SHSAT Sunset

Practice Test

7.NS.1d properties for adding and subtracting

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Question 1 of 6

2026 Practice Test D Math, question 15

Question

111+122+133\frac{1}{11} + \frac{1}{22} + \frac{1}{33}

What is the value of the expression shown above?

  • A122\frac{1}{22}
  • B111\frac{1}{11}
  • C16\frac{1}{6}
  • D322\frac{3}{22}
Show the answer

Correct answer: C

To add fractions, find a common denominator and rename the fractions using the common denominator. The least common denominator for 11, 22, and 33 is 66. After adding the fractions, the final fraction can be reduced to lowest terms.

111=1×611×6=666\frac{1}{11} = \frac{1 \times 6}{11 \times 6} = \frac{6}{66}

122=1×322×3=366\frac{1}{22} = \frac{1 \times 3}{22 \times 3} = \frac{3}{66}

133=1×233×2=266\frac{1}{33} = \frac{1 \times 2}{33 \times 2} = \frac{2}{66}

666+366+266=1166=16\frac{6}{66} + \frac{3}{66} + \frac{2}{66} = \frac{11}{66} = \frac{1}{6}

Question analysis

This fractions question asks for the value of 1/11 + 1/22 + 1/33. Students find a common denominator for three fractions, rename each one, add and simplify. The denominators are all multiples of 11, and adding the tops and bottoms straight across, or forgetting to rename a numerator, leads to tempting wrong choices.

NYS standards
  • 7.NS.1d properties for adding and subtracting
  • 6.NS.4 common factors and multiples
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Below grade 7 Adding fractions with unlike denominators is taught before grade 7. Finding a common denominator for three related numbers and simplifying is short, routine work, lighter than typical grade 7 problems with signed rational numbers.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 9, 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

The question checks whether students rename fractions before adding, and each wrong choice comes from a shortcut that skips that step or stops early. Seeing that 22 and 33 are multiples of 11 points to 66 as the common denominator. A size check helps too: the sum must be more than 1/11, its largest term, which rules out two of the wrong choices at once.

  1. Error-prone arithmetic moderate Working it out

    Three unlike denominators must be renamed over 66, not over the product of all three, and the numerators 6, 3 and 2 have to be found one by one before adding. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

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

    The sum comes out as 11/66, which must be reduced to match a choice; its simplest form has a denominator, 6, unlike any in the expression. (Enright, Morley & Sheehan, 2002)

Strategy: Notice that 22 and 33 are multiples of 11, rename all three fractions over their least common multiple, add the numerators, and simplify the result.

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.
  • Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Equivalent fractions
ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}
a/b is a fraction, and n is any number except 0. Multiplying the top and the bottom by the same n gives an equal fraction. It works because n/n is 1, and multiplying by 1 changes nothing. Use it to give fractions the same bottom.
Adding fractions with the same denominator
ad+bd+cd=a+b+cd\frac{a}{d} + \frac{b}{d} + \frac{c}{d} = \frac{a + b + c}{d}
d is the denominator, the bottom number, which tells the size of each piece. a, b and c are the numerators, the top numbers, which count the pieces. When the pieces are the same size, add the tops and keep the bottom.

  1. Sort out the question

    You need to add three fractions: 1/11, 1/22 and 1/33. Their denominators (bottom numbers) are all different. You can only add fractions when the pieces are the same size. So first give all three the same denominator.

  2. Find a common denominator

    A common denominator is a number that 11, 22 and 33 all divide into. Notice that 22 = 2 × 11 and 33 = 3 × 11. So the smallest number that all three divide into is:

    2×3×11=662 \times 3 \times 11 = 66

  3. Rename each fraction

    Multiply the top and bottom of each fraction by the number that makes the bottom 66:

    111=1×611×6=666\frac{1}{11} = \frac{1 \times 6}{11 \times 6} = \frac{6}{66}

    122=1×322×3=366\frac{1}{22} = \frac{1 \times 3}{22 \times 3} = \frac{3}{66}

    133=1×233×2=266\frac{1}{33} = \frac{1 \times 2}{33 \times 2} = \frac{2}{66}

  4. Add the tops

    Now every piece is a sixty-sixth. Add the tops and keep the bottom:

    666+366+266=1166\frac{6}{66} + \frac{3}{66} + \frac{2}{66} = \frac{11}{66}

    Watch out: never add the bottoms. Adding straight across gives 3/66 = 1/22, choice A. Putting the three 1s over 33 gives 3/33 = 1/11, choice B.

