SHSAT Sunset

Kim can repair 3 bicycles every 4 hours

2020 Handbook

2020 Test B Math · Question 28 of 57

Question

Kim can repair 3 bicycles every 4 hours. At this rate, how long will it take her to repair 5 bicycles?

  • A6 hr
  • B6 hr 40 min
  • C6 hr 45 min
  • D7 hr 15 min
Show the answer

Correct answer: B

3 bicycles every 4 hours = 3 bicycles every 240 minutes (4×60 min4 \times 60\text{ min})

240 min÷3 bicycles=80 min per bicycle240\text{ min} \div 3\text{ bicycles} = 80\text{ min per bicycle}

5 bicycles×80 min=400 min5\text{ bicycles} \times 80\text{ min} = 400\text{ min}

400 min60 min/hr=203 hr=623 hr\frac{400\text{ min}}{60\text{ min/hr}} = \frac{20}{3}\text{ hr} = 6\frac{2}{3}\text{ hr}

23×60 min=40 min\frac{2}{3} \times 60\text{ min} = 40\text{ min}

623 hr=6 hr 40 min6\frac{2}{3}\text{ hr} = 6\text{ hr }40\text{ min}

Question analysis

A rate word problem: Kim repairs 3 bicycles every 4 hours, and students find how long she needs for 5 bicycles at the same rate. The time per bicycle is not a whole number of hours, so the total must be written in hours and minutes, and choices only minutes apart punish a careless conversion.

NYS standards
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Unit-rate problems with a change of time units are grade 6 work. This one needs a rate per bicycle, one multiplication and a conversion of part of an hour into minutes, a short task below typical grade 7 demand.
Estimated difficulty
Straightforward: two steps or one careful reading.
Solution
3 steps
Reading level
Question and choices: about grade 2, 25 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 main risk is thinking in differences: 2 more bicycles than 3 seems to call for 2 more hours than 4, a tidy wrong answer of 6 hours. Students who scale correctly still have to turn a fraction of an hour into minutes, and a careless 45 for the 2/3 hour lands on the key's neighbor. Working in minutes, 80 per bicycle, keeps every number whole and leaves one simple conversion at the end.

  1. An answer in an unexpected form moderate Choosing the answer

    The rate gives 5 × 4/3 = 20/3, or 6 2/3 hours, but every choice is in hours and minutes, so the 2/3 of an hour must become 40 minutes before the answer can be recognized. (Enright, Morley & Sheehan, 2002)

  2. Error-prone arithmetic minor Working it out

    Each bicycle takes 4/3 of an hour, not a whole number; changing 4 hours to 240 minutes first gives a whole 80 minutes per bicycle and avoids the fraction. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  3. A very close wrong answer minor Choosing the answer

    The choices 6 hr 40 min and 6 hr 45 min are five minutes apart, so an estimate of a little under 7 hours fits both and only an exact conversion decides. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Change the 4 hours to minutes first so the time per bicycle comes out whole, multiply by 5, and convert the total minutes back into hours and minutes.

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.
  • Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Unit rate
time for one=total time÷number done\text{time for one} = \text{total time} \div \text{number done}
A unit rate tells how much goes with just 1 of something. Here it is the time to fix 1 bicycle. Total time is how long the work took, and number done is how many bicycles were fixed in that time. Once you know the time for one, multiply to scale up.
Hours and minutes
1 hour=60 minutes1 \text{ hour} = 60 \text{ minutes}
Each hour has 60 minutes. To change hours to minutes, multiply by 60. To change minutes back to hours, divide by 60. Any remainder is the minutes left over.

  1. Sort out the question

    Kim fixes 3 bicycles every 4 hours. She keeps working at that same rate. You need the time for 5 bicycles. The choices are in hours and minutes, so your answer must be too.

  2. Change the hours to minutes

    Splitting 4 hours among 3 bicycles does not give a whole number of hours. Minutes keep the numbers whole.

    1 hour=60 minutes1 \text{ hour} = 60 \text{ minutes}

    4×60=2404 \times 60 = 240

    So Kim fixes 3 bicycles in 240 minutes.

  3. Find the time for one bicycle

    Next, find the unit rate. Share the 240 minutes equally among the 3 bicycles.

    time for one=total time÷number done\text{time for one} = \text{total time} \div \text{number done}

    time for one=240÷3=80\text{time for one} = 240 \div 3 = 80

    Each bicycle takes her 80 minutes.

  4. Find the time for 5 bicycles

    5×80=4005 \times 80 = 400

    Five bicycles take 400 minutes. Watch out: don’t just add. Two more bicycles does not mean two more hours. That thinking gives 6 hours, choice A.

  5. Change back to hours and minutes

    Divide by 60 to find how many full hours fit in 400 minutes.

    6×60=3606 \times 60 = 360

    400−360=40400 - 360 = 40

    That is 6 full hours with 40 minutes left over. Watch out: the leftover is 23\frac{2}{3} of an hour, which is 40 minutes. 45 minutes would be 34\frac{3}{4} of an hour, choice C.

    So the answer is B, 6 hr 40 min.

Check your answer

Work backward. 6 hr 40 min is 400 minutes. Split among 5 bicycles, that is 80 minutes each. Then 3 bicycles take 3×80=2403 \times 80 = 240 minutes, which is exactly 4 hours. It matches.

More questions like this

The original DOE question

Question 85 as printed on page 194 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2021 Admissions, page 194, question 85. Open page 194 of the handbook (PDF)