A paste used to cover a billboard

2018 Handbook

2018 Test A Math · Question 57 of 57

Question

A paste used to cover a billboard is made by mixing the following ingredients by weight: 4 parts powder, 3 parts water, 2 parts resin, and 1 part hardener. To cover one billboard requires 30 pounds of this paste. How many total pounds of resin are required to cover 4 billboards?

Show the answer

Correct answer: C

The ratio is $4:3:2:1$ , so the total parts is 10.

Since there are two parts resin, the fraction of resin is

$\frac{2}{10}=\frac{1}{5}$

So the amount of resin in 30 lb of paste (for 1 billboard) is

$\frac{1}{5}\times 30=6$ lb.

For 4 billboards, that would be 6 × 4 = 24 lb.

Question analysis

This ratio word problem gives a billboard paste recipe of 4 parts powder, 3 parts water, 2 parts resin and 1 part hardener by weight, with 30 pounds of paste per billboard. It asks how many pounds of resin four billboards need. Students must find resin's share of the total parts and scale up to four billboards.

NYS standards: 7.RP.3 multistep ratio and percent problems
6.RP.1 ratio concepts and language
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Part-to-whole ratio problems with several ingredients are typical grade 7 work. This one adds a final scaling step for four billboards, which keeps it a routine multistep ratio problem at grade 7 level.
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: Many steps minor
Solution: 3 steps
Reading level: Question and choices: about grade 8, 45 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 recipe lists four ingredients but only resin's share matters, so the work is part over whole and then scaling. The question asks for four billboards only in the last sentence, after the per-billboard amount has been set up, which is why stopping early is the main risk. Finding the total paste first, 120 pounds, and taking a fifth of it reaches the answer in one step.

Many steps minor Working it out
Add the parts to 10, take resin's 2/10 of 30 pounds, then multiply by 4 billboards: three short steps. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Add the ratio parts to get the total, find resin's fraction of one batch of paste, take that fraction of 30 pounds, and multiply by the number of billboards.

Traps

Stops one step early: A choice is a number from partway through the solution.
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 Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds the resin for one billboard, 2/10 of 30 = 6 pounds, and stops without multiplying by four billboards.
B Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: moderate
Treats resin's 2 parts as 2 pounds and multiplies by 4 billboards, ignoring the 30 pounds of paste each one needs.
D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: weak
Uses powder's 4 parts instead of resin's 2, finding 4/10 of 120 pounds, which is the powder needed for four billboards.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Fraction from a ratio: $f = \frac{p}{T}$; f is the fraction of the mix that is one ingredient. p is that ingredient’s number of parts. T is the total number of parts, found by adding all the ratio numbers. Use it to turn a ratio into a fraction of the whole.

Sort out what you know
The paste is 4 parts powder, 3 parts water, 2 parts resin and 1 part hardener, by weight. A part is just an equal share of the mix.
One billboard needs 30 pounds of paste. You need the pounds of resin for 4 billboards.
Find resin’s fraction of the paste
$4 + 3 + 2 + 1 = 10$
The mix has 10 parts in all, and resin is 2 of them:
$f = \frac{2}{10} = \frac{1}{5}$
So resin is one fifth of the paste. Watch out: 2 parts is not 2 pounds. Treating it that way gives 2 × 4 = 8, choice B.
Find the resin for one billboard
$\frac{1}{5} \times 30 = 6$
One billboard needs 6 pounds of resin. Watch out: 6 is choice A, but that covers only one billboard.
Multiply by four billboards
Each billboard needs the same 6 pounds of resin, so multiply by 4:
$6 \times 4 = 24$
Four billboards need 24 pounds of resin. So the answer is C, 24.

Check your answer

Work it another way. Four billboards need 4 × 30 = 120 pounds of paste. One fifth of 120 is 24. Choice D, 48, is the powder, which has 4 parts out of 10.

The original DOE question

Question 114 as printed on page 92 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 92, question 114. Open page 92 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).