AMC 8 · 2007 · #2

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

ratio-proportiongraph-readingfraction-arithmetic identify-subproblems ↑ Prerequisites: graph-readingfraction-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

Problem

$650$ students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti?

$\mathrm{(A)} \frac{2}{5} \qquad \mathrm{(B)} \frac{1}{2} \qquad \mathrm{(C)} \frac{5}{4} \qquad \mathrm{(D)} \frac{5}{3} \qquad \mathrm{(E)} \frac{5}{2}$

Pick an answer.

(A)

$$\frac{2}{5}$$

(B)

$$\frac{1}{2}$$

(C)

$$\frac{5}{4}$$

(D)

$$\frac{5}{3}$$

(E)

$$\frac{5}{2}$$

View mode:

Toolkit + CCSS Solution

Understand

Restated: A bar graph shows how $650$ students voted on four pasta types: lasagna, manicotti, ravioli, and spaghetti. Find the ratio of students who preferred spaghetti to students who preferred manicotti.

Givens: $650$ students were surveyed in total; The four choices are lasagna, manicotti, ravioli, and spaghetti; The bar graph's vertical axis is marked at $50, 100, 150, 200, 250$; Bar heights: Lasagna $= 150$, Manicotti $= 100$, Ravioli $= 150$, Spaghetti $= 250$; Answer choices: (A) $\tfrac{2}{5}$, (B) $\tfrac{1}{2}$, (C) $\tfrac{5}{4}$, (D) $\tfrac{5}{3}$, (E) $\tfrac{5}{2}$

Unknowns: The ratio $\dfrac{\text{spaghetti voters}}{\text{manicotti voters}}$ in lowest terms

Understand

Plan

Primary tool: #15 Visualize / Read the Picture

Secondary: #2 List Out Cases

The whole problem lives inside the picture, so Tool #15 (Visualize) is primary — read each bar's top against the labeled gridline. Tool #2 (List Out) helps you write down the four bar heights cleanly so you do not confuse spaghetti with manicotti. Once both numbers are on the page, the ratio is a simple fraction to reduce.

Execute — Answer: E

#15 Visualize / Read the Picture 3.MD.B.3 Step 1

Read each bar against the gridlines and list the values.
The axis labels $50, 100, 150, 200, 250$ make it easy to line up each bar's top.

Lasagna $= 150$, Manicotti $= 100$, Ravioli $= 150$, Spaghetti $= 250$

💡 Reading a scaled bar graph is a Grade 3 measurement-and-data skill: match the top of the bar to the labeled gridline.

#2 List Out Cases 3.NBT.A.2 Step 2

Quick consistency check: the four bar heights should add to the total $650$ students stated in the problem.
This is a free sanity check before computing.

$150 + 100 + 150 + 250 = 650$ \checkmark

💡 Listing all four values lets us verify the totals match, which confirms we read the bars correctly.

#2 List Out Cases 6.RP.A.1 Step 3

Write the ratio in the order the problem asks: spaghetti first, manicotti second.
Express it as a fraction.

$$\dfrac{\text{spaghetti}}{\text{manicotti}} = \dfrac{250}{100}$$

💡 A ratio $a$ to $b$ is the fraction $\tfrac{a}{b}$. Keep the order — swapping would give the wrong answer.

#2 List Out Cases 4.NF.A.1 Step 4

Reduce the fraction by the GCD.
Both $250$ and $100$ are divisible by $50$.

$$\dfrac{250}{100} = \dfrac{250 \div 50}{100 \div 50} = \dfrac{5}{2} \;\Rightarrow\; \textbf{(E)}$$

💡 Dividing top and bottom by the same number gives an equivalent fraction in lowest terms.

[1] #15 3.MD.B.3 Read each bar against the gridlines and list the values. The axis labels $50, 10

[2] #2 3.NBT.A.2 Quick consistency check: the four bar heights should add to the total $650$ stud

[3] #2 6.RP.A.1 Write the ratio in the order the problem asks: spaghetti first, manicotti second

[4] #2 4.NF.A.1 Reduce the fraction by the GCD. Both $250$ and $100$ are divisible by $50$.

Review

Reasonableness: Spaghetti's bar ($250$) is taller than manicotti's bar ($100$), so the ratio must be greater than $1$. That immediately rules out (A) $\tfrac{2}{5}$ and (B) $\tfrac{1}{2}$. Also, the spaghetti bar is $2.5$ times the manicotti bar by eye — exactly $\tfrac{5}{2}$. The answer (E) matches.

Alternative: Tool #12 (Use Modular / Divisibility): notice $250 = 5 \cdot 50$ and $100 = 2 \cdot 50$, so the ratio is $5:2$ once the common factor $50$ cancels. Same answer, fewer arithmetic steps.

CCSS standards used (min grade 6)

3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set; solve problems using information presented in scaled bar graphs (Reading each bar's height against the labeled gridlines to extract the spaghetti and manicotti counts.)
3.NBT.A.2 Fluently add and subtract within $1000$ using strategies and algorithms based on place value (Verifying $150 + 100 + 150 + 250 = 650$ as a consistency check on the bar readings.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Writing 'spaghetti to manicotti' as the fraction $\tfrac{250}{100}$ in the correct order.)
4.NF.A.1 Explain why a fraction $a/b$ is equivalent to a fraction $(n \times a)/(n \times b)$ (Reducing $\tfrac{250}{100}$ to $\tfrac{5}{2}$ by dividing numerator and denominator by $50$.)

⭐ Read the two bars you need, write the ratio in the order asked, then reduce the fraction. The picture does most of the work.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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