A roofing contractor uses shingles

2018 Handbook

2018 Test B Math · Question 9 of 57

Also in 2026 Practice Test B Math, question 8; 2025 Practice Test B Math, question 9.

Question

A roofing contractor uses shingles at a rate of 3 bundles for every 96 square feet of roof covered. At this rate, how many bundles of shingles will he need in order to cover a roof that is 416 square feet?

Show the answer

Correct answer: C

Set up a proportion:

$\frac{x}{416}=\frac{3}{96}$

$96x=1{,}248$

$x=13\text{ bundles}$

Question analysis

This rate word problem says a roofing contractor uses 3 bundles of shingles for every 96 square feet of roof and asks how many bundles a 416-square-foot roof needs. Students set up a proportion or find a unit rate. Care is needed to answer in bundles rather than in 96-square-foot sections of roof.

NYS standards: 6.RP.3b unit rate problems
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Using a rate to scale up a quantity is grade 6 ratio reasoning. Here it takes two short steps with whole numbers, easier than the grade 7 proportion problems that involve fractions or percents.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: A very close wrong answer minor
Error-prone arithmetic minor
Solution: 2 steps
Reading level: Question and choices: about grade 8, 38 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 rate is simple once it is turned into 32 square feet per bundle, and 416 ÷ 32 comes out exactly. Trouble starts for students who divide 416 by 96 first: 4.33 sections of roof invites rounding, and rounding before multiplying by 3 gives a neighbor of the right answer. Keeping the fraction exact, or using the unit rate, avoids the problem.

A very close wrong answer minor Choosing the answer
12 and 14 bundles sit on either side of the right 13, so a rough estimate of 416 ÷ 32 cannot pick among them; only the exact division, 32 × 13 = 416, does. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Error-prone arithmetic minor Working it out
The division 416 ÷ 96 does not come out even, about 4.33, so students who use it rather than 32 square feet per bundle must carry a repeating decimal. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Find how many square feet one bundle covers, then divide the roof's 416 square feet by that amount.

Traps

Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: moderate
Divides 416 by 96 and rounds 4.33 up to 5, giving the number of 96-square-foot sections instead of the number of bundles.
B Rounds too early (King et al., 2004)
Reveals partial knowledge Expected pull: strong
Rounds 416 ÷ 96 down to 4 sections before multiplying by 3 bundles, leaving the last 32 square feet uncovered.
D An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: weak
Slips in dividing 1,248 by 96, or adds a spare bundle to be safe, landing one bundle above the exact count.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Unit rate: $\text{area per bundle} = \frac{\text{area covered}}{\text{bundles used}}$; The area covered is in square feet. Bundles used is how many bundles covered that area. Dividing gives how much just one bundle covers. An amount for just one is called a unit rate.
Bundles needed: $\text{bundles} = \frac{\text{roof area}}{\text{area per bundle}}$; Roof area is the whole roof in square feet. Area per bundle is what one bundle covers. Dividing tells how many bundles it takes to cover the whole roof.

Write down what you know
3 bundles cover 96 square feet. A bundle is a pack of shingles. The new roof is 416 square feet.
You need the number of bundles for that roof, using the same rate.
Find what one bundle covers
Finding the amount for one bundle makes it easy to scale up to any roof.
$\text{area per bundle} = \frac{96}{3} = 32$
One bundle covers 32 square feet.
Divide the roof by 32
Now see how many 32-square-foot pieces fit in the roof.
$\text{bundles} = \frac{416}{32} = 13$
To check the division, $32 \times 13 = 320 + 96 = 416$ . So exactly 13 bundles cover the roof.
Watch out: 416 ÷ 96 is about 4.33. That counts groups of 96 square feet, not bundles. Rounding it to 5 gives choice A. Rounding down to 4 groups gives 12 bundles, choice B, and leaves part of the roof bare.
So the answer is C, 13.

Check your answer

Use the rate from the question. 13 bundles is 4 groups of 3 bundles plus 1 more bundle. That covers $4 \times 96 + 32 = 384 + 32 = 416$ square feet, the whole roof.

The original DOE question

Question 66 as printed on page 154 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 154, question 66. Open page 154 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).