AMC 10 · 2013 · #8
Grade 6 rate-ratioPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The whole problem lives in the units "miles per gallon." Tool #8 (Analyze the Units) fixes the meaning: a combined rate is total miles divided by total gallons, so the job is to add up the miles and add up the gallons. To keep the gallons whole and avoid fractions, Tool #9 (Solve an Easier Related Problem) picks one convenient distance that both 40 and 10 divide evenly, works the numbers there, and trusts that equal distances give the same rate no matter which distance is chosen.
Set up the combined rate
Miles per gallon means miles divided by gallons, so the combined rate is total miles over total gallons — not the average of 40 and 10.
A rate is one thing per another thing, so combining rates means totalling each thing separately, not averaging the rates.
Combining rates means totalling each quantity separately, not averaging the rates.
▸ Why?
An overall rate is a total shared over a count, so both totals must be rebuilt first.
▸ Why?
Each car's fuel is its distance divided by its own rate, so the two contribute unequally.
Pick one convenient distance
Let each car drive 40 miles: Ray's burns 40 ÷ 40 = 1 gallon, Tom's burns 40 ÷ 10 = 4 gallons, four times as much.
Turning gas mileage into gallons for a picked distance replaces an abstract rate with countable amounts you can add.
6.RP.A.3Solve An Easier Related ProblemAdd the totals and divide
Together they cover 40 + 40 = 80 miles on 1 + 4 = 5 gallons, so the rate is 80 ÷ 5 = 16 miles per gallon, choice (B).
Dividing the summed miles by the summed gallons gives the honest overall rate because it weighs each car by the gas it actually used.
6.RP.A.3Analyze The UnitsTo combine two rates, add up the total miles and the total gallons and then divide — never just average the two rates, because the gas-guzzler counts for more.
- Set up the combined rate
- Pick one convenient distance
- Add the totals and divide