READING ASSIGNMENTS

2022 Handbook

2022 Test B Math · Question 35 of 57

Question

READING ASSIGNMENTS

Assignment	Pages Assigned
I	3–17, inclusive
II	25–38, inclusive
III	45–60, inclusive

Over a two-week period, students are assigned a total of 250 pages to read from a science textbook. The table above shows the first three reading assignments within the larger assignment. What percentage of the 250 pages is the total number of pages in these first three assignments?

A16.8%
B18.0%
C22.8%
D24.0%

Show the answer

Correct answer: B

First determine the pages read for each assignment. Since the pages per assignment are inclusive, subtract the first page from the last page assigned and then add 1 to determine the pages read for each assignment.

Assignment I: $(17-3)+1=15$

Assignment II: $(38-25)+1=14$

Assignment III: $(60-45)+1=16$

The student was assigned a total of 45 pages, since $15+14+16=45$ .

To determine the percent, express 45 pages out of 250 pages as a fraction.

$\frac{45}{250}=\frac{45\div 5}{250\div 5}=\frac{9}{50}$

$\frac{9}{50}=\frac{18}{100}=18\%$

Question analysis

A table lists the first three reading assignments from a science textbook as page ranges marked inclusive, out of 250 pages assigned over two weeks. Students find what percentage of the 250 pages the three assignments make up in total. It tests percents, but the harder part comes first: counting the pages in a range when both end pages are included.

NYS standards: 7.RP.3 multistep ratio and percent problems
6.RP.3c percent as a rate per 100
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: Above grade 7 Expressing one number as a percent of another is familiar from grades 6 and 7. Here that number must first be counted from three inclusive page ranges, a step most grade 7 percent problems leave out.
Estimated difficulty: Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Something you must already know moderate
Many steps moderate
Reading a figure, table or graph minor
Solution: 5 steps
Reading level: Question and choices: about grade 6, 56 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

Counting pages in a range is the real test; the percent at the end is routine once the three counts are right. Subtracting first page from last feels natural, and the 42 pages it gives lead straight to a listed choice. The table also invites shortcuts, the span from page 3 to page 60 or the last page alone, and each lands on a choice too. Checking the rule on a tiny range, pages 1 to 3, which hold 3 pages and not 2, fixes the count.

Something you must already know moderate Working it out
Each row says inclusive, but nothing says that pages 3 to 17 hold 17 − 3 + 1 = 15 pages; subtracting alone loses one page from each of the three assignments. (Anderson & Krathwohl, 2001)
Many steps moderate Working it out
Three page counts, their sum, a fraction of the 250 pages and a change to a percent come in sequence, and a miscount in the first step carries through to the choices. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
Reading a figure, table or graph minor Understanding the question
The table’s ranges leave gaps, pages 18 to 24 and 39 to 44, so the first and last pages listed, 3 and 60, do not bound the pages actually assigned. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Work out how many pages each range holds when both end pages count, total the three, then write that total as a fraction of 250 and change it to a percent.

Traps

Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
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

A Counting the ends (Brown & Burton, 1978)
Reveals partial knowledge Expected pull: strong
Subtracts the first page from the last in each row, 14 + 13 + 15 = 42 pages, leaving out one end of each inclusive range, and 42/250 = 16.8%.
C Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: moderate
Treats pages 3 to 60 as one stretch, 60 − 3 = 57, counting the unassigned pages between assignments, and 57/250 = 22.8%.
D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Takes the last assigned page, 60, as the number of pages read, as if the assignments covered pages 1 to 60, and 60/250 = 24%.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Counting pages in an inclusive range: $\text{pages} = \text{last page} - \text{first page} + 1$; Inclusive means the first and last pages are both read. Subtracting counts the steps from the first page to the last. That misses one page, so add 1. Test it: pages 1 to 3 are 3 pages, and 3 − 1 + 1 = 3.
Percent of a total: $\text{percent} = \frac{\text{part}}{\text{whole}} \times 100$; The part is the amount you have, here the pages in the three assignments. The whole is the amount you compare with, here 250 pages. Dividing gives a fraction or decimal. Multiplying by 100 turns it into a percent, a number out of 100.

Sort out the question
The whole assignment is 250 pages. The table gives three page ranges:
- I: pages 3 to 17
- II: pages 25 to 38
- III: pages 45 to 60
Each range is inclusive. That means its first and last pages are read too. You need the percent of 250 pages these three make up.
Count the pages in each range
Use the rule: last page minus first page, plus 1.
$17 - 3 + 1 = 15$
$38 - 25 + 1 = 14$
$60 - 45 + 1 = 16$
Watch out: subtracting alone gives 14 + 13 + 15 = 42 pages. That leads to 16.8%, choice A.
Add the three ranges
$15 + 14 + 16 = 45$
The three assignments hold 45 pages in all.
Watch out: don’t count pages 3 to 60 as one stretch. Pages 18 to 24 and 39 to 44 are not assigned. That mistake gives 57 pages and 22.8%, choice C. Using 60 pages gives 24.0%, choice D.
Change to a percent
$\text{percent} = \frac{45}{250} \times 100$
Simplify first. Divide the top and bottom by 5, then multiply them both by 2.
$\frac{45}{250} = \frac{9}{50} = \frac{18}{100}$
That is 18 out of 100, or 18%. So the answer is B, 18.0%.

Check your answer

Work backward. 10% of 250 pages is 25 pages, so 1% is 2.5 pages. Then 18% is $18 \times 2.5 = 45$ pages. That matches the count of 45.

The original DOE question

Question 92 as printed on page 206 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2023 Admissions, page 206, question 92. Open page 206 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).