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By what percent did the price

2024 Handbook

2024 Test A Math · Question 38 of 57

Question

By what percent did the price of a cup of coffee increase if its price was increased from $1.25 to $1.35?

  • A7%
  • B8%
  • C10%
  • D12%
Show the answer

Correct answer: B

Subtract to determine the amount of the increase.

1.35−1.25=0.101.35 - 1.25 = 0.10

Divide the increase by the original price.

0.10÷1.25=0.080.10 \div 1.25 = 0.08

Multiply the quotient by 100 to express the decimal answer as a percent.

0.08×100=80.08 \times 100 = 8

Question analysis

This percent increase question gives the price of a cup of coffee before and after a raise, from $1.25 to $1.35, and asks by what percent the price went up. Students find the change in price and compare it with a price. Deciding which price to compare the change with is the tricky part.

NYS standards
  • 7.RP.3 multistep ratio and percent problems
  • 6.NS.3 multi-digit decimal operations
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Percent increase is grade 7 content, and this is a standard case with two prices. What makes it harder than routine is deciding which price the change is measured against and dividing small decimals accurately.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 6, 19 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

Most students see that the price rose by 10 cents; the question is what to compare that with. Percent change is measured from the original amount, and dividing by the new price gives a number close enough to the answer that rounding hides the mistake. Reading the 10 cents directly as 10 percent is the other quick trap, since 10 cents is only 10 percent of a dollar.

  1. A very close wrong answer moderate Choosing the answer

    Dividing the 10-cent rise by either price gives a single-digit percent, 8% from $1.25 and about 7.4% from $1.35, so only the choice of base separates the two listed answers. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Error-prone arithmetic moderate Working it out

    Without a calculator, 0.10 ÷ 1.25 is a division by a decimal; rewriting it in cents as 10/125, which simplifies to 8/100, is the clean way through. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Subtract to get the increase, then divide it by the price before the raise and multiply by 100; working in cents keeps the decimals out of the way.

Traps

  • Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.
  • Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
  • 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).

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent increase
percent increase=new−originaloriginal×100\text{percent increase} = \frac{\text{new} - \text{original}}{\text{original}} \times 100
New is the price after the change, and original is the price before it. The top is the amount of the increase. Dividing by the original compares the increase with where you started. Multiplying by 100 turns that decimal into a percent.

  1. Sort out the question

    The coffee cost $1.25 before. That is the original price. Now it costs $1.35, the new price. You need the percent increase. That compares how much the price went up with the original price.

  2. Find the amount of the increase

    1.35−1.25=0.101.35 - 1.25 = 0.10

    The price went up $0.10, or 10 cents. Watch out: 10 cents is not 10%. It would be 10% only if the price started at $1.00. That mistake gives choice C.

  3. Divide by the original price

    percent increase=new−originaloriginal×100\text{percent increase} = \frac{\text{new} - \text{original}}{\text{original}} \times 100

    1.35−1.251.25×100=0.101.25×100\frac{1.35 - 1.25}{1.25} \times 100 = \frac{0.10}{1.25} \times 100

    Dividing by a decimal is easier in cents: 10 cents out of 125 cents. Divide the top and bottom by 5. Then multiply both by 4 to get hundredths.

    10125=225=8100=0.08\frac{10}{125} = \frac{2}{25} = \frac{8}{100} = 0.08

  4. Change to a percent

    0.08×100=80.08 \times 100 = 8

    The price went up 8%. Watch out: divide by the original price, $1.25, not the new one. Dividing by $1.35 gives about 7.4%, which rounds to choice A.

    So the answer is B, 8%.

Check your answer

Work forward from the answer. 8% of $1.25 is 0.08×1.25=0.100.08 \times 1.25 = 0.10, or 10 cents. Then 1.25+0.10=1.351.25 + 0.10 = 1.35. That is the new price, so 8% is right.

More questions like this

The original DOE question

Question 95 as printed on page 86 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 86, question 95. Open page 86 of the handbook (PDF)