AMC 10 · 2004 · #15
Grade 7 rate-ratioPick an answer.
Sketching the circle turns the word problem into distances you can add. Once you see how far the two runners cover together before each meeting, constant speed lets you compare the two stretches by simple proportion instead of heavy algebra.
Distance covered together at first meeting
Starting opposite each other, their two distances fill exactly half a lap at the first meeting.
Two people closing a gap together cover the whole gap between them.
6.RP.A.3Draw A DiagramDistance covered together between meetings
Meeting again means they together cover one full lap between the meetings.
On a loop, each new meeting means the pair together has run one more full lap.
7.RP.A.2Draw A DiagramThe second stretch is twice the first
Constant speeds make the second stretch exactly twice the first for each runner.
At a steady speed, twice the time means twice the distance.
At a steady speed, twice the time covers twice the distance.
▸ Why?
With the rate fixed, the distance is the rate times the time, so time and distance move together.
▸ Why?
A fixed ratio between two quantities means doubling one doubles the other, whatever the starting values.
Add the second-stretch distances
Adding the two second-stretch distances gives the lap: 350, choice (C).
The full lap between meetings is just the two second-stretch distances added together.
7.RP.A.3Introduce A VariableOn a circular track, two runners together cover half a lap before their first meeting and a full lap between each meeting after.
- Distance covered together at first meeting
- Distance covered together between meetings
- The second stretch is twice the first
- Add the second-stretch distances