  5. Simplify the fraction

    To simplify, divide the top and bottom by a number that goes into both. Here 11 goes into 11 and into 66:

    1166=11÷1166÷11=16\frac{11}{66} = \frac{11 \div 11}{66 \div 11} = \frac{1}{6}

    Watch out: add all three fractions. Stopping after the first two gives 3/22, choice D.

    So the answer is C, 16\frac{1}{6}.

Check your answer

Estimate with decimals. 1/11 is about 0.091, 1/22 is about 0.045 and 1/33 is about 0.030. They add to about 0.166. And 1/6 is 0.1666…, a match. The other choices are all 0.136 or less.

More questions like this

The original DOE question
Question 15 as shown on the DOE’s digital test, page 88 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D Math, page 88, question 15.

Question 2 of 6

2026 Practice Test D Math, question 18

Question

Which of the following decimals is equivalent to 123+2491\frac{2}{3} + 2\frac{4}{9}?

  • A3.13.\overline{1}
  • B3.63.\overline{6}
  • C4.14.\overline{1}
  • D4.94.\overline{9}
Show the answer

Correct answer: C

First, add the mixed numbers. Then, convert the fraction 19\frac{1}{9} to a decimal.

123+2491\frac{2}{3} + 2\frac{4}{9}

169+2491\frac{6}{9} + 2\frac{4}{9}

3+1093 + \frac{10}{9}

3+1193 + 1\frac{1}{9}

419=4.111…=4.14\frac{1}{9} = 4.111\ldots = 4.\overline{1}

Question analysis

This question asks which repeating decimal equals the sum of the mixed numbers 1 2/3 and 2 4/9. Students rename the fractions with a common denominator, add, and convert the fraction part to a decimal with a bar over its repeating digit. The choices differ in both the whole number and the repeating digit, so each step has to be right.

NYS standards
  • 7.NS.2d fractions to decimals by long division
  • 7.NS.1d properties for adding and subtracting
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Adding mixed numbers and writing a fraction as a repeating decimal are standard grade 7 skills. This combines them in three short steps; renaming a fraction and reading the bar notation are the only places that need care.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 13, 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 fraction moves and one conversion stand between the sum and the choices. The carry is the main hazard: 10/9 has to become 1 1/9, and students who keep only the 1/9 land on a choice with the wrong whole number. Estimating first, about 1.7 + 2.4 = 4.1, rules out both choices that begin with 3 and the one that equals 5.

  1. Error-prone arithmetic moderate Working it out

    Adding 2/3 and 4/9 needs 2/3 renamed as 6/9, and the sum 10/9 is more than 1, so a whole must be carried into the 3 before converting. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

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

    The choices are repeating decimals with bars, so a mixed number in ninths has to be recognized as a digit repeating forever; knowing that ninths repeat their numerator makes the match quick. (Enright, Morley & Sheehan, 2002)

Strategy: Rename 2/3 as ninths, add the whole numbers and the fractions separately, regroup any fraction of 9/9 or more as a whole, then divide the leftover numerator by 9.

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.
  • 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

Equivalent fractions
ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}
a/b is a fraction, and n is any number except 0. Multiplying the top and the bottom by the same n gives an equal fraction, because n/n is 1. Use it to give two fractions the same bottom before you add them.
Ninths as repeating decimals
19=0.111…=0.1\frac{1}{9} = 0.111\ldots = 0.\overline{1}
1 divided by 9 is 0.111…, with 1s that go on forever. The bar over a digit means that digit repeats forever. Ninths from 1/9 to 8/9 repeat their top digit, so 4/9 is 0.444… and 2/9 is 0.222…

  1. Sort out the question

    You need to add two mixed numbers, 1 2/3 and 2 4/9. A mixed number is a whole number plus a fraction. Then write the sum as a decimal. The choices are repeating decimals, where the bar over a digit means it repeats forever.

  2. Rename two thirds as ninths

    The fractions 2/3 and 4/9 have different bottoms. Since 3 × 3 = 9, multiply the top and bottom of 2/3 by 3:

    23=2×33×3=69\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}

    So 1 2/3 is the same as 1 6/9.

    Watch out: adding the tops 2 and 4 over 9 without renaming gives 3 6/9, choice B.

