×13)(53+14÷34)=\frac{\left(\frac{2}{3} - \frac{4}{5} \times \frac{1}{3}\right)}{\left(\frac{5}{3} + \frac{1}{4} \div \frac{3}{4}\right)} =

  • A−2115-\frac{2}{115}
  • B−145-\frac{1}{45}
  • C29\frac{2}{9}
  • D15\frac{1}{5}
Show the answer

Correct answer: D

First, simplify the numerator and the denominator separately.

To simplify the numerator, multiply the last two fractions and then subtract from the first fraction.

23−(45×13)\frac{2}{3} - \left(\frac{4}{5} \times \frac{1}{3}\right)

23−415\frac{2}{3} - \frac{4}{15}

1015−415\frac{10}{15} - \frac{4}{15}

615=25\frac{6}{15} = \frac{2}{5}

To simplify the denominator, divide the last two fractions and then add to the first fraction.

53+(14÷34)\frac{5}{3} + \left(\frac{1}{4} \div \frac{3}{4}\right)

53+(14×43)\frac{5}{3} + \left(\frac{1}{4} \times \frac{4}{3}\right)

53+13\frac{5}{3} + \frac{1}{3}

53+13=63=2\frac{5}{3} + \frac{1}{3} = \frac{6}{3} = 2

Finally, divide the numerator by the denominator.

252=25÷2=25×12=15\frac{\frac{2}{5}}{2} = \frac{2}{5} \div 2 = \frac{2}{5} \times \frac{1}{2} = \frac{1}{5}

Question analysis

This fraction arithmetic question gives a complex fraction: a numerator with a subtraction and a multiplication over a denominator with an addition and a division, all with fractions. Students choose its value. It tests order of operations and the four operations with fractions, and working left to right instead of following the order of operations leads to a wrong result.

NYS standards
  • 7.NS.3 four operations with rational numbers
  • 6.NS.1 dividing fractions by fractions
  • 6.EE.2c evaluating expressions and order of operations
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students compute with fractions, usually two or three operations at a time. This expression needs five, in an order set by the rules rather than read left to right, so it is longer and more trap-prone than typical work.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
What makes it hard
Solution
5 steps
Reading level
Question and choices: about grade 16+, 5 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Nothing here is beyond grade 7, but the expression stacks five fraction operations, and the top and the bottom each hide an order-of-operations decision. Working left to right in the numerator alone gives a negative value, and doing so in both parts gives an even stranger fraction; both are listed. The sign is a quick check: 4/15 is less than 2/3, so the numerator is positive, and so is the value.

  1. Many steps moderate Working it out

    The numerator and the denominator are each simplified on their own, to 2/5 and 2, before the final division, so five operations stand between the expression and its value. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  2. Error-prone arithmetic moderate Working it out

    Every step is fraction work: 2/3 − 4/15 needs fifteenths, 1/4 ÷ 3/4 needs the divisor flipped, and the last step, 2/5 ÷ 2, is easily turned into 2/5 × 2. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  3. Something you must already know minor Working it out

    No parentheses inside the numerator or the denominator show the order, so students must know to multiply before subtracting and to divide before adding. (Anderson & Krathwohl, 2001)

Strategy: Simplify the numerator and the denominator separately, doing the multiplication and the division before the subtraction and the addition, then divide the two results.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Order of operations
× and ÷ before + and −\times \text{ and } \div \text{ before } + \text{ and } -
When no parentheses show the order, multiply and divide first. Then add and subtract. Everyone follows the same order, so an expression has only one value.
Multiplying fractions
ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
a and c are the top numbers, and b and d are the bottom numbers. Multiply top times top and bottom times bottom.
Dividing fractions
ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
To divide by a fraction, multiply by its reciprocal, the fraction flipped upside down. The reciprocal of c/d is d/c. It works because c/d times d/c is 1. So multiplying by d/c undoes c/d, just as dividing by c/d does.

  1. Sort out the question

    The big fraction bar means divide: the top part divided by the bottom part. Work out the top and the bottom on their own. Then divide.

    Inside each part, no parentheses show the order. So follow the order of operations.

  2. Multiply first in the top

    The top is 23−45×13\frac{2}{3} - \frac{4}{5} \times \frac{1}{3}. Multiply before you subtract.

    45×13=4×15×3=415\frac{4}{5} \times \frac{1}{3} = \frac{4 \times 1}{5 \times 3} = \frac{4}{15}

    Watch out: subtracting first, 2/3 − 4/5, makes the top negative. That leads to −1/45, choice B. Working left to right in the bottom as well gives −2/115, choice A.

