AMC 10 · 2010 · #7
Grade 7 rate-ratioPick an answer.
Two unknown time chunks share one total time and one total distance, so Tool #4 (Introduce a Variable) is the natural fit: name the rain time, write the sun time as "what's left," and turn the distance fact into one equation. Tool #8 (Analyze the Units) keeps the arithmetic honest — speeds are per hour but the answer is wanted in minutes, so units must be converted at the start and again at the end.
Match the units
The total time becomes a fraction of an hour.
Distance = speed × time only works when the time unit matches the speed's unit, so switch minutes into hours first.
Distance equals speed times time only when the time is measured in the speed's own unit.
▸ Why?
A speed says how far you go in one unit of time, so the time has to be counted in those same units.
▸ Why?
An hour is a fixed bundle of minutes, so trading one for the other is exact and loses nothing.
Name the unknown and set up
One unknown names both stretches.
One variable is enough because the sun time is forced to be the leftover once the rain time is chosen.
7.EE.B.4Introduce A VariableSolve for the rain time
The distances add to a single equation.
Peeling operations off one at a time strips the equation down to the value of r.
7.EE.B.4Introduce A VariableConvert back to minutes
Converting back gives 24 minutes, choice (C).
Return to the unit the question actually asked for — the answer is 24 minutes, choice (C).
6.RP.A.3Analyze The UnitsName the rain time as one variable, make the sun time the leftover, and add up the distances — the whole race problem shrinks to a single Grade 7 equation.
- Match the units
- Name the unknown and set up
- Solve for the rain time
- Convert back to minutes