  3. Add the mixed numbers

    Add the whole numbers, then add the fractions:

    1+2=369+49=1091 + 2 = 3 \qquad \frac{6}{9} + \frac{4}{9} = \frac{10}{9}

    10/9 is more than 1. It is 9/9 plus 1/9, or 1 1/9. Carry the 1 into the whole number:

    3+119=4193 + 1\frac{1}{9} = 4\frac{1}{9}

    Watch out: forgetting to carry the 1 gives 3 1/9, choice A.

  4. Change it to a decimal

    One ninth is 0.111…, with the 1 repeating forever. So:

    419=4.111…=4.14\frac{1}{9} = 4.111\ldots = 4.\overline{1}

    Watch out: ninths repeat the top digit, not the 9. Choice D, 4.999…, does not match 4 1/9.

    So the answer is C, 4.14.\overline{1}.

Check your answer

Use rough decimals. 1 2/3 is about 1.667, and 2 4/9 is about 2.444. Their sum is about 4.111. That matches 4.111… and rules out both choices that start with 3. And 4.999… is almost 5, which is too big.

More questions like this

The original DOE question
Question 18 as shown on the DOE’s digital test, page 91 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D Math, page 91, question 18.

Question 3 of 6

2024 Test A Math, question 52

Question

What is the value of 1012+(−513)−(−234)10\frac{1}{2} + \left(-5\frac{1}{3}\right) - \left(-2\frac{3}{4}\right)?

  • A25122\frac{5}{12}
  • B711127\frac{11}{12}
  • C1311213\frac{1}{12}
  • D1871218\frac{7}{12}
Show the answer

Correct answer: B

Rewrite the mixed numbers so that they all have equivalent denominators.

1012+(−513)−(−234)10\frac{1}{2} + \left(-5\frac{1}{3}\right) - \left(-2\frac{3}{4}\right)

10612+(−5412)−(−2912)10\frac{6}{12} + \left(-5\frac{4}{12}\right) - \left(-2\frac{9}{12}\right)

Subtracting a negative is the same as adding.

10612−5412+291210\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}

Group the whole numbers together and the fractions together.

10612−5412+291210\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}

(10−5+2)+(612−412+912)(10 - 5 + 2) + \left(\frac{6}{12} - \frac{4}{12} + \frac{9}{12}\right)

711127\frac{11}{12}

Question analysis

This signed-number question asks for the value of 10 1/2 plus negative 5 1/3 minus negative 2 3/4. It tests adding and subtracting negative mixed numbers with unlike denominators. Turning each sign combination into a plain addition or subtraction, and then working in twelfths, is where students most often go wrong.

NYS standards
  • 7.NS.1c rational number subtraction
  • 7.NS.1d properties for adding and subtracting
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Adding and subtracting negative mixed numbers is central grade 7 work. This three-term expression is typical in length, but it needs a common denominator of 12 and two different sign rules: adding a negative and subtracting one.
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 10, 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 is short, but it packs both sign rules for negative numbers into one line, and each misapplication has its own choice. Subtracting a negative is the rule most often broken, and the result is a reasonable-looking mixed number. Estimating with decimals, 10.5 − 5.3 + 2.8 is about 8, separates the answer from every wrong choice.

  1. Error-prone arithmetic moderate Working it out

    Two different sign rules meet: adding −5 1/3 means subtracting 5 1/3, while subtracting −2 3/4 means adding 2 3/4, and mixing them up gives a listed value. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    After the signs, halves, thirds and quarters become twelfths, and the wholes, 10 − 5 + 2, are combined apart from the fractions, 6/12 − 4/12 + 9/12. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Rewrite adding a negative as subtraction and subtracting a negative as addition, convert the fractions to twelfths, then combine the whole numbers and the fractions separately.

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.)

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Adding and subtracting negatives
a+(−b)=a−banda−(−b)=a+ba + (-b) = a - b \quad \text{and} \quad a - (-b) = a + b
a is any number, and b is a positive number, so −b is negative. Adding a negative moves you left on the number line, just like subtracting. Subtracting a negative is like taking away a debt, so you end up with more. It is the same as adding.
Equivalent fractions
ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}
a is the top of a fraction and b is the bottom. n is any number except 0. Multiplying the top and bottom by the same number gives an equal fraction, cut into smaller pieces. Use it to give two fractions the same bottom before you add or subtract.