  3. Subtract to finish the top

    To subtract, the bottoms must match. Change 2/3 to fifteenths by multiplying its top and bottom by 5.

    23−415=1015−415=615=25\frac{2}{3} - \frac{4}{15} = \frac{10}{15} - \frac{4}{15} = \frac{6}{15} = \frac{2}{5}

    To simplify 6/15, divide its top and bottom by 3. The top part is 2/5.

  4. Divide first in the bottom

    The bottom is 53+14÷34\frac{5}{3} + \frac{1}{4} \div \frac{3}{4}. Divide before you add. Flip 3/4 and multiply.

    14÷34=14×43=412=13\frac{1}{4} \div \frac{3}{4} = \frac{1}{4} \times \frac{4}{3} = \frac{4}{12} = \frac{1}{3}

    To simplify 4/12, divide its top and bottom by 4.

  5. Add to finish the bottom

    The bottoms already match, so add the tops and keep the bottom.

    53+13=63=2\frac{5}{3} + \frac{1}{3} = \frac{6}{3} = 2

    Six thirds make 2 wholes. The bottom part is 2.

  6. Divide the top by the bottom

    Now divide 2/5 by 2. Flip 2, which is 2/1, to get 1/2.

    25÷2=25×12=210=15\frac{2}{5} \div 2 = \frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = \frac{1}{5}

    Dividing by 2 is the same as taking half, and half of 2/5 is 1/5. So the answer is D, 15\frac{1}{5}.

Check your answer

Look at the signs. 4/15 is less than 2/3, so the top is positive. The bottom is positive too. So the answer must be positive, which rules out A and B. Then multiply back: 15×2=25\frac{1}{5} \times 2 = \frac{2}{5}, the top. It matches.

More questions like this

The original DOE question
Question 83 as printed on page 204 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 204, question 83. Open page 204 of the handbook (PDF)

Question 20 of 23

2018 Test A Math, question 7

Question

7+(3n+6)−(4n+8)=7+\left ( 3n+6 \right )-\left ( 4n+8 \right )=

  • A5−n5-n
  • B5+n5+n
  • C21−n21-n
  • D21+n21+n
Show the answer

Correct answer: A

7+3n+6−4n−8=(7+6−8)+(3n−4n)=5−n7+3n+6-4n-8=(7+6-8)+(3n-4n)=5-n

Question analysis

An expression question: simplify a sum and difference of two linear expressions in n. Students remove the parentheses, subtract every term inside the second group, and combine like terms. The minus sign in front of the second parentheses is the trap, since it has to change the sign of both terms inside.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
  • 7.NS.1c rational number subtraction
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Simplifying linear expressions with a subtracted group is standard grade 7 work. This is a typical textbook exercise in length, with the usual sign trap and no extra steps or real-world context.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 16+, 5 words.
Corrected from the DOE handbook
  • The handbook’s answer key for this question was corrected.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

There are only five terms, but two signs decide the answer: the minus in front of the second parentheses and the sign left after 3n − 4n. Each wrong choice comes from one or both of those sign decisions, so all of them look plausible, and checking needs care rather than time. Even the DOE handbook keyed the version with the wrong sign on n.

  1. Error-prone arithmetic moderate Working it out

    The minus sign before (4n + 8) has to reach both terms, making −4n and −8, and 3n − 4n leaves a negative n term. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    5 − n and 5 + n are written alike and share the constant 5; only the sign of the n term, from 3n − 4n, separates them, and the DOE handbook's own key picked the wrong one. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Rewrite the subtraction as adding the opposite of each term in the second parentheses, then group the constants and the n terms separately.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Answer key: The handbook (its question 64) keys this question as F, which is 5 + n (choice B here), in its answer key and in the heading of its explanation. Its own working gives 7 + 3n + 6 − 4n − 8 = 5 − n, which is the handbook’s choice E and choice A here, and that is correct. The key here is A.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Subtracting a group in parentheses
−(a+b)=−a−b-(a + b) = -a - b
a and b are the terms inside the parentheses. A term is a number, a letter, or a number times a letter. The minus sign in front means you subtract the whole group. So you subtract every term inside, not just the first one.
Combining like terms
3n−4n=(3−4)n3n - 4n = (3 - 4)n
n stands for an unknown number, and 3n means 3 times n. Like terms have the same letter part, like 3n and 4n. To combine them, add or subtract the numbers in front and keep the letter. Plain numbers combine only with plain numbers.

