AMC 10 · 2002 · #17
Grade 6 geometry-2dPick an answer.
The question hides a rate problem. Tool #8 (Analyze the Units) supplies the one relationship that settles everything: a mowing speed is area cut per unit time, so time to finish = lawn area ÷ mowing speed. Bigger lawn means more time; faster mower means less time. Because only ratios are given, Tool #4 (Introduce a Variable) lets us pin down convenient stand-in numbers that respect every ratio — pick Andy's area as a multiple of both 2 and 3, and pick Carlos' speed as the base 1. Tool #13 (Convert to Algebra) then turns each 'finishes when?' into a single division, and comparing the three quotients names the winner. Choosing numbers is legitimate here precisely because the answer depends only on ratios, not on the raw sizes.
Turn the area ratios into numbers
Pick Andy's area as 6 so the ratios give whole numbers: areas 6, 3, 2.
Choosing the biggest lawn as a multiple of both ratio numbers keeps every area a whole number.
6.RP.A.3Introduce A VariableTurn the speed ratios into numbers
Taking the slowest mower as 1 gives speeds 3, 2, 1.
Anchoring the slowest mower at 1 makes the faster mowers simple whole-number multiples.
6.RP.A.3Introduce A VariableCompute each finishing time
Time is area divided by speed, giving 2, 1.5, 2.
Dividing how much work by how fast you work tells you how long the job takes.
Each mower's time is how much lawn there is spread over how fast the machine cuts.
▸ Why?
At a steady rate the amount done is the rate times the time, so any two of the three fix the third.
▸ Why?
Once all three times are plain numbers, comparing them two at a time chains into one full ranking.
Compare the times
The other two tie at 2, so the smallest time belongs to Beth, choice (B).
Least time wins, and Beth's mid-size lawn with a fast-enough mower beats both bigger and slower matchups.
6.RP.A.3Convert To AlgebraWhen something happens at a rate, time equals the amount of work divided by the speed — so a smaller job with a fast-enough tool can beat a bigger job every time.
- Turn the area ratios into numbers
- Turn the speed ratios into numbers
- Compute each finishing time
- Compare the times