On a map, the distance

2023 Handbook

2023 Test B Math · Question 15 of 57

Question

On a map, the distance between Deshawn’s house and Xin’s house is 5.4 inches. If 2 inches on the map represents 3 miles, what is the distance between these two houses, in miles?

A2.6
B3.6
C7.5
D8.1

Show the answer

Correct answer: D

Let d represent the distance between the two houses, in miles. Write a proportion to solve for d:

$\frac{d}{5.4} = \frac{3}{2}$

$d = \frac{3(5.4)}{2} = \frac{16.2}{2} = 8.1$

Question analysis

A map shows Deshawn's and Xin's houses 5.4 inches apart, with 2 inches on the map standing for 3 miles. Students find the actual distance between the houses in miles. This scale question uses a decimal map length, so the multiplication must be exact and the inches-to-miles ratio kept the right way up.

NYS standards: 7.G.1 scale drawings
7.RP.2b constant of proportionality
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Converting a map distance with a given scale is routine grade 7 work. This one needs a single proportion and a decimal multiplication, with no rounding or second step, so it matches a typical classroom exercise.
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: Error-prone arithmetic minor
A very close wrong answer minor
Solution: 2 steps
Reading level: Question and choices: about grade 6, 30 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

Unlike most map questions, this one asks for an exact distance, so the decimal 5.4 has to be carried all the way through. The two tempting wrong answers come from opposite places: an upside-down scale shrinks the distance, and a rounded map length gives a result close to the true one. Checking that the answer is one and a half times 5.4 catches both.

Error-prone arithmetic minor Working it out
The map distance is the decimal 5.4, and 5.4 × 3 = 16.2 then halved is a decimal multiplication where rounding 5.4 to make it easier changes the answer. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
A very close wrong answer minor Choosing the answer
7.5 and 8.1 miles are close enough that a rough estimate, 5 × 1.5, lands on the wrong one; only the exact product separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Set up miles over inches equal to 3 over 2, then multiply 5.4 by 3 and divide by 2, keeping the decimal exact throughout.

Traps

Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.

The wrong answers, classified

A Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No common working gives 2.6, which sits just below half of 5.4 and matches no step with the scale, so it mostly draws guesses.
B Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Reveals a misconception Expected pull: moderate
Inverts the scale, multiplying 5.4 by 2/3 instead of 3/2, as if each mile took up 1.5 inches on the map.
C Rounds too early (King et al., 2004)
Reveals partial knowledge Expected pull: moderate
Rounds 5.4 inches down to 5 before scaling, so 5 × 3 ÷ 2 = 7.5 miles instead of the exact distance.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Scale proportion: $\frac{\text{real distance}}{\text{map distance}} = \frac{\text{real distance}}{\text{map distance}}$; A map shrinks every distance by the same amount. So real distance over map distance makes the same fraction everywhere on the map. Put miles on top on both sides and inches on the bottom. Then solve for the missing distance.

Sort out the question
On the map, Deshawn’s house and Xin’s house are 5.4 inches apart. The scale says 2 inches on the map stand for 3 miles. You need the real distance between the houses, in miles.
Set up the proportion
Let d be the real distance, in miles. Put miles on top and inches on the bottom, on both sides.
$\frac{d}{5.4} = \frac{3}{2}$
Solve for d
To get d alone, multiply both sides by 5.4.
$d = \frac{3 \times 5.4}{2} = \frac{16.2}{2} = 8.1$
The houses are 8.1 miles apart. Watch out: keep 5.4 exact. Rounding it to 5 gives 7.5 miles, choice C.
Give the answer
Watch out: don’t flip the scale. Multiplying 5.4 by $\frac{2}{3}$ gives 3.6, choice B. Each inch stands for more than a mile, so the miles must be more than 5.4. So the answer is D, 8.1.

Check your answer

Each inch stands for $3 \div 2 = 1.5$ miles. So 5 inches is 7.5 miles, and 0.4 inch is $0.4 \times 1.5 = 0.6$ mile. Together, $7.5 + 0.6 = 8.1$ miles.

The original DOE question

Question 72 as printed on page 195 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 195, question 72. Open page 195 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).