Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #8
Grade 6 rate-ratioPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a rate problem, so Tool #8 (Analyze the Units) keeps the bookkeeping honest: miles divided by mph gives hours, which we then convert to minutes. The trick that simplifies the whole thing is Tool #15 (Reorganize Information): instead of tracking two moving people, switch to Emerson's frame of reference — pretend Emerson is standing still and Emily approaches him at the relative speed 12 - 8 = 4 mph. Now there is one moving object covering a fixed total distance of 1 mile (close the 1/2-mile gap, then open another 1/2-mile gap).
Find the relative speed
Both move forward, but only the gap matters — freeze Emerson and let Emily close in at the relative speed 12 - 8 = 4 mph.
Switching to a single-mover frame is the Tool #15 move: same data, simpler picture.
Because Emily and Emerson travel the same direction, the gap between them closes at 12 - 8 = 4 miles per hour, so we can picture Emerson standing still and Emily rolling toward him at 4 mph.
▸ Why?
In one hour Emily rides 12 miles and Emerson skates 8 miles along the same road, because holding one steady speed lays down the same distance every hour.
▸ Why?
Heading the same way, Emily's forward miles eat into the gap while Emerson's forward miles feed it, so the two combine by subtraction and 12 - 8 = 4 miles of gap vanish each hour.
▸ Why?
Emily rolling forward removes road from the gap while Emerson rolling forward adds road to it, opposite actions that cancel part for part.
▸ Why?
Emily's 12 miles for the hour split with no overlap into the 8 that merely keep pace with Emerson and the 4 that truly close the gap, so 12 = 8 + 4 pins the closing at 4 miles.
Find the relative distance
Emily starts mile behind, passes him, and ends mile ahead — in his frame that is + = 1 mile of relative travel.
Adding the "closing" leg and the "opening" leg captures the whole window in one number.
4.MD.A.2Organize Information In More WaysDivide distance by speed
Units are consistent, so time = distance / speed = = hour.
Dividing miles by miles-per-hour cancels "miles" and leaves "hours" — Tool #8 unit check at work.
6.RP.A.3Analyze The UnitsConvert hours to minutes
The choices are in minutes, so convert hour to 15 minutes → (D).
Same time, just expressed in the unit the question wants — a Grade 5 unit-conversion step.
5.MD.A.1Analyze The UnitsSame-direction chase problems become easy when you pretend the slower mover is standing still — then it's just one distance over one speed, the Grade 6 rate idea you already know.
- Find the relative speed
- Find the relative distance
- Divide distance by speed
- Convert hours to minutes
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