The distance from W to X

2023 Handbook

2023 Test B Math · Question 33 of 57

Question

$A line segment with point W at its left end and point Y at its right end. Point X lies on the segment between W and Y, closer to Y.$

The distance from W to X is 70% of the distance from W to Y. If the distance from X to Y is 15 miles, what is the distance, in miles, from W to X?

A10.5
B35
C45
D50

Show the answer

Correct answer: B

Since the distance from W to X is 70% of the distance from W to Y, the distance from X to Y is 30% of the distance from W to Y. Let d represent the distance from W to Y. Then 15 miles is 30% of d.

$0.30d=15$

$d=\frac{15}{0.30}=50$

So, the distance from W to X, in miles, is $50-15=35$ .

Question analysis

A percent question with a line segment: point X lies between W and Y, the distance from W to X is 70% of the whole, and the distance from X to Y is 15 miles. Students find the distance from W to X. Matching the 15 miles to the right percent before finding the whole is the step students most often skip.

NYS standards: 7.RP.3 multistep ratio and percent problems
6.RP.3c percent as a rate per 100
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 percent problems often give a part and its percent and ask for the whole. This one gives the distance for the part the percent does not describe and asks for the other part, so the whole has to be found on the way.
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: Turning words into math moderate
Many steps minor
Solution: 3 steps
Reading level: Question and choices: about grade 5, 33 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 percent and the given distance describe different pieces of the road, which is the whole difficulty. Students who apply the 70% straight to the 15 miles get a small number fast, and students who solve correctly for the whole of WY must still remember to take XY away. Checking that 35 is 70% of 50 confirms the answer and catches both slips.

Turning words into math moderate Understanding the question
The 70% describes WX, but the 15 miles given is XY, so the sentence has to be recast as “XY is 30% of WY” before the 15 miles can be used. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Many steps minor Working it out
Finding the complement, 30%, then the whole, 15 ÷ 0.30 = 50, then the part, 50 − 15, gives three results, and the middle one is itself a choice. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Work out what percent of WY the 15-mile stretch from X to Y must be, use it to find the whole of WY, then find the part from W to X.

Traps

Stops one step early: A choice is a number from partway through the solution.
Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.

The wrong answers, classified

A Percent of the wrong amount (Parker & Leinhardt, 1995)
Reveals a misconception Expected pull: moderate
Takes 70% of the 15 miles, 0.7 × 15 = 10.5, applying the percent to XY instead of to the whole distance from W to Y.
C Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No common working gives 45; it would make XY a quarter of WY instead of 30%, so it mostly draws guesses.
D Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds the whole distance from W to Y, 15 ÷ 0.30 = 50 miles, and stops before subtracting the 15 miles from X to Y.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent of a number: $\text{part} = \frac{\text{percent}}{100} \times \text{whole}$; The whole is the full amount. The part is a piece of it. The percent tells how many hundredths of the whole the part is. For example, 30% means 30 out of every 100, or 0.30. Use it whenever a percent describes a piece of a whole.

Sort out the distances
Point X sits between W and Y. So the trip from W to Y has two pieces: W to X, then X to Y.
W to X is 70% of the whole distance from W to Y. X to Y is 15 miles. You need the distance from W to X, in miles.
Find the percent for X to Y
The whole distance from W to Y is 100%. W to X uses 70% of it. The rest is X to Y.
$100 - 70 = 30$
So the 15 miles from X to Y is 30% of the whole distance.
Find the whole distance
Let d be the distance from W to Y. Use the percent formula. The part is 15, and the percent is 30.
$15 = \frac{30}{100} \times d = 0.30 \times d$
Divide both sides by 0.30 to get d by itself.
$d = 15 \div 0.30 = 50$
The distance from W to Y is 50 miles.
Find the distance from W to X
Take the 15 miles from X to Y away from the whole.
$50 - 15 = 35$
W to X is 35 miles.
Watch out: 50 miles is the whole distance from W to Y, choice D. You still have to subtract. Also, the 70% is of the whole distance, not of the 15 miles. Taking 70% of 15 gives 10.5, choice A.
So the answer is B, 35 miles.

Check your answer

Is 35 really 70% of 50? $0.70 \times 50 = 35$ . Yes. And 35 + 15 = 50, the whole distance. The figure also shows X closer to Y. So W to X should be longer than 15 miles, and it is.

The original DOE question

Question 90 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 90. 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).