  1. Sort out the question

    You need to simplify the expression, which means write it in a shorter form. An expression is a math phrase made of numbers, letters and signs. Here n stands for an unknown number. Find the choice that matches the short form.

  2. Remove the parentheses

    The first group has a plus sign in front. Adding a group changes nothing, so +(3n+6)+(3n + 6) is just +3n+6+3n + 6. The second group has a minus sign in front. You must subtract both terms inside it.

    −(4n+8)=−4n−8-(4n + 8) = -4n - 8

    7+3n+6−4n−87 + 3n + 6 - 4n - 8

    Watch out: the minus reaches the 8 too. Adding the 8 instead gives 21 for the plain numbers. That leads to choice C or D.

  3. Combine the plain numbers

    Group the numbers without a letter. Work them out from left to right.

    7+6−8=57 + 6 - 8 = 5

    The plain numbers make 5.

  4. Combine the n terms

    You have 3 n’s, and you take away 4 n’s. The numbers in front give 3 − 4 = −1. So you are left with −1n, which is written −n.

    3n−4n=(3−4)n=−1n=−n3n - 4n = (3 - 4)n = -1n = -n

    Watch out: it is easy to write +n here. That gives 5 + n, choice B.

  5. Put the two parts together

    Join the number part and the n part. Adding a negative is the same as subtracting.

    5+(−n)=5−n5 + (-n) = 5 - n

    So the answer is A, 5 − n.

Check your answer

Pick any number for n and test it. Try n = 1. The original: 7+(3+6)−(4+8)=7+9−12=47 + (3 + 6) - (4 + 8) = 7 + 9 - 12 = 4. Choice A: 5−1=45 - 1 = 4. They match. Choice B gives 5+1=65 + 1 = 6, which does not.

More questions like this

The original DOE question
Question 64 as printed on page 82 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 82, question 64. Open page 82 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D). This is the DOE’s version, before the corrections listed in the analysis.

Question 21 of 23

2018 Test A Math, question 50

Question

100(2+0.1)2−100=100\left ( 2+0.1 \right )^{2}-100=

  • A101
  • B200
  • C301
  • D341
Show the answer

Correct answer: D

100(2+0.1)2−100=100(2+0.1)^2-100=

100(2.1)2−100=100(4.41)−100=441−100=341100(2.1)^2-100=100(4.41)-100=441-100=341

Question analysis

This order of operations question asks students to evaluate 100(2 + 0.1)² − 100, an expression with a decimal, a square and a subtraction. Students must add inside the parentheses first, square the result, multiply by 100, and then subtract. Squaring the two parts of the sum separately is the trap.

NYS standards
  • 6.EE.1 whole-number exponent expressions
  • 6.NS.3 multi-digit decimal operations
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Evaluating an expression with an exponent and decimals in the correct order is grade 6 content. This is a short routine calculation, so it asks less than typical grade 7 work with negative numbers and multi-step equations.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 16+, 1 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

This is a direct order-of-operations calculation with no word problem around it, so the only real risk is the square of a sum. Students who apply the exponent to each part, or only to the 0.1, find a listed answer waiting. Adding inside the parentheses before squaring, and then squaring 2.1 exactly, is the whole task.

  1. A very close wrong answer moderate Choosing the answer

    The choices 301 and 341 differ by 40, and squaring 2.1 roughly as 4 lands nearer 301; only the exact square, 4.41, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Error-prone arithmetic minor Working it out

    Squaring 2.1 by hand to get 4.41 puts two decimal places in the product, an easy place to misplace the point. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Add inside the parentheses to get a single decimal, square it carefully, then multiply by 100 and subtract 100, following the order of operations.

Traps

  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Order of operations
parentheses→exponents→×,÷→+,−\text{parentheses} \rightarrow \text{exponents} \rightarrow \times, \div \rightarrow +, -
This is the agreed order for doing math steps. Work inside parentheses first, then exponents, then multiply and divide, then add and subtract. Everyone follows it, so a problem has only one answer.
Squaring a number
a2=a×aa^2 = a \times a
a is any number. The small 2, called an exponent, means multiply the number by itself. So 2.1 squared means 2.1 times 2.1.

  1. Sort out the problem

    You need to work out 100(2+0.1)2−100100(2 + 0.1)^2 - 100. The 100 right next to the parentheses means 100 times. Follow the order of operations, one step at a time.

