What is the value of 10 1/2 + (-5 1/3) - (-2 3/4) ?

10 1/2 + (−5 1/3) − (−2 3/4)

2024 Handbook

2024 Test A Math · Question 52 of 57

Question

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

A $2\frac{5}{12}$
B $7\frac{11}{12}$
C $13\frac{1}{12}$
D $18\frac{7}{12}$

Show the answer

Correct answer: B

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

$10\frac{1}{2} + \left(-5\frac{1}{3}\right) - \left(-2\frac{3}{4}\right)$

$10\frac{6}{12} + \left(-5\frac{4}{12}\right) - \left(-2\frac{9}{12}\right)$

Subtracting a negative is the same as adding.

$10\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}$

Group the whole numbers together and the fractions together.

$10\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}$

$(10 - 5 + 2) + \left(\frac{6}{12} - \frac{4}{12} + \frac{9}{12}\right)$

$7\frac{11}{12}$

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.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Error-prone arithmetic moderate
Many steps minor
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.

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)
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

A Negatives and direction (Brown & Burton, 1978)
Reveals a slip Expected pull: strong
Handles the added negative but treats subtracting −2 3/4 as subtracting 2 3/4, so 5 1/6 − 2 3/4 = 2 5/12.
C Negatives and direction (Brown & Burton, 1978)
Reveals a misconception Expected pull: moderate
Drops the negative signs inside the parentheses and keeps the operations, 10 1/2 + 5 1/3 − 2 3/4 = 13 1/12.
D Negatives and direction (Brown & Burton, 1978)
Reveals a surface strategy Expected pull: weak
Ignores every sign and adds all three mixed numbers, 10 1/2 + 5 1/3 + 2 3/4 = 18 7/12.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Adding and subtracting negatives: $a + (-b) = a - b \quad \text{and} \quad a - (-b) = a + b$; 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: $\frac{a}{b} = \frac{a \times n}{b \times n}$; 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.

Sort out the expression
You need $10\frac{1}{2} + \left(-5\frac{1}{3}\right) - \left(-2\frac{3}{4}\right)$ . 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.
Fix the signs
Adding $-5\frac{1}{3}$ means subtracting $5\frac{1}{3}$ . Subtracting $-2\frac{3}{4}$ means adding $2\frac{3}{4}$ .
$a + (-b) = a - b \quad \text{and} \quad a - (-b) = a + b$
$10\frac{1}{2} - 5\frac{1}{3} + 2\frac{3}{4}$
Watch out: subtracting a negative is adding. Subtracting $2\frac{3}{4}$ instead gives $2\frac{5}{12}$ , choice A. Dropping the minus signs gives C or D.
Use twelfths for the fractions
The bottoms are 2, 3 and 4. All three go into 12, so change each fraction to twelfths.
$\frac{a}{b} = \frac{a \times n}{b \times n}$
$\frac{1}{2} = \frac{6}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{3}{4} = \frac{9}{12}$
Now the problem is $10\frac{6}{12} - 5\frac{4}{12} + 2\frac{9}{12}$ .
Combine whole numbers, then fractions
Work with the whole numbers and the fractions one group at a time. Keep each sign.
$10 - 5 + 2 = 7$
$\frac{6}{12} - \frac{4}{12} + \frac{9}{12} = \frac{11}{12}$
The fraction part is less than 1, so just put them together: $7\frac{11}{12}$ .
So the answer is B, $7\frac{11}{12}$ .

Check your answer

Estimate with decimals. $10.5 - 5.33 + 2.75$ is about 7.92. And $7\frac{11}{12}$ is also about 7.92, since $\frac{11}{12}$ is almost 1. They match.

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)