In Center City

2023 Handbook

2023 Test B Math · Question 35 of 57

Question

In Center City, the ratio of people to dogs is $4:1$ , and the ratio of dogs to cats is $3:2$ . What is the ratio of people to cats?

A $2:1$
B $7:3$
C $6:1$
D $12:1$

Show the answer

Correct answer: C

Take each ratio and solve for dogs:

$\frac{p}{d}=\frac{4}{1}$

$4d=p$

$d=\frac{1}{4}p$

And

$\frac{d}{c}=\frac{3}{2}$

$d=\frac{3}{2}c$

Set the two expressions for d equal to each other and solve for $\frac{p}{c}$ :

$\frac{1}{4}p=\frac{3}{2}c$

$p=6c$

$\frac{p}{c}=\frac{6}{1}$

So, the ratio of people to cats is $6:1$ .

Question analysis

A ratio question about Center City, where the ratio of people to dogs and the ratio of dogs to cats are both given. Students link the two ratios through the dogs to find the ratio of people to cats. It tests combining ratios, a step that goes wrong when the given numbers are paired up as written.

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 Grade 7 ratio problems usually work with a single ratio. Here two ratios share the dogs but give them different numbers of parts, so one ratio must be rescaled before people and cats can be compared.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: No obvious method moderate
Much to keep track of minor
Solution: 3 steps
Reading level: Question and choices: about grade 5, 32 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 rewards seeing that the dogs link the two ratios but are counted differently in each. Reading the people’s 4 and the cats’ 2 straight off the page gives a tidy 2 : 1, and it is the trap most students fall into. Rewriting people to dogs as 12 : 3 makes one chain, 12 : 3 : 2, from which people to cats is 12 : 2.

No obvious method moderate Working it out
Dogs are 1 part in the first ratio and 3 parts in the second, so the ratios cannot be joined until 4 : 1 is rescaled to 12 : 3; nothing in the stem says to do this. (Schoenfeld, 1985)
Much to keep track of minor Working it out
Three groups, people, dogs and cats, run through two ratios, and students must keep track of which number belongs to which group while scaling. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Rewrite both ratios with the same number for dogs, so that people to dogs to cats becomes one three-part ratio, then read off people to cats and simplify.

Traps

Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.
Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: strong
Pairs the people’s 4 from the first ratio with the cats’ 2 from the second, 4 : 2 = 2 : 1, without first matching the dogs’ parts.
B Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Adds the two ratios term by term, (4 + 3) : (1 + 2) = 7 : 3, treating ratio terms as amounts that can be combined.
D Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: moderate
Multiplies people per dog by the dogs’ 3, 4 × 3 = 12, and stops before dividing by the cats’ 2, giving 12 : 1.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Equivalent ratios: $a : b = (a \times k) : (b \times k)$; a and b are the two amounts in a ratio, in order. k is any number you choose, except 0. Multiplying both amounts by the same number keeps the comparison the same. For example, 4 : 1 and 12 : 3 both mean 4 people for every dog.

Sort out the question
A ratio compares two amounts. People to dogs is 4 : 1. That means 4 people for every 1 dog. Dogs to cats is 3 : 2. That means 3 dogs for every 2 cats.
You need people to cats. Dogs show up in both ratios, so they are the link.
Make the dog numbers match
The dogs are 1 in the first ratio and 3 in the second. To join the ratios, the dog numbers must be the same. So multiply both parts of the first ratio by 3.
$4 : 1 = (4 \times 3) : (1 \times 3) = 12 : 3$
Now there are 12 people for every 3 dogs.
Join the two ratios
Both ratios now have 3 dogs. So you can write all three groups in one ratio.
$\text{people} : \text{dogs} : \text{cats} = 12 : 3 : 2$
For every 12 people, there are 3 dogs and 2 cats.
Read off and simplify
People to cats is 12 : 2. Divide both parts by 2.
$12 : 2 = (12 \div 2) : (2 \div 2) = 6 : 1$
Watch out: Taking the 4 people and the 2 cats straight from the problem gives 4 : 2 = 2 : 1, choice A. That skips matching the dogs. Stopping at 12 : 1, without using the 2 cats, gives choice D.
So the answer is C, 6 : 1.

Check your answer

Build a small town. Start with 2 cats. The second ratio gives 3 dogs. The first ratio gives 4 people for each dog, so 4 × 3 = 12 people. That is 12 people for 2 cats, or 6 people for each cat.

The original DOE question

Question 92 as printed on page 201 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 201, question 92. Open page 201 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).