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Mikah’s history assignment is to read 420 pages

2019 Handbook

2019 Test A Math · Question 15 of 57

Question

Mikah’s history assignment is to read 420 pages. He planned to do the assignment in 6 hours. He read the first 160 pages in 2 hours. What is the mean number of pages he must read per hour during the next 4 hours in order to complete the assignment according to plan?

  • A60
  • B65
  • C70
  • D75
Show the answer

Correct answer: B

First, calculate how many pages Mikah has left to read:

420−160=260420 - 160 = 260

Then, divide 260 by 4 to calculate the mean number of pages he must read per hour to finish the assignment according to his plan:

2604=65\frac{260}{4} = 65

Question analysis

Mikah must read 420 pages of history in 6 hours and has finished 160 pages in the first 2. This reading-rate word problem asks for the mean number of pages per hour he needs over the next 4 hours to keep to his plan. The stem offers several totals, so the care lies in pairing the right pages with the right hours.

NYS standards
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Below grade 7 Unit rates are grade 6 content, and this needs only a subtraction and a division with numbers that divide evenly. A grade 7 rate problem would usually add a conversion, a fraction or a percent, so this is lighter.
Estimated difficulty
Straightforward: two steps or one careful reading.
What makes it hard
Solution
2 steps
Reading level
Question and choices: about grade 5, 47 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

Every operation is easy, so the question tests whether students pair each page count with the right stretch of time. Dividing 420 by 6 gives the plan's own rate, which feels like the answer because the stem ends with “according to plan”. Others work from Mikah's 20-page head start and adjust the rate the wrong way, or balance his 80 pages an hour against the next stretch as if both lasted 2 hours.

  1. Turning words into math moderate Understanding the question

    “The mean number of pages he must read per hour during the next 4 hours” asks for a rate on the 260 pages still unread, not on all 420, so the 160 already read comes off first. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  2. Much to keep track of minor Working it out

    Five numbers sit in four short sentences, 420 pages, 6 hours, 160 pages, 2 hours and the next 4 hours, and only three of them enter the division. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Subtract the pages already read from 420, then divide what is left by the 4 hours remaining; the 6-hour plan only tells you how many hours are left.

Traps

  • 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.
  • 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.
  • Information you don't need: The question gives numbers or facts that play no part in the answer.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Mean rate
rate=pageshours\text{rate} = \frac{\text{pages}}{\text{hours}}
The rate is the number of pages read in each hour. Pages is how many pages there are to read. Hours is the time to read them. This rate is a mean, or average. It is the same amount every hour that adds up to the total.

  1. Sort out the question

    Here is what you know:

    • The whole assignment is 420 pages.
    • The plan is 6 hours in all.
    • Mikah read 160 pages in the first 2 hours.

    That leaves 6−2=46 - 2 = 4 hours. You need the mean number of pages per hour for those 4 hours. The mean is the even share: the same number of pages each hour.

  2. Find the pages left

    Only the pages he has not read yet matter now. Take the pages he already read away from the whole assignment.

    420−160=260420 - 160 = 260

    Mikah has 260 pages left to read.

  3. Divide by the hours left

    Use the rate formula with the pages left and the hours left.

    rate=pageshours\text{rate} = \frac{\text{pages}}{\text{hours}}

    rate=2604=65\text{rate} = \frac{260}{4} = 65

    He must read 65 pages per hour for the next 4 hours.

  4. Compare with the plan

    The plan’s own rate is 420÷6=70420 \div 6 = 70 pages per hour. But Mikah read 160 pages in 2 hours. That is 80 pages per hour, so he is ahead. Now he needs fewer than 70 pages per hour, and 65 is fewer.

    Watch out: 70, choice C, is the rate for the whole plan, not for the pages left. He is 20 pages ahead, which is 5 fewer pages each hour. Adding the 5 instead of taking it away gives 75, choice D. At 60 pages per hour, choice A, he would read only 240 pages, 20 short.

    So the answer is B, 65.

Check your answer

Put it back together. In 4 hours at 65 pages per hour, he reads 4×65=2604 \times 65 = 260 pages. Add the first 160 pages: 160+260=420160 + 260 = 420. That is the whole assignment, in 2+4=62 + 4 = 6 hours, just as planned.

More questions like this

The original DOE question

Question 72 as printed on page 99 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2019–2020, page 99, question 72. Open page 99 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).