SHSAT Sunset

3 1/4 − 1 2/3 − (−4 1/2)

2023 Handbook

2023 Test B Math · Question 6 of 57

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)