AMC 10 · 2013 · #4
Grade 6 rate-ratioPick an answer.
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
The combined rate is totals over totals.
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 two rates.
▸ Why?
An overall rate is one total shared over the other total, which is what an average really is.
▸ Why?
A rate says how much of one thing goes with one unit of the other, so each car's amounts come from its own rate.
Pick one convenient distance
One convenient distance makes both amounts whole.
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
Dividing gives 16, 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