AMC 10 · 2004 · #17
Grade 7 rate-ratioBrenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
Pick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Two runners start at opposite ends of a circular track and run in opposite directions at constant speeds. Given how far each has run by their first and second meetings, find the length of the track.
Givens: Brenda and Sally start at diametrically opposite points on a circular track.; They run in opposite directions, each at a constant speed.; At their first meeting, Brenda has run 100 meters.; At their second meeting, Sally has run 150 meters past the first meeting point.
Unknowns: The length of the track, in meters.
Understand
Restated: Two runners start at opposite ends of a circular track and run in opposite directions at constant speeds. Given how far each has run by their first and second meetings, find the length of the track.
Givens: Brenda and Sally start at diametrically opposite points on a circular track.; They run in opposite directions, each at a constant speed.; At their first meeting, Brenda has run 100 meters.; At their second meeting, Sally has run 150 meters past the first meeting point.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #8 Analyze the Units, #4 Introduce a Variable
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.
Execute — Answer: C
6.RP.A.3 Step 1 Distance covered together at first meeting
- Draw the circle and put the two starting points at opposite ends, half a lap apart.
- Because the runners head toward each other, when they first meet the distance Brenda has run plus the distance Sally has run together fill exactly that gap — one half of the track.
💡 Two people closing a gap together cover the whole gap between them.
7.RP.A.2 Step 2 Distance covered together between meetings
- After the first meeting they keep running and eventually meet again.
- On a loop, meeting once more means that together they have covered one full extra lap.
- So between the two meetings, Brenda's distance plus Sally's distance add up to the whole track length.
💡 On a loop, each new meeting means the pair together has run one more full lap.
7.RP.A.2 Step 3 The second stretch is twice the first
- Together they ran half a lap to reach the first meeting, but a full lap between the two meetings — twice as far.
- Since neither runner changes speed, that second stretch simply took twice as long, so each girl ran exactly twice as far as she did in the first stretch.
💡 At a steady speed, twice the time means twice the distance.
7.RP.A.3 Step 4 Add the second-stretch distances
- Brenda ran 100 meters in the first stretch, so she ran 200 meters between the meetings.
- Sally ran 150 meters between the meetings, which is what the problem states.
- Those two distances together make one full lap, so the track is 200 + 150 = 350 meters.
- The answer is (C).
💡 The full lap between meetings is just the two second-stretch distances added together.
6.RP.A.3 Draw the circle and put the two starting points at opposite ends, half a lap apa 7.RP.A.2 After the first meeting they keep running and eventually meet again. On a loop, 7.RP.A.2 Together they ran half a lap to reach the first meeting, but a full lap between 7.RP.A.3 Brenda ran 100 meters in the first stretch, so she ran 200 meters between the me Review
Reasonableness: Check that the speed ratio stays constant. In the first stretch Brenda runs 100 m and Sally runs 175 - 100 = 75 m, a ratio of 4:3. In the second stretch Brenda runs 200 m and Sally runs 150 m, also 4:3. The ratio matches in both stretches, and 350 is one of the answer choices, so the result holds together.
Alternative: Skip the doubling shortcut and use the constant speed ratio directly. Let the track length be L. At the first meeting Brenda:Sally = 100:(L/2 - 100); between meetings the ratio is (L - 150):150. Setting these equal and simplifying gives L^2/2 = 175L, so L = 350.
CCSS standards used (min grade 7)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Seeing that the two runners together close the half-lap gap by the time they first meet.)7.RP.A.2Recognize and represent proportional relationships between quantities (Using constant speed so distance is proportional to time — twice the time means twice the distance.)7.RP.A.3Use proportional relationships to solve multi-step ratio problems (Chaining the meeting facts to find each runner's second-stretch distance and adding them to get the track length.)
⭐ On a circular track, two runners together cover half a lap before their first meeting and a full lap between each meeting after.
⭐ On a circular track, two runners together cover half a lap before their first meeting and a full lap between each meeting after.
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