  1. Sort out the expression

    You need 1012+(−513)−(−234)10\frac{1}{2} + \left(-5\frac{1}{3}\right) - \left(-2\frac{3}{4}\right). There are three mixed numbers, and two of them are negative. A mixed number is a whole number plus a fraction. Plan: fix the signs, give the fractions the same bottom, then work with the whole numbers and the fractions one group at a time.

  2. Fix the signs

    Adding −513-5\frac{1}{3} means subtracting 5135\frac{1}{3}. Subtracting −234-2\frac{3}{4} means adding 2342\frac{3}{4}.

    a+(−b)=a−banda−(−b)=a+ba + (-b) = a - b \quad \text{and} \quad a - (-b) = a + b

    1012−513+23410\frac{1}{2} - 5\frac{1}{3} + 2\frac{3}{4}

    Watch out: subtracting a negative is adding. Subtracting 2342\frac{3}{4} instead gives 25122\frac{5}{12}, choice A. Dropping the minus signs gives C or D.

  3. Use twelfths for the fractions

    The bottoms are 2, 3 and 4. All three go into 12, so change each fraction to twelfths.

    ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}

    12=612,13=412,34=912\frac{1}{2} = \frac{6}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{3}{4} = \frac{9}{12}

    Now the problem is 10612−5412+291210\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}.

  4. Combine whole numbers, then fractions

    Work with the whole numbers and the fractions one group at a time. Keep each sign.

    10−5+2=710 - 5 + 2 = 7

    612−412+912=1112\frac{6}{12} - \frac{4}{12} + \frac{9}{12} = \frac{11}{12}

    The fraction part is less than 1, so just put them together: 711127\frac{11}{12}.

    So the answer is B, 711127\frac{11}{12}.

Check your answer

Estimate with decimals. 10.5−5.33+2.7510.5 - 5.33 + 2.75 is about 7.92. And 711127\frac{11}{12} is also about 7.92, since 1112\frac{11}{12} is almost 1. They match.

More questions like this

The original DOE question
Question 109 as printed on page 89 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 89, question 109. Open page 89 of the handbook (PDF)

Question 4 of 6

2023 Test A Math, question 7

Question

In one week, 1341\frac{3}{4} inches of rain fell on Monday, 2232\frac{2}{3} inches fell on Tuesday, and 78\frac{7}{8} inch fell on Wednesday. How many inches of rain fell during those three days?

  • A57245\frac{7}{24}
  • B51245\frac{1}{24}
  • C3453\frac{4}{5}
  • D3123\frac{1}{2}
Show the answer

Correct answer: A

To solve, add the three given fractions:

134+223+781\frac{3}{4} + 2\frac{2}{3} + \frac{7}{8}

The common denominator is 24:

=11824+21624+2124= 1\frac{18}{24} + 2\frac{16}{24} + \frac{21}{24}

=41024+2124= 4\frac{10}{24} + \frac{21}{24}

=5724= 5\frac{7}{24}

Question analysis

A rainfall word problem gives the inches of rain on Monday, Tuesday and Wednesday as two mixed numbers and a fraction, and asks for the total over the three days. Students add fractions with unlike denominators. Finding a common denominator for fourths, thirds and eighths, and regrouping the fractional part into wholes, is where errors come in.

NYS standards
  • 7.NS.3 four operations with rational numbers
  • 7.NS.1d properties for adding and subtracting
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Below grade 7 Adding fractions and mixed numbers with unlike denominators is taught before grade 7, and this is a direct word problem. It takes more care than a typical exercise only because three different denominators must be rewritten over a common one.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 5, 35 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 setup is plain addition, so the difficulty is all in the fractions. Students who know to rename over a common denominator rarely miss it, while those who add the fraction parts straight across, or put the numerators over 24 without scaling them, reach choices with a whole part of 3. Since 1 3/4 + 2 2/3 alone is more than 4, any total below 4 can be ruled out at a glance.

  1. Error-prone arithmetic moderate Working it out

    The fractional parts have denominators 4, 3 and 8, so they must be renamed as 18, 16 and 21 twenty-fourths, and their sum, 55/24, regrouped as 2 7/24 before the wholes are added. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    Renaming, adding the numerators, regrouping the improper fraction and adding the whole numbers are four separate moves, and losing the 2 carried from 55/24 drops the total by two inches. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Rewrite the three fractional parts over a common denominator of 24, add the whole numbers and the fractions separately, and regroup any extra twenty-fourths as a whole.

