AMC 10 · 2007 · #2
Grade 6 rate-ratioPick an answer.
Miles per gallon is a rate — miles divided by gallons — so Tool #8 (Analyze the Units) tells you that overall mileage must be total miles over total gallons, not the average of 30 and 20. Tool #7 (Identify Subproblems) splits the work into finding the gas each leg burns before combining. Tool #3 (Eliminate Possibilities) flags the trap answer: naively averaging the two mileages gives (30+20)/2=25, which is choice (C) and is wrong.
Gas used driving home
The outbound leg burns 4.
Dividing the miles by the mileage cancels the miles and leaves the gallons.
6.RP.A.3Analyze The UnitsGas used driving back
The return leg burns 6, more than the first.
Worse mileage means the same distance eats more gallons.
6.RP.A.3Analyze The UnitsTotal miles and total gallons
Distances and fuel both add directly.
The whole trip is just the two legs stacked end to end, in both miles and gallons.
4.OA.A.3Identify SubproblemsAverage mileage for the round trip
Dividing gives 24, below the naive midpoint, choice (B).
Overall mileage is all the miles shared out over all the gallons, and the thirstier leg counts for more gallons.
Overall mileage is all the miles shared out over all the gallons, so the thirstier leg counts for more.
▸ Why?
An average is a total shared out over a count, not the plain average of the two separate rates.
▸ Why?
A rate says how many miles fit in one gallon, so dividing miles by that rate returns the gallons used.
Average mileage is all your miles divided by all your gallons, not the average of the two speeds — so the gas-guzzling leg pulls it down.
- Gas used driving home
- Gas used driving back
- Total miles and total gallons
- Average mileage for the round trip