The Barnes family and the Ramirez family

2023 Handbook

2023 Test B Math · Question 45 of 57

Question

The Barnes family and the Ramirez family each have a pizza of equal size. The first pizza is cut into 18 equal slices, and the second pizza is cut into 15 equal slices. If the Barnes family eats 11 slices from the first pizza, what is the greatest number of whole slices from the second pizza that the Ramirez family can eat without eating a greater percentage of a pizza than the Barnes family ate?

Show the answer

Correct answer: D

The fraction of the pizza that the Barnes family eats is $\frac{11}{18}$ . Let x represent the number of slices of pizza that the Ramirez family eats. Then the fraction of the pizza that the Ramirez family eats is $\frac{x}{15}$ . Write an inequality and solve for x:

$\frac{x}{15}\leq\frac{11}{18}$

$x\leq\frac{11\times 15}{18}$

$x\leq\frac{165}{18}$

$x\leq 9\frac{1}{6}$

So, the greatest number of whole slices of pizza the Ramirez family can eat is 9.

Question analysis

Two pizzas of the same size are cut into 18 and 15 equal slices, and one family eats 11 slices of the first. Students find the most whole slices of the second pizza the other family can eat without eating a larger share. It tests comparing fractions with different denominators and rounding to a whole number in the right direction.

NYS standards: 7.EE.4b two-step inequalities
6.RP.3 ratio and rate reasoning
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: Above grade 7 Grade 7 inequality problems usually ask for the greatest whole number that fits a budget. This one wraps that idea in a long stem about shares of two pizzas sliced differently, with several conditions to keep straight at once.
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: Complex sentences moderate
Turning words into math moderate
Error-prone arithmetic minor
Solution: 3 steps
Reading level: Question and choices: about grade 11, 72 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 math is one inequality between fractions, but the long final sentence hides it, so students who skim fall back on comparing slice counts. Reasoning that the second pizza has 3 fewer slices, so the Ramirez family should eat 3 fewer, is the strongest pull. Comparing 11/18 with x/15, or turning 11/18 into about 61% and taking 61% of 15, keeps the comparison about shares of a pizza.

Complex sentences moderate Understanding the question
The question is one 42-word sentence ending in a nested condition, “without eating a greater percentage of a pizza than the Barnes family ate”, a reading hurdle rather than more math. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)
Turning words into math moderate Understanding the question
The limit of no “greater percentage of a pizza” must become the inequality x/15 ≤ 11/18, comparing fractions of two pizzas cut into different numbers of slices, not slice counts. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Error-prone arithmetic minor Working it out
Solving gives x ≤ 11 × 15 ÷ 18 = 165/18, a division by 18 that leaves 9 1/6, which must then be cut back to whole slices. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Write the Barnes family's share as a fraction of their pizza, then find the largest whole number of fifteenths that does not exceed it, by cross-multiplying or by converting to decimals.

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.
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.

The wrong answers, classified

A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: weak
Finds that 9 slices is the limit but gives the slices left on the second pizza, 15 − 9 = 6, answering a different question.
B Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Subtracts the stem’s numbers, 18 − 11 = 7, the slices left on the Barnes family’s pizza, without comparing shares of the two pizzas.
C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: strong
Reasons that the second pizza has 3 fewer slices, so the Ramirez family should eat 3 fewer, 11 − 3 = 8, comparing by subtraction instead of by fractions.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Fraction of a pizza: $\text{fraction eaten} = \frac{\text{slices eaten}}{\text{slices in the pizza}}$; The top counts the slices eaten. The bottom counts all the equal slices the pizza was cut into. The fraction tells how much of the whole pizza was eaten. Use fractions to compare pizzas cut into different numbers of slices.

Sort out the question
The two pizzas are the same size. The Barnes pizza has 18 slices, and they eat 11. The Ramirez pizza has 15 slices, so its slices are bigger. You need the most whole slices the Ramirez family can eat without eating a bigger share of their pizza.
Find the Barnes family’s share
Their share is the slices they ate out of all the slices in their pizza. Use the fraction formula.
$\text{fraction eaten} = \frac{11}{18}$
The Barnes family ate $\frac{11}{18}$ of their pizza.
Write an inequality
Let x be the number of Ramirez slices. Their share is $\frac{x}{15}$ . It must be no more than the Barnes share. The sign ≤ means less than or equal to.
$\frac{x}{15} \leq \frac{11}{18}$
Solve for x
Multiply both sides by 15 to get x by itself.
$x \leq \frac{11 \times 15}{18} = \frac{165}{18}$
Now divide. 18 goes into 165 nine times, since 18 × 9 = 162. There are 3 left over.
$x \leq 9\frac{3}{18} = 9\frac{1}{6}$
Pick the whole number
x must be a whole number no bigger than $9\frac{1}{6}$ . So the most is 9 slices. 10 slices would be too many.
Watch out: The Ramirez pizza has 3 fewer slices, but that does not mean they eat 3 fewer. Their slices are bigger. Eating 11 − 3 = 8 slices is choice C, which is not the most they can eat.
So the answer is D, 9.

Check your answer

Use decimals. The Barnes share is $11 \div 18 \approx 0.611$ . Nine Ramirez slices are $9 \div 15 = 0.6$ , just under it. Ten slices are $10 \div 15 \approx 0.667$ , too much. So 9 is the most.

The original DOE question

Question 102 as printed on page 204 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 204, question 102. Open page 204 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).