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.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Equivalent fractions
ab=a×kb×k\frac{a}{b} = \frac{a \times k}{b \times k}
a over b is a fraction, and k is any whole number except 0. Multiplying the top and the bottom by the same number gives an equal fraction. You cut each piece into k smaller pieces. So you have k times as many pieces, each k times smaller.
Adding fractions with the same bottom
ad+bd+cd=a+b+cd\frac{a}{d} + \frac{b}{d} + \frac{c}{d} = \frac{a + b + c}{d}
d is the denominator, the bottom number. It tells the size of the pieces. a, b and c are the numerators, the top numbers. They count the pieces. When the pieces are the same size, add the counts. The bottom stays the same.
Fraction to mixed number
ab=qrb\frac{a}{b} = q\frac{r}{b}
a over b is a fraction whose top is bigger than its bottom. q is the quotient, how many whole times b fits into a. r is the remainder, what is left over. The leftover stays over the same bottom number.

  1. Sort out the question

    Rain fell on three days: 1341\frac{3}{4} inches, 2232\frac{2}{3} inches and 78\frac{7}{8} inch. You need the total rain, in inches. So add the three amounts. Add the whole numbers and the fractions separately. The whole numbers are 1 and 2, so they add to 3 inches.

  2. Find a common denominator

    The fractions have denominators, or bottom numbers, 4, 3 and 8. You can only add pieces of the same size. So find the smallest number that 4, 3 and 8 all divide into. Count by 8s: 8, 16, 24. 24 works, since 24 = 4 × 6 and 24 = 3 × 8. So use 24ths.

  3. Rename each fraction in 24ths

    Multiply the top and the bottom by the same number.

    34=3×64×6=1824\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24}

    23=2×83×8=1624\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}

    78=7×38×3=2124\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}

    Watch out: change the tops too. Keeping 3, 2 and 7 over 24 gives 1224=12\frac{12}{24} = \frac{1}{2} and a total of 3123\frac{1}{2}, choice D.

  4. Add the fractions

    Now the pieces are the same size, so add the tops.

    1824+1624+2124=5524\frac{18}{24} + \frac{16}{24} + \frac{21}{24} = \frac{55}{24}

    That is more than 1 whole. 24 goes into 55 twice, with 7 left over.

    5524=2724\frac{55}{24} = 2\frac{7}{24}

    Watch out: never add the bottoms. Adding straight across gives 1215=45\frac{12}{15} = \frac{4}{5} and a total of 3453\frac{4}{5}, choice C.

  5. Put the parts together

    Add the 3 whole inches from before.

    3+2724=57243 + 2\frac{7}{24} = 5\frac{7}{24}

    So the answer is A, 57245\frac{7}{24} inches.

Check your answer

Use decimals. 1341\frac{3}{4} is 1.75, 2232\frac{2}{3} is about 2.67, and 78\frac{7}{8} is 0.875. They add to about 5.29. And 724\frac{7}{24} is about 0.29. So 57245\frac{7}{24} fits, and 51245\frac{1}{24} is too small.

More questions like this

The original DOE question
Question 64 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 64. Open page 77 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 5 of 6

2023 Test B Math, question 6

Question

What is the value of 314−123−(−412)3\frac{1}{4} - 1\frac{2}{3} - \left(-4\frac{1}{2}\right)?

  • A−3112-3\frac{1}{12}
  • B−21112-2\frac{11}{12}
  • C61126\frac{1}{12}
  • D95129\frac{5}{12}
Show the answer

Correct answer: C

The first step is to rewrite the problem replacing all subtraction with adding the opposite:

314−123−(−412)3\frac{1}{4} - 1\frac{2}{3} - \left(-4\frac{1}{2}\right)

314+(−123)+4123\frac{1}{4} + \left(-1\frac{2}{3}\right) + 4\frac{1}{2}

Then, rewrite all the fractions using a common denominator of 12, which is the least common multiple of the fractions’ denominators, 4, 3, and 2:

3312+(−1812)+46123\frac{3}{12} + \left(-1\frac{8}{12}\right) + 4\frac{6}{12}

Next, combine the positive numbers and, finally, add the negative number to the sum by subtracting the number with the least absolute value from the number with the greatest absolute value and using the sign of the number with the greatest absolute value:

7912+(−1812)7\frac{9}{12} + \left(-1\frac{8}{12}\right)

61126\frac{1}{12}

Question analysis

What is 3 1/4 − 1 2/3 − (−4 1/2)? This question on rational-number arithmetic combines mixed numbers with denominators of 4, 3 and 2 and ends by subtracting a negative number. Students must turn that last subtraction into an addition and find a common denominator before combining.