  2. Add inside the parentheses

    Parentheses come first in the order of operations. So add the numbers inside them:

    2+0.1=2.12 + 0.1 = 2.1

    Now the problem is:

    100×2.12−100100 \times 2.1^2 - 100

  3. Square 2.1

    2.12=2.1×2.1=4.412.1^2 = 2.1 \times 2.1 = 4.41

    Work out 21 × 21 = 441, then put in two decimal places, one for each 2.1.

    Watch out: do not square the 2 and the 0.1 one at a time. That gives 4 + 0.01 = 4.01, which is wrong. It leads to 301, choice C.

  4. Multiply, then subtract

    The exponent is done, so multiply next. Multiplying by 100 moves the decimal point two places to the right. Subtracting comes last:

    100×4.41=441100 \times 4.41 = 441

    441−100=341441 - 100 = 341

    So the answer is D, 341.

Check your answer

Estimate. 2.1 is a bit more than 2, so its square is a bit more than 4. 100 times that is a bit more than 400. Take away 100 and you get a bit more than 300. 341 fits. Choices A and B are too small.

More questions like this

The original DOE question
Question 107 as printed on page 90 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 90, question 107. Open page 90 of the handbook (PDF)

Question 22 of 23

2018 Test B Math, question 31

Question

Simplify this expression:

4(7−3x)−(5−x)4\left ( 7-3x \right )-\left ( 5-x \right )

  • A23−4x23-4x
  • B23−11x23-11x
  • C28−4x28-4x
  • D28−12x28-12x
Show the answer

Correct answer: B

4( 7 − 3x ) −( 5 − x) = 28 − 12x − 5 + x = 23 − 11x

Question analysis

This question asks students to simplify an algebraic expression by distributing a number into one set of parentheses, subtracting another set, and combining like terms. It checks the distributive property with a minus sign in front of parentheses. The usual slip is subtracting only the first term inside the second parentheses, which changes the sign of an x term.

NYS standards
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Expanding and combining linear expressions is core grade 7 work. This is a standard example, two distributions and one subtraction, with nothing beyond what a grade 7 lesson on the distributive property would cover.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 16+, 8 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every step is routine, but subtracting a whole group is where students go wrong: the minus sign has to reach the −x as well as the 5. Each choice pairs a constant with an x coefficient, so getting the constant right only narrows the field to two. The question rewards writing out both distributions in full rather than doing them mentally.

  1. Error-prone arithmetic moderate Working it out

    The minus sign in front of (5 − x) must change both terms, so −x becomes +x and then combines with −12x from the first group. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    Two distributions, then collecting the constants 28 and −5 and the x terms separately: four small operations, each a place where a sign can slip. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  3. A very close wrong answer minor Choosing the answer

    23 − 4x shares the right constant with 23 − 11x, so getting 23 does not settle it; only the x coefficient, −12x + x rather than −3x − x, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Rewrite the subtraction as adding −1 times the second group, distribute both multipliers, then collect the constant terms and the x terms separately.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

The distributive property
a(b−c)=a×b−a×ca(b - c) = a \times b - a \times c
a is the number in front of the parentheses. b and c are the terms inside. You multiply a by each term inside, not only the first one. This works because a groups of (b − c) is a groups of b take away a groups of c.
Subtracting a group
−(b−c)=−b+c-(b - c) = -b + c
b and c are the terms inside the parentheses. A minus sign in front means take away the whole group. So you change the sign of each term inside. Taking away a minus c is the same as adding c.

  1. Sort out the question

    You need to simplify 4(7−3x)−(5−x)4(7-3x)-(5-x). Simplify means write it in the shortest way. A term is one piece of the expression, like 7 or −3x-3x. Like terms have the same letter part, so −12x-12x and xx are like terms. Plain numbers are like terms too.

  2. Multiply the 4 into the first group

    Use the distributive property. The 4 multiplies both terms inside.

    4(7−3x)=4×7−4×3x=28−12x4(7-3x) = 4 \times 7 - 4 \times 3x = 28 - 12x

    Watch out: the 4 must multiply the 3x3x too. Writing only 28−3x28-3x is the first slip that leads to choice A.

  3. Take away the second group

    The minus sign in front of (5−x)(5-x) means subtract the whole group. So change the sign of each term inside.

    −(5−x)=−5+x-(5-x) = -5 + x

    Taking away a minus xx gives plus xx.

    Watch out: if you stop before this step, you get 28−12x28-12x. That is choice D.

  4. Combine the like terms

    Now you have 28−12x−5+x28-12x-5+x. Put the plain numbers together. Then put the xx terms together.

