AMC 8 · 2006 · #13

Grade 6 rate-ratio

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Constant-Rate Closing Speed

rateunit-conversionlinear-equations-one-var identify-subproblems ↑ Prerequisites: ratefraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

Cassie leaves Escanaba at 8 : 30 AM biking toward Marquette at 12 mph. Brian leaves Marquette at 9 : 00 AM biking toward Escanaba at 16 mph. The route is 62 miles long. At what time do they meet?

Pick an answer.

(A)

10: 00

(B)

10: 15

(C)

10: 30

(D)

11: 00

(E)

11: 30

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

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.

1STEP 1

Draw the 62-mile route with Cassie and Brian as arrows facing each other; she starts 30 minutes earlier.

Escanaba → ● Brian 16 mph Marquette, gap = 62 mi

2STEP 2

Change the 30-minute head start into $\frac{1}{2}$ hour, then multiply by her speed: Cassie covers 6 miles before Brian starts.

12 mi/hr ×

\frac{1}{2}

hr = 6 mi

3STEP 3

Subtract the 6-mile lead so 62 - 6 = 56 miles remain, and riding toward each other they close at 12 + 16 = 28 mph.

gap at 9 : 00 = 62 - 6 = 56 mi, closing speed = 12 + 16 = 28 mph

4STEP 4

Divide 56 by 28 to get 2 hours after Brian's 9 : 00 AM start, so they meet at 11 : 00 AM.

(56 mi)/(28 mph) = 2 hr, 9 : 00 + 2 : 00 = 11 : 00 AM → (D)

Answer

11: 00

Check both riders' positions at 11 : 00 AM. Cassie has been riding 2.5 hours, so she has covered 12 × 2.5 = 30 miles from Escanaba. Brian has been riding 2 hours, so he has covered 16 × 2 = 32 miles from Marquette. Together: 30 + 32 = 62 miles — exactly the route length, so they are at the same point. The answer is consistent. Also, the other choices fail this check: at 10 : 30 AM (choice C), Cassie has gone 24 mi and Brian 24 mi, totaling only 48 ＜ 62, so they have not met yet.

💡Key takeaway

When 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.

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