NYS standards
  • 7.NS.1c rational number subtraction
  • 7.NS.1d properties for adding and subtracting
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Subtracting negative mixed numbers is core grade 7 content. This expression is typical in length, but it needs a common denominator of 12 and careful handling of the rule for subtracting a negative.
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 10, 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 is short, but it mixes a subtraction that stays a subtraction with one that becomes an addition, and that difference is what most wrong answers miss. The fractions add a second load, since twelfths are needed before anything combines. Estimating first, 3 − 1.7 + 4.5 is about 6, shows the result is positive, which rules out both negative values.

  1. Error-prone arithmetic moderate Working it out

    Three mixed numbers with denominators 4, 3 and 2 must become twelfths, and the last term, −(−4 1/2), changes sign while the middle subtraction does not. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Many steps minor Working it out

    Rewriting the double negative, renaming each fraction, adding 3 1/4 and 4 1/2, then taking away 1 2/3 are four steps, each able to carry a slip forward. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Change subtracting −4 1/2 into adding 4 1/2, rename every fraction in twelfths, add the two positive numbers, then take away 1 2/3.

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.)

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Subtracting a negative number
a−(−b)=a+ba - (-b) = a + b
a and b stand for any numbers. Taking away a negative number is the same as adding its opposite, the same number with the other sign. Think of it this way: taking away a debt leaves you richer.
Equivalent fractions
ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}
a is the top and b is the bottom of a fraction. Multiplying both by the same number n does not change its value. Use it to give fractions the same bottom number, so you can add or subtract them.

  1. Sort out the question

    You need the value of 314−123−(−412)3\frac{1}{4} - 1\frac{2}{3} - \left(-4\frac{1}{2}\right). Each number is a mixed number, a whole number plus a fraction. There are two subtractions. The second one takes away a negative number.

  2. Change the double negative

    Use the rule for subtracting a negative. Taking away −412-4\frac{1}{2} is the same as adding 4124\frac{1}{2}.

    314−123+4123\frac{1}{4} - 1\frac{2}{3} + 4\frac{1}{2}

    The first minus sign stays as it is. Watch out: if you subtract 4124\frac{1}{2} instead, you get −21112-2\frac{11}{12}, choice B.

  3. Give the fractions one denominator

    The denominator is the bottom number of a fraction. Here they are 4, 3 and 2. The smallest number all three divide into is 12. So change each fraction to twelfths.

    14=31223=81212=612\frac{1}{4} = \frac{3}{12} \qquad \frac{2}{3} = \frac{8}{12} \qquad \frac{1}{2} = \frac{6}{12}

    The problem is now:

    3312−1812+46123\frac{3}{12} - 1\frac{8}{12} + 4\frac{6}{12}

  4. Add the two positive numbers

    You may add and subtract in any order, as long as each number keeps its sign. Add the two positive numbers first.

    3312+4612=79123\frac{3}{12} + 4\frac{6}{12} = 7\frac{9}{12}

    That is 3 + 4 = 7 wholes and 3 + 6 = 9 twelfths.

  5. Take away the last number

    7912−1812=61127\frac{9}{12} - 1\frac{8}{12} = 6\frac{1}{12}

    That is 7 − 1 = 6 wholes and 9 − 8 = 1 twelfth. So the answer is C, 61126\frac{1}{12}.

Check your answer

Estimate with decimals: 3.25−1.67+4.5≈6.083.25 - 1.67 + 4.5 \approx 6.08. And 61126\frac{1}{12} is about 6.08 too. The answer must be positive, so A and B are out. Choice D comes from adding all three numbers.

More questions like this

The original DOE question
Question 63 as printed on page 193 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 193, question 63. Open page 193 of the handbook (PDF)

Question 6 of 6

2018 Test A Math, question 28

Question

215+3310+425+5122\frac{1}{5}+3\frac{3}{10}+4\frac{2}{5}+5\frac{1}{2}

What is the value of the expression shown above?

