A hiker plans on hiking 17 miles

2024 Handbook

2024 Test A Math · Question 48 of 57

Question

A hiker plans on hiking 17 miles in 3 days. Which equation describes the relationship between the number of days hiked, x, and the number of miles traveled, y?

A $y = \frac{3}{17}x$
B $y = 3x$
C $y = \frac{17}{3}x$
D $y = 17x$

Show the answer

Correct answer: C

There is a proportional relationship between the number of miles traveled, y, and the number of days hiked, x.

$\frac{y}{x} = \frac{17}{3}$

$y = \frac{17}{3}x$

So the equation $y = \frac{17}{3}x$ describes the relationship.

Question analysis

A hiker plans to cover 17 miles in 3 days, and this proportional relationship question asks for the equation linking days hiked, x, to miles traveled, y. Students find the constant rate and write it as an equation of the form y = kx. Putting the right quantity on top of the rate is the main decision.

NYS standards: 7.RP.2c proportional relationship equations
7.RP.2b constant of proportionality
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Writing an equation y = kx for a proportional relationship is a core grade 7 skill, and this is the textbook form. The only extra demand is keeping the rate's direction straight: miles per day or days per mile.
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 moderate
Turning words into math minor
Solution: 2 steps
Reading level: Question and choices: about grade 7, 31 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

Writing y = kx is routine, so the question rests on putting the right number on top of the rate. Two choices use one number from the stem with no division at all, and one inverts the rate. Substituting x = 3 into each equation is the fastest check, since only one of them gives y = 17.

A very close wrong answer moderate Choosing the answer
The coefficients 17/3 and 3/17 use the same two numbers, one the other inverted, so only checking that 3 days gives 17 miles separates the two equations. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Turning words into math minor Understanding the question
With x as days and y as miles, "17 miles in 3 days" becomes miles per day, 17 ÷ 3, the number that multiplies x. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find the constant of proportionality as miles per day, since y is miles and x is days, then write y equals that constant times x; check with x = 3.

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).
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.

The wrong answers, classified

A Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Reveals partial knowledge Expected pull: strong
Divides days by miles, 3/17, the number of days per mile, turning the rate upside down.
B Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Uses the 3 days as the rate, writing y = 3x, a number from the question with no division.
D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: moderate
Uses the 17 total miles as if they were covered in one day, writing y = 17x without dividing by the 3 days.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Constant of proportionality: $k = \frac{y}{x}$; x and y are two amounts that change together. k is the constant of proportionality: how much y goes with 1 of x. In a proportional relationship, y divided by x gives the same k every time.
Proportional relationship: $y = kx$; kx means k times x. In a proportional relationship, y is always k times x. Here x is the number of days and y is the miles. k is the miles for each day. Use it to write the equation once you know k.

Sort out the question
The hiker plans 17 miles in 3 days. x is the number of days, and y is the number of miles. You need an equation that links them. Assume the hiker goes the same distance each day. Then y is always the same number times x.
Find the miles for each day
That number is the constant of proportionality, k. It is miles divided by days.
$k = \frac{y}{x}$
$k = \frac{17}{3}$
That is $5\frac{2}{3}$ miles each day. Watch out: miles go on top, because y is miles. $\frac{3}{17}$ is days per mile, upside down. That is choice A.
Write the equation
$y = kx$
$y = \frac{17}{3}x$
Watch out: 17 is the miles for all 3 days, not for 1 day. So $y = 17x$ , choice D, is too fast. And $y = 3x$ , choice B, uses the days as the rate.
So the answer is C, $y = \frac{17}{3}x$ .

Check your answer

Put in x = 3, for 3 days: $y = \frac{17}{3} \times 3 = 17$ . That is 17 miles, which matches the plan. Try choice A the same way: $\frac{3}{17} \times 3 = \frac{9}{17}$ . That is less than 1 mile, far too short.

The original DOE question

Question 105 as printed on page 88 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 88, question 105. Open page 88 of the handbook (PDF)