Carolyn walked 3 miles from her house

2024 Handbook

2024 Test A Math · Question 49 of 57

Question

Carolyn walked 3 miles from her house to the library and then $2\frac{1}{2}$ miles farther to the grocery store. Returning home by the same route, she walked $1\frac{2}{3}$ miles before stopping at a friend’s house. How many miles did Carolyn have left to walk home?

A $3\frac{5}{6}$
B $4\frac{1}{6}$
C $4\frac{2}{3}$
D $7\frac{1}{6}$

Show the answer

Correct answer: A

The distance from Carolyn’s house to the grocery store is $3 + 2\frac{1}{2} = 5\frac{1}{2}$ miles.

Carolyn walked $1\frac{2}{3}$ miles back, so she still had $5\frac{1}{2} - 1\frac{2}{3}$ miles left to walk home.

$5\frac{1}{2} - 1\frac{2}{3} = \frac{11}{2} - \frac{5}{3}$

$\frac{33}{6} - \frac{10}{6} = \frac{23}{6} = 3\frac{5}{6}$

So Carolyn had $3\frac{5}{6}$ miles left to walk home.

Question analysis

Carolyn walks 3 miles from home to the library and 2 1/2 miles more to the grocery store, then heads back the same way and stops at a friend's house after 1 2/3 miles. This fractions word problem asks how far she still has to walk home. Students add and subtract mixed numbers, and the subtraction needs regrouping across unlike denominators.

NYS standards: 7.EE.3 multi-step rational number problems
7.NS.3 four operations with rational numbers
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 students solve multi-step word problems with mixed numbers. This one has only positive numbers, but it needs a clear picture of the route and a subtraction that requires regrouping across unlike denominators.
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: Error-prone arithmetic moderate
A very close wrong answer moderate
Turning words into math minor
Solution: 3 steps
Reading level: Question and choices: about grade 6, 48 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 story needs a quick sketch, but the real test is one mixed-number subtraction in which the second fraction is larger than the first. Many students subtract the fractions the easy way round, which gives a choice close to the right one. Converting both numbers to sixths, 33/6 − 10/6, sidesteps the regrouping, and adding the result to 1 2/3 to get back 5 1/2 is a quick check.

Error-prone arithmetic moderate Working it out
The subtraction 5 1/2 − 1 2/3 needs sixths, 5 3/6 − 1 4/6, and the fraction on top is the smaller one, so a whole must be regrouped. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
A very close wrong answer moderate Choosing the answer
Subtracting the fraction parts the wrong way round gives 4 1/6, only 1/3 away from the answer, so a rough estimate cannot tell the two apart. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Turning words into math minor Understanding the question
The route has to be pictured: the walk back starts at the grocery store, 5 1/2 miles from home, not at the library. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Sketch the route as a line from home to the library to the store, find the total distance out, then subtract the distance already walked back, using sixths.

Traps

A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
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

B A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: strong
Subtracts the whole numbers, 5 − 1 = 4, then the fractions the easy way round, 2/3 − 1/2 = 1/6, giving 4 1/6.
C Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: weak
On the store-to-library stretch, uses the 1 2/3 miles already walked instead of the 5/6 mile left, then adds the 3 miles home.
D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: moderate
Adds all three distances, 3 + 2 1/2 + 1 2/3 = 7 1/6, which is how far she has walked so far, not how far is left.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Mixed number to fraction: $a\frac{b}{c} = \frac{a \times c + b}{c}$; a is the whole number part, b is the top of the fraction, and c is the bottom. Each whole holds c pieces, so a wholes hold a times c pieces. Then add the b extra pieces.
Equivalent fractions: $\frac{a}{b} = \frac{a \times n}{b \times n}$; a is the top of a fraction and b is the bottom. n is any number except 0. Multiplying the top and bottom by the same number gives an equal fraction, cut into smaller pieces. Use it to give two fractions the same bottom before you add or subtract.
Subtracting fractions with the same bottom: $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$; c is the shared bottom number. a and b are the tops. When the pieces are the same size, you just take away how many pieces you have. The size of the pieces, c, stays the same.

Picture the route
Draw a line: home, then the library, then the store. Home to the library is 3 miles. The library to the store is $2\frac{1}{2}$ miles more. Carolyn walks back the same way. After $1\frac{2}{3}$ miles she stops at a friend’s house. You need the miles still left to get home.
Find the distance from home to the store
$3 + 2\frac{1}{2} = 5\frac{1}{2}$
The store is $5\frac{1}{2}$ miles from home. She starts back from the store, so this is the full trip home. Watch out: adding all three distances gives $7\frac{1}{6}$ , choice D. That is how far she has walked, not how far is left.
Change to improper fractions
The miles left are $5\frac{1}{2} - 1\frac{2}{3}$ . An improper fraction has a top bigger than its bottom. Change each mixed number to one.
$a\frac{b}{c} = \frac{a \times c + b}{c}$
$5\frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{11}{2}$
$1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$
Use sixths for both
To subtract, the fractions need the same bottom. 2 and 3 both go into 6, so change both to sixths.
$\frac{a}{b} = \frac{a \times n}{b \times n}$
$\frac{11}{2} = \frac{11 \times 3}{2 \times 3} = \frac{33}{6}$
$\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}$
Subtract and simplify
$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$
$\frac{33}{6} - \frac{10}{6} = \frac{23}{6}$
23 sixths is 3 wholes (18 sixths) and 5 sixths more. So $\frac{23}{6} = 3\frac{5}{6}$ . Watch out: $\frac{1}{2}$ is less than $\frac{2}{3}$ . Subtracting the fraction parts the wrong way round gives $4\frac{1}{6}$ , choice B.
So the answer is A, $3\frac{5}{6}$ miles.

Check your answer

Add back: $3\frac{5}{6} + 1\frac{2}{3} = 3\frac{5}{6} + 1\frac{4}{6} = 4\frac{9}{6} = 5\frac{3}{6} = 5\frac{1}{2}$ . That is the full distance from the store to home. Also estimate: 5.5 minus about 1.7 is about 3.8, and $3\frac{5}{6}$ is about 3.83.

The original DOE question

Question 106 as printed on page 89 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 89, question 106. Open page 89 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).