SHSAT Sunset

A seller at an Internet site

2020 Handbook

2020 Test B Math · Question 18 of 57

Question

A seller at an Internet site reports that 99.7% of his sales have resulted in satisfied customers. Based on the report, what fraction of the sales have resulted in unsatisfied customers?

  • A1997\frac{1}{997}
  • B31,000\frac{3}{1{,}000}
  • C3997\frac{3}{997}
  • D1300\frac{1}{300}
Show the answer

Correct answer: B

Since 100% means all customers,

100%−99.7%=0.3%100\% - 99.7\% = 0.3\%

Rewrite the percent as a decimal.

0.3%=0.0030.3\% = 0.003

Now rewrite the decimal as a fraction.

0.003=31,0000.003 = \frac{3}{1{,}000}

Question analysis

An online seller reports that 99.7% of his sales ended with satisfied customers, and students must find what fraction of the sales left customers unsatisfied. The question combines finding the remaining part of a whole with turning a percent that contains a decimal into a fraction, and keeping track of what the fraction is out of is where errors creep in.

NYS standards
  • 6.RP.3c percent as a rate per 100
  • 6.NS.3 multi-digit decimal operations
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Finding the rest of a whole and writing a percent as a fraction are grade 6 skills. The decimal inside the percent is the only twist, and the work takes a few quick steps, below typical grade 7 demand.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 8, 34 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 subtraction is easy; the care goes into what the fraction is out of. A student who pictures 997 satisfied sales for every 3 unsatisfied ones writes 3/997, which compares the two groups instead of the unsatisfied with all sales, and it sits beside the key in both form and size. Turning 0.3% into a fraction also tests decimal place value, since 0.3 means three tenths, not one third.

  1. A very close wrong answer moderate Choosing the answer

    Two choices share the numerator 3, and 3/1,000 and 3/997 are almost equal, so estimating cannot separate them; only knowing that the fraction is out of all sales does. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Error-prone arithmetic minor Working it out

    The percent holds a decimal, 0.3%, which becomes 0.003 only after dividing by 100, and 0.3 itself is easily taken for one third. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  3. Combines ideas minor Working it out

    Two ideas meet: the unsatisfied share is what is left of 100% after 99.7%, and that percent must then be rewritten as a fraction of the whole. (Embretson, 1998; Daniel & Embretson, 2010)

Strategy: Subtract the satisfied percent from 100% to find the unsatisfied percent, write it as a decimal by dividing by 100, and turn that decimal into a fraction by its place value.

Traps

  • 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.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Parts of a whole
unsatisfied=100%−satisfied\text{unsatisfied} = 100\% - \text{satisfied}
100% stands for all of the sales. Satisfied and unsatisfied are the percents of sales of each kind. Every sale is one kind or the other, so the two percents add up to 100%. The unsatisfied part is whatever is left over.
Changing a percent to a decimal
p%=p÷100p\% = p \div 100
p is the number in front of the percent sign. Percent means out of 100, so p% is p hundredths. Dividing by 100 moves the decimal point two places to the left.

  1. Sort out the question

    The seller says 99.7% of his sales left customers satisfied, or happy. You need the fraction of sales that left customers unsatisfied. A fraction of the sales means a part out of all the sales.

  2. Find the unsatisfied percent

    Every sale is either satisfied or unsatisfied. So subtract the satisfied part from the whole.

    unsatisfied=100%−satisfied\text{unsatisfied} = 100\% - \text{satisfied}

    unsatisfied=100%−99.7%=0.3%\text{unsatisfied} = 100\% - 99.7\% = 0.3\%

    So 0.3% of the sales left customers unsatisfied.

  3. Change the percent to a decimal

    p%=p÷100p\% = p \div 100

    0.3%=0.3÷100=0.0030.3\% = 0.3 \div 100 = 0.003

    Watch out: 0.3 is three tenths, not one third. Treating 0.3% as a third of a percent gives 1300\frac{1}{300}, choice D.

  4. Write the decimal as a fraction

    Say 0.003 out loud: three thousandths. The 3 sits in the thousandths place, three places after the decimal point. So it means 3 out of 1,000.

    0.003=31,0000.003 = \frac{3}{1{,}000}

  5. Pick the answer

    Watch out: picture 1,000 sales. Then 997 are satisfied and 3 are not. The fraction 3997\frac{3}{997} compares the unsatisfied sales with the satisfied ones. That is choice C. The question wants unsatisfied out of all 1,000 sales.

    So the answer is B, 31,000\frac{3}{1{,}000}.

Check your answer

Turn the answer back into a percent. 3÷1,000=0.0033 \div 1{,}000 = 0.003, which is 0.3%. Now add the satisfied part: 0.3%+99.7%=100%0.3\% + 99.7\% = 100\%. Together they make all the sales, so the answer fits.

More questions like this

The original DOE question

Question 75 as printed on page 192 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2021 Admissions, page 192, question 75. Open page 192 of the handbook (PDF)