AMC 8 · 2009 · #2

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionmulti-digit-arithmetic ratio-proportionpattern-recognition ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

112

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: At a dealership, for every $4$ sports cars sold, $7$ sedans are sold. Next month they expect to sell $28$ sports cars. How many sedans should they expect to sell at the same rate?

Givens: Sales ratio is constant: sports cars to sedans $= 4 : 7$; Predicted sports car sales next month $= 28$; Answer choices: (A) $7$, (B) $32$, (C) $35$, (D) $49$, (E) $112$

Unknowns: The expected number of sedans sold next month

Understand

Restated: At a dealership, for every $4$ sports cars sold, $7$ sedans are sold. Next month they expect to sell $28$ sports cars. How many sedans should they expect to sell at the same rate?

Givens: Sales ratio is constant: sports cars to sedans $= 4 : 7$; Predicted sports car sales next month $= 28$; Answer choices: (A) $7$, (B) $32$, (C) $35$, (D) $49$, (E) $112$

Plan

Primary tool: #5 Look for a Pattern

Secondary: #6 Guess and Check

The phrase "for every $4$ sports cars, $7$ sedans" sets up a repeating pattern: $(4, 7), (8, 14), (12, 21), \ldots$ — each step adds one more $4 : 7$ group. Tool #5 (Look for a Pattern) makes that ladder visible and shows that the sports-car column is multiplied by some whole number to reach $28$, so the sedan column gets multiplied by the same number. Tool #6 (Guess and Check) is the natural backup: test each answer choice against the $4 : 7$ ratio and keep the one that matches.

Execute — Answer: D

#5 Look for a Pattern 6.RP.A.3 Step 1

Write out the ratio as a small table to see the pattern.
Each row is one $4 : 7$ "group," and adding another group adds $4$ to the sports-car column and $7$ to the sedan column.

$$\begin{array}{c|c} \text{sports} & \text{sedans} \\ \hline 4 & 7 \\ 8 & 14 \\ 12 & 21 \\ \vdots & \vdots \end{array}$$

💡 Listing a few rows turns the abstract ratio into something you can literally count up.

#5 Look for a Pattern 6.RP.A.3 Step 2

Find the scaling factor.
To reach $28$ sports cars from the $4$ in one group, multiply by $28 \div 4 = 7$.
So $28$ sports cars represents $7$ groups of $4 : 7$.

$$28 \div 4 = 7 \text{ groups}$$

💡 The number of groups is the same for both columns — that is what "constant ratio" means.

#5 Look for a Pattern 5.NF.B.5 Step 3

Apply the same scaling factor to the sedan column.
Each group contributes $7$ sedans, and there are $7$ groups, so the total is $7 \times 7$.

$$\text{sedans} = 7 \times 7 = 49 \;\Rightarrow\; \textbf{(D)}$$

💡 Scaling both parts of a ratio by the same whole number keeps the ratio equivalent.

[1] #5 6.RP.A.3 Write out the ratio as a small table to see the pattern. Each row is one $4 : 7$

[2] #5 6.RP.A.3 Find the scaling factor. To reach $28$ sports cars from the $4$ in one group, mu

[3] #5 5.NF.B.5 Apply the same scaling factor to the sedan column. Each group contributes $7$ se

Review

Reasonableness: Check the ratio: $28 : 49$. Divide both by their common factor $7$ to get $4 : 7$ — exactly the original ratio. Also, sedans should outnumber sports cars (since $7 > 4$), and $49 > 28$ confirms the direction. Choices smaller than $28$ (A: $7$) or only slightly above (B: $32$, C: $35$) would not preserve the $4 : 7$ shape, and (E) $112$ is the sports-car count times $4$, which is too aggressive.

Alternative: Tool #6 (Guess and Check) on the choices: for each candidate $s$, simplify $28 : s$ and see if it equals $4 : 7$. (A) $28 : 7 = 4 : 1$, no. (B) $28 : 32 = 7 : 8$, no. (C) $28 : 35 = 4 : 5$, no. (D) $28 : 49 = 4 : 7$, yes. (E) $28 : 112 = 1 : 4$, no. Only (D) matches.

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Recognizing the $4 : 7$ ratio as a constant relationship and finding the scaling factor $28 \div 4 = 7$ that links the two months.)
5.NF.B.5 Interpret multiplication as scaling (resizing) (Scaling the sedan side of the ratio by the same factor of $7$ to compute $7 \times 7 = 49$ sedans.)

⭐ When two quantities grow at a constant ratio, find how many times one side is scaled up — then scale the other side by the same number.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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