Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #13
Grade 6 rate-ratioPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a textbook rate/distance/time problem, so Tool #8 (Analyze Units) is the natural primary. Speeds are in mph and times are in minutes — convert minutes to hours so that speed × time gives miles cleanly. Tool #1 (Draw a Diagram) turns the word problem into a 62-mile segment with Cassie and Brian as arrows pointing at each other; once drawn, the head-start distance and the closing speed are easy to read off. We do not reach for Tool #13 (Algebra) because the picture plus unit-tracking gives the answer in three short steps.
Draw the route
Draw the 62-mile route with Cassie and Brian as arrows facing each other; she starts 30 minutes earlier.
Drawing the two riders with arrows toward each other makes the "closing the gap" idea visible — a Grade 4 multi-step word-problem move.
4.OA.A.3Draw A DiagramTurn the head start into miles
Change the 30-minute head start into hour, then multiply by her speed: Cassie covers 6 miles before Brian starts.
Grade 5 unit conversion: minutes to hours, then the "hr" units cancel so the answer comes out in miles, exactly what the problem needs.
5.MD.A.1Analyze The UnitsFind the gap and closing speed
Subtract the 6-mile lead so 62 - 6 = 56 miles remain, and riding toward each other they close at 12 + 16 = 28 mph.
Two riders heading toward each other close the gap at the sum of their speeds — like two arrows squeezing inward on the diagram.
When Brian sets out at 9:00, the two riders are 56 miles apart, and from then on the gap between them shrinks by their two speeds added together, 28 miles every hour.
▸ Why?
At 9:00 Cassie has ridden 6 of the route's 62 miles and Brian has not yet moved, so the route is just those 6 covered miles plus the open gap; the gap is therefore 62 - 6 = 56 miles.
▸ Why?
The route stays 62 miles and always equals Cassie's covered miles plus the open gap plus Brian's covered miles; each hour Cassie's part grows by 12 and Brian's by 16, so with the total pinned the gap must lose 12 + 16 = 28 miles each hour.
▸ Why?
At every moment the road has no gaps or overlaps: the stretch behind Cassie, the open gap, and the stretch behind Brian together tile the whole 62 miles.
▸ Why?
A steady 12 miles per hour adds one hour's worth — 12 miles — to Cassie's covered stretch each hour, and likewise 16 miles for Brian.
Divide to get the meeting time
Divide 56 by 28 to get 2 hours after Brian's 9 : 00 AM start, so they meet at 11 : 00 AM.
Grade 6 rate reasoning: time = distance ÷ rate. The miles cancel and hours pop out, which is exactly what the clock-time question wants.
6.RP.A.3Analyze The UnitsWhen two travelers move toward each other, peel off any head-start distance first, then divide the remaining gap by the sum of their speeds — the answer falls out in one step.
- Draw the route
- Turn the head start into miles
- Find the gap and closing speed
- Divide to get the meeting time
A parent dashboard for the family lives at sensimlab.com.