AMC 8 · 2009 · #2
Grade 6 rate-ratioProblem
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.
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
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.
💡 Listing a few rows turns the abstract ratio into something you can literally count up.
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 $4 : 7$.
💡 The number of groups is the same for both columns — that is what "constant ratio" means.
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 $7 \times 7$.
💡 Scaling both parts of a ratio by the same whole number keeps the ratio equivalent.
6.RP.A.3 Write out the ratio as a small table to see the pattern. Each row is one $4 : 7$ 6.RP.A.3 Find the scaling factor. To reach $28$ sports cars from the $4$ in one group, mu 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.3Use 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.5Interpret 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.
⭐ 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.