AMC 10 · 2004 · #17
Grade 7 rate-ratioPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Draw the circle with the two starts half a lap apart. Running toward each other, their first-meeting distances fill exactly that gap.
Two people closing a gap together cover the whole gap between them.
6.RP.A.3Draw A DiagramDistance covered together between meetings
On a loop they meet again only after covering one more full lap together, so the between-meetings distances add to the whole track.
On a loop, each new meeting means the pair together has run one more full lap.
On a loop, each new meeting means the pair together has run one more full lap.
▸ Why?
After a full lap everything returns to where it began, so meetings repeat on that period.
▸ Why?
Running toward each other closes the gap at the sum of their speeds, so the pair covers the lap together.
The second stretch is twice the first
Half a lap versus a full lap means the second stretch took twice as long, so at a fixed speed each girl ran twice as far.
At a steady speed, twice the time means twice the distance.
7.RP.A.2Analyze The UnitsAdd the second-stretch distances
Brenda's 100 doubles to 200, Sally ran 150, and one lap is their sum: the track is 350 meters, 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