Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #2
Grade 6 rate-ratioPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "for every 4 sports cars, 7 sedans" sets up a repeating pattern: (4, 7), (8, 14), (12, 21), … — 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.
Set up the ratio table
List the ratio as a table — each row is one 4 : 7 group, and each new group adds 4 sports cars and 7 sedans.
Listing a few rows turns the abstract ratio into something you can literally count up.
6.RP.A.3Look For A PatternFind the scaling factor
Find the scaling factor: 28 ÷ 4 = 7, so 28 sports cars is 7 groups of 4 : 7.
The number of groups is the same for both columns — that is what "constant ratio" means.
6.RP.A.3Look For A PatternScale the sedan column
Scale the sedan side the same way: 7 groups × 7 sedans each = 49 → (D).
Scaling both parts of a ratio by the same whole number keeps the ratio equivalent.
The predicted number of sedans is 7 multiplied by the factor that scales the 4 sports cars up to the predicted 28, because scaling both parts of the 4 : 7 ratio by that same factor keeps the ratio equal.
▸ Why?
The factor that turns the original 4 sports cars into the predicted 28 is 7, found by 28 divided by 4, because dividing undoes the multiplying and reveals the missing factor.
▸ Why?
Writing the rate as the fraction 4/7 (sports cars over sedans) and multiplying both its top and bottom by that same 7 gives the equal fraction 28/49, so the predicted sedan count is 7 times 7.
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.
- Set up the ratio table
- Find the scaling factor
- Scale the sedan column
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