AMC 10 · 2008 · #7
Grade 6 rate-ratioPick an answer.
The numbers come in different units: miles, miles per hour, gallons, and gallons per minute. Tool #8 (Analyze the Units) says line the units up first — turn the trip into a number of minutes, then everything can be measured in gallons per minute. Because the question asks for the slowest safe rate, Tool #14 (Extreme Principle) points to the boundary case where the water builds up to exactly the 30-gallon limit and not a drop more. Tool #4 (Introduce a Variable) names the bailing rate so the balance between water coming in and going out becomes one clean inequality.
Find how long the trip takes
The trip takes 15 minutes.
Going 1 mile at 4 miles each hour uses a quarter of an hour, and a quarter of 60 minutes is 15.
6.RP.A.3Analyze The UnitsSet the largest net gain that stays safe
The limit over that time gives the fastest safe net rate.
If 30 gallons is the whole budget for 15 minutes, then 2 gallons a minute is all the boat can afford to gain.
With thirty gallons the whole budget for fifteen minutes, two gallons a minute is all the boat can afford to gain.
▸ Why?
Water pours in and is bailed out at once, so the level changes at the difference of the two rates.
▸ Why?
At a steady rate the total is the rate multiplied by the time, so the budget divides cleanly by the minutes.
Balance water in against water out
Balancing in against out gives 8, choice (D).
Bailing has to erase all but 2 of the 10 gallons pouring in each minute, so it must clear at least 8.
6.EE.B.8Introduce A VariableTurn everything into gallons-per-minute over the 15-minute trip: the boat can only gain 2 of the 10 gallons leaking in each minute, so LeRoy must bail at least 8.
- Find how long the trip takes
- Set the largest net gain that stays safe
- Balance water in against water out