The total cost of a book

2022 Handbook

2022 Test B Math · Question 4 of 57

Question

The total cost of a book is given by the expression $1.08x$ , where x is the price of the book before tax. The tax rate is m%. What is the value of m?

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: 8

The price of a book before tax is x and after tax is $1.08x$ . Subtract to determine the tax rate.

$1.08x - x = 0.08x$ , so the tax rate is 0.08.

The tax rate m% can be written as $\frac{m}{100}$ .

Solve for m:

$\frac{m}{100} = 0.08$

$100\left(\frac{m}{100}\right) = 100(0.08)$

$m = 8$

Question analysis

The total cost of a book, tax included, is given by the expression 1.08x, where x is the price before tax, and the tax rate is m%. Students find the value of m and type it in. It tests reading a percent increase from an algebraic expression, and the rate has to be entered as a number of percent, not as a decimal.

NYS standards: 7.EE.2 rewriting expressions to show relationships
7.RP.3 multistep ratio and percent problems
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Reading a percent increase from an expression such as 1.08x is a named grade 7 idea. It takes one short step, but students must turn a decimal into a percent and type the value with no choices to check.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly understand
What makes it hard: An answer in an unexpected form moderate
Something you must already know minor
Variables instead of numbers minor
Solution: 2 steps
Reading level: Question and choices: about grade 3, 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 expression already holds the answer, since 1.08x is the price, 1x, plus a tax of 0.08x. The work lies in pulling the tax apart from the price and then writing it in the form the stem asks for, the number in front of a percent sign. Students who stop at the decimal, or who read the whole 1.08 as 108 percent, grid a value that looks reasonable. Trying a price of $100, which costs $108 with tax, makes the 8 visible.

An answer in an unexpected form moderate Choosing the answer
The rate is asked for as the m in m%, so the 0.08 found from the expression must become 8, and the grid takes 0.08 or 108 just as readily. (Enright, Morley & Sheehan, 2002)
Something you must already know minor Working it out
Nothing in the stem says that multiplying a price by 1.08 adds 8% of it; students must split 1.08x into 1x for the price and 0.08x for the tax. (Anderson & Krathwohl, 2001)
Variables instead of numbers minor Understanding the question
The price is the letter x, never a dollar amount, so the tax has to be read from the coefficient 1.08 rather than worked out from a number. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

Strategy: Split 1.08x into the price before tax plus the tax added to it, then write the tax part's coefficient as a percent.

Traps

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

0.08 Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: strong
Finds the tax as 0.08x and enters the decimal rate 0.08, one step short of multiplying by 100 to give m as a number of percent.
108 Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: moderate
Reads 1.08 as 108% and enters it, giving the whole cost as a percent of the price instead of only the tax added to it.
1.08 Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Copies the coefficient 1.08 from the expression as the rate, without separating the original price, 1x, from the tax added on top.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent as a fraction: $m\% = \frac{m}{100}$; Percent means out of 100. So m% is m hundredths, the fraction m over 100. To change a decimal to a percent, multiply it by 100. To change a percent to a decimal, divide it by 100.

Sort out the question
x is the price of the book before tax. The total cost with tax is 1.08x, which means 1.08 times x. The tax rate is m%. You need the number m.
The tax rate is the percent of the price that is added on as tax.
Split the total into price and tax
The total is the price plus the tax. Split 1.08 into 1 and 0.08.
$1.08x = 1x + 0.08x = x + 0.08x$
The x part is the price you started with. So the extra 0.08x is the tax. The tax is 0.08 times the price.
Change the decimal to a percent
The rate is 0.08 as a decimal. Use the formula and set it equal to 0.08.
$\frac{m}{100} = 0.08$
To get m alone, multiply both sides by 100.
$m = 0.08 \times 100 = 8$
So the tax rate is 8%.
Give the answer
The question asks for m, the number in front of the % sign. That is 8.
Watch out: 0.08 is the rate as a decimal, one step short. And 1.08, or 108%, counts the price as well as the tax. So you would grid 8.

Check your answer

Try a price of $100. The total is $1.08 \times 100 = 108$ dollars. The tax is 108 − 100 = 8 dollars. And $8 out of $100 is 8%. So m = 8.

The original DOE question

Question 61 as printed on page 198 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 198, question 61. Open page 198 of the handbook (PDF)