  • A1472014\frac{7}{20}
  • B142514\frac{2}{5}
  • C1572015\frac{7}{20}
  • D152515\frac{2}{5}
Show the answer

Correct answer: D

215+3310+425+5122\frac{1}{5}+3\frac{3}{10}+4\frac{2}{5}+5\frac{1}{2}

Convert all the fractions to a common denominator (10):

2210+3310+4410+55102\frac{2}{10}+3\frac{3}{10}+4\frac{4}{10}+5\frac{5}{10}

=(2+3+4+5)+(2+3+4+510)=(2+3+4+5)+\left(\frac{2+3+4+5}{10}\right)

=14+1410=1525=14+1\frac{4}{10}=15\frac{2}{5}

Question analysis

A fractions question: add four mixed numbers whose fractions have denominators of 5, 10, 5 and 2. Students rewrite the fractions over a common denominator, add the whole numbers and the fractions, regroup any extra whole, and simplify. Regrouping and lowest terms both matter, since the choices differ in exactly those places.

NYS standards
  • 7.NS.1d properties for adding and subtracting
  • 7.EE.3 multi-step rational number problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Below grade 7 Adding mixed numbers with unlike denominators is grade 5 content. With four terms and a regrouping step it takes care, but it asks less than typical grade 7 work with negative rational numbers.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 9, 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

The numbers are friendly: every fraction converts to tenths, and the whole parts add to 14. The risk is in the bookkeeping, either forgetting to carry the extra whole from the fractions or adding numerators over a made-up denominator. Converting to decimals, 2.2 + 3.3 + 4.4 + 5.5, is a fast check that avoids both.

  1. Error-prone arithmetic moderate Working it out

    Four mixed numbers with denominators 5, 10, 5 and 2 must be rewritten as tenths, 2 + 3 + 4 + 5 tenths, and the 14/10 regrouped into a whole and a fraction. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    15 2/5 and 14 2/5 are written alike and share the fraction 2/5; only the whole carried from 14/10 separates them, so a missed carry is invisible at a glance. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

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

    The fractions total 1 4/10, but every choice is in lowest terms, so the 4/10 has to be reduced to 2/5 before it matches. (Enright, Morley & Sheehan, 2002)

Strategy: Convert each mixed number to a decimal, since these fractions all convert exactly, add them, and turn the decimal part back into a fraction in lowest terms.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Equal fractions
ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}
a is the top number, called the numerator. b is the bottom number, called the denominator. n is any whole number except zero. Multiplying the top and bottom by the same n gives an equal fraction. Use it to give fractions the same denominator so you can add them.

  1. Sort out the question

    You need to add four mixed numbers. A mixed number is a whole number plus a fraction, like 2152\frac{1}{5}. The fractions have different denominators, or bottom numbers: 5, 10, 5 and 2. To add fractions, they need the same denominator.

  2. Add the whole numbers

    Add the whole numbers first. You will add the fractions on their own after that.

    2+3+4+5=142 + 3 + 4 + 5 = 14

    The whole numbers make 14.

  3. Rewrite each fraction in tenths

    Use 10 as the denominator, since 5 and 2 both go into 10.

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

    25=2×25×2=410\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}

    12=1×52×5=510\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}

    310\frac{3}{10} is already in tenths. Watch out: when you change a bottom number, change the top too. Adding the old tops, 1 + 3 + 2 + 1 = 7, and putting them over 20 gives 7/20. That is in choices A and C.

  4. Add the fractions

    210+310+410+510=1410\frac{2}{10} + \frac{3}{10} + \frac{4}{10} + \frac{5}{10} = \frac{14}{10}

    14 tenths is more than 1 whole. Ten tenths make 1, so 14 tenths is 1 whole and 4 tenths.

    1410=1410=125\frac{14}{10} = 1\frac{4}{10} = 1\frac{2}{5}

    To simplify 4/10, divide the top and bottom by 2.

  5. Put it all together

    Add the 14 wholes to the sum of the fractions.

    14+125=152514 + 1\frac{2}{5} = 15\frac{2}{5}

    Watch out: carry the extra whole into the 14. Missing it gives 142514\frac{2}{5}, choice B. So the answer is D, 152515\frac{2}{5}.

Check your answer

Change each mixed number to a decimal: 2.2+3.3+4.4+5.5=15.42.2 + 3.3 + 4.4 + 5.5 = 15.4. The 0.4 is 4 tenths, which is 2/5. So the sum is 152515\frac{2}{5}.

More questions like this

The original DOE question
Question 85 as printed on page 86 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 86, question 85. Open page 86 of the handbook (PDF)