The Chens spend $5 of every $8

2018 Handbook

2018 Test B Math · Question 36 of 57

Also in 2026 Practice Test B Math, question 30; 2025 Practice Test B Math, question 36.

Question

The Chens spend $5 of every $8 they earn on planned expenses. If the family earns $29,600 in one year, how much will they spend on planned expenses that year?

A$1,850
B$3,700
C$5,920
D$18,500

Show the answer

Correct answer: D

Let $x$ be the amount spent on planned expense in one year:

$\frac{x}{29{,}600}=\frac{5}{8}$

$x=\frac{5}{8}(29{,}600)=18{,}500$

Question analysis

The Chens spend $5 out of every $8 they earn on planned expenses, and students apply that ratio to a yearly income of $29,600. It tests using a part-to-whole ratio with large numbers. The trap is using only part of the ratio, or using its two numbers the wrong way round.

NYS standards: 6.RP.3 ratio and rate reasoning
6.RP.1 ratio concepts and language
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Applying a ratio to a total is grade 6 ratio reasoning. This is a two-step calculation with friendly numbers, easier than the multistep ratio and percent problems typical of grade 7.
Estimated difficulty: Routine: one familiar step, little to misread.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Turning words into math minor
Error-prone arithmetic minor
Solution: 2 steps
Reading level: Question and choices: about grade 4, 27 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

This is a direct fraction-of-a-quantity problem, and most students who read the ratio as five-eighths will finish it. The common loss is stopping after the division by 8, since that one-eighth share is also offered. Checking that the result is a little more than half the income catches both that error and a misplaced zero.

Turning words into math minor Understanding the question
"$5 of every $8" has to be read as the fraction 5/8 of the income, not as an instruction to divide by 5 or by 8 alone. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Error-prone arithmetic minor Working it out
Dividing $29,600 by 8 and multiplying by 5 works with five-digit numbers, where a dropped zero gives an answer ten times too small. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Treat 5 of every 8 dollars as the fraction five-eighths of the income: divide the yearly earnings by 8, then multiply by 5.

Traps

Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

A An arithmetic slip (Junker & Sijtsma, 2001)
Reveals a slip Expected pull: weak
Divides by 8 and multiplies by 5 with the right digits but drops a zero, ending with one tenth of the planned expenses.
B Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Divides $29,600 by 8 to get one-eighth of the income and stops before multiplying by 5.
C Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Reveals a surface strategy Expected pull: moderate
Divides $29,600 by 5, using only the 5 from the ratio, as if the family spent one-fifth of its income.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Fraction of a number: $\text{part} = \frac{a}{b} \times \text{whole}$; The whole is the full amount, here the money earned. The fraction a over b tells what share you want: a dollars out of every b dollars. Dividing the whole by b gives one share. Multiplying by a gives a of those shares.

Sort out what you know
The Chens earn $29,600 in one year. They spend “$5 of every $8” on planned expenses. That means 5 out of every 8 dollars. As a fraction, that is $\frac{5}{8}$ of what they earn. You need to find how much they spend.
Find one eighth of the money
Split the money into 8 equal parts.
$29{,}600 \div 8 = 3{,}700$
One eighth is $3,700.
Watch out: this is not the answer yet. Stopping here gives choice B.
Take five of those parts
They spend 5 of the 8 parts. Use the fraction formula.
$\text{part} = \frac{5}{8} \times 29{,}600 = 5 \times 3{,}700 = 18{,}500$
They spend $18,500.
Watch out: dividing only by 5 gives $5,920, choice C. Dropping a zero gives $1,850, choice A.
Give the answer
The planned expenses are 5 of the 8 equal parts, or $18,500. So the answer is D, $18,500.

Check your answer

$\frac{5}{8}$ is a little more than half. Half of $29,600 is $14,800. So the answer should be a bit more than $14,800. Only $18,500 fits. The other three choices are all less than half.

The original DOE question

Question 93 as printed on page 159 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 159, question 93. Open page 159 of the handbook (PDF)