    28−5=2328 - 5 = 23

    −12x+x=−11x-12x + x = -11x

    So the expression simplifies to 23−11x23-11x.

  5. Give the answer

    The plain number is 23, and the xx term is −11x-11x. So the answer is B, 23−11x23-11x.

Check your answer

Put in an easy number, like x=1x = 1. The original gives 4(7−3)−(5−1)=16−4=124(7-3)-(5-1) = 16-4 = 12. Choice B gives 23−11=1223-11 = 12. They match. Choice A would give 23−4=1923-4 = 19, which does not match.

More questions like this

The original DOE question
Question 88 as printed on page 158 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 158, question 88. Open page 158 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 23 of 23

2018 Test B Math, question 34

Question

5x3+3x+9+1x25x^{3}+3x+9+\frac{1}{x^{2}}

If x=10x=10, what is the value of the expression above?

  • A2,539.01
  • B5,039.01
  • C5,039.1
  • D5,139
Show the answer

Correct answer: B

5x3+3x+9+1x25x^3+3x+9+\frac{1}{x^2}

=5(103)+3(10)+9+1102=5(10^3)+3(10)+9+\frac{1}{10^2}

=5,000+30+9+1100=5,039.01=5{,}000+30+9+\frac{1}{100}=5{,}039.01

Question analysis

Students substitute 10 for x in an expression with a cube, a linear term, a constant and a fraction with x squared in its denominator, then add the results. It tests evaluating expressions with exponents and decimals. The fraction term is where most slips happen, because its place value is easy to get wrong.

NYS standards
  • 6.EE.2c evaluating expressions and order of operations
  • 6.EE.1 whole-number exponent expressions
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Evaluating expressions with whole-number exponents is grade 6 work. The only extra care here is a reciprocal term that turns into a decimal, a single routine substitution that typical grade 7 students handle comfortably.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 7, 11 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Substituting 10 is easy; the risk lies in the two unlike terms at the ends. The cube must be applied to x before multiplying by 5, and the reciprocal of 100 must become a hundredth, not a tenth. With choices that differ by a single decimal place, the question rewards writing each term as a number before adding.

  1. Error-prone arithmetic moderate Working it out

    The last term, 1/x² at x = 10, is 1/100, which must be written as 0.01 and lined up beside 5,039; one place too far gives 0.1. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    5,039.1 sits beside the right 5,039.01 and differs only in the hundredths place; only writing 1/100 as 0.01, not 0.1, tells them apart, and estimating cannot. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Work term by term: compute 10 cubed, then the reciprocal of 10 squared as a decimal, and line up the place values carefully before adding.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Exponents
x3=x×x×xx^3 = x \times x \times x
x is the base number. The small raised 3 is the exponent. It tells how many times to use x in a multiplication. In the same way, x squared, written with a raised 2, means x times x.

  1. Sort out the question

    You need the value of 5x3+3x+9+1x25x^3+3x+9+\frac{1}{x^2} when x=10x = 10. Put 10 in place of every xx. Then work out each term, one at a time. A term is a piece joined to the others by a plus sign.

  2. Work out the first term

    Do the exponent first. Then multiply by 5.

    103=10×10×10=1,00010^3 = 10 \times 10 \times 10 = 1{,}000

    5×1,000=5,0005 \times 1{,}000 = 5{,}000

    Watch out: the exponent belongs only to the xx, not to the 5. Squaring 5×105 \times 10 gives 2,500, which leads to choice A.

  3. Work out the middle terms

    The next term means 3 times xx.

    3×10=303 \times 10 = 30

    The 9 has no xx in it, so it stays 9.

  4. Work out the last term

    First find the bottom. Then divide.

    102=10×10=10010^2 = 10 \times 10 = 100

    1100=0.01\frac{1}{100} = 0.01

    One hundredth is 0.01, with the 1 two places after the decimal point.

    Watch out: 0.1 is one tenth, not one hundredth. That slip gives choice C. Adding 100 instead of 1100\frac{1}{100} gives choice D.

  5. Add the four terms

    Line up the decimal points and add. The 0.01 goes in the hundredths place.

    5,000+30+9+0.01=5,039.015{,}000 + 30 + 9 + 0.01 = 5{,}039.01

    So the answer is B, 5,039.01.

Check your answer

The first three terms are whole numbers. They add up to 5,039. The last term is a small part of 1. So the answer is just over 5,039. Only B and C fit. The last term is hundredths, so B is right.

More questions like this

The original DOE question
Question 91 as printed on page 159 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 159, question 91. Open page 159 of the handbook (PDF)