AMC 10 · 2025 · #1

Grade 6 rate-ratio

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

ratedimensional-analysislinear-equations-one-var convert-to-algebra ↑ Prerequisites: rate

📏 Medium solution 💡 1 insight

Problem

Andy leaves a starting point at 1{:}30, biking due north at a steady 8 mph. Betsy leaves the same point at 2{:}30, biking due east at a steady 12 mph. Find the clock time at which Andy and Betsy are the same distance from the starting point.

Pick an answer.

(A)

$3{:}30$

(B)

$3{:}45$

(C)

$4{:}00$

(D)

$4{:}15$

(E)

$4{:}30$

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

Perpendicular directions tempt you to reach for the Pythagorean theorem, but that would give the distance BETWEEN the riders. The question asks how far each is FROM the start, and since each travels a single straight line, the units tell the whole story: miles = (miles per hour) × (hours). That is Tool #8 — let the units pick the model. Once distance is just speed × time, Tool #16 (change focus) makes it fast: instead of tracking two growing distances, watch the single gap between them and how quickly it closes. Tool #4 (introduce a variable) gives the same answer through a set-the-distances-equal equation, kept as a backup.

1STEP 1

Pick the model from the units

Each rider goes straight out from home, so distance from the start is speed × time: miles = (miles/hour) × hours.

distance from start = speed × time, mi = mi/hr × hr

2STEP 2

Andy's one-hour head start

Andy rides the full hour from 1{:}30 to 2{:}30 before Betsy starts, covering 8 × 1 = 8 miles while she is still at 0.

8 mi/hr × 1 hr = 8 mi

3STEP 3

Track the gap, not both distances

Watch the gap, not two growing distances: Betsy gains 12 miles an hour and Andy 8, so the gap closes by 12 - 8 = 4 miles per hour.

12 mi/hr - 8 mi/hr = 4 mi/hr

4STEP 4

When does the gap reach zero?

The 8-mile gap closing at 4 miles per hour vanishes after 8 ÷ 4 = 2 hours of Betsy riding, when the two distances match.

(8 mi)/(4 mi/hr) = 2 hr after 2{:}30

5STEP 5

Convert to clock time

Betsy starts at 2{:}30, so 2 hours later the clock reads 4{:}30, when both riders are 24 miles from home. That is choice (E).

2{:}30 + 2 hr = 4{:}30 → (E)

Answer

4{:}30

Check the tie directly: at 4{:}30 Andy has ridden 3 hours and Betsy 2 hours, giving 8 × 3 = 24 miles and 12 × 2 = 24 miles — an exact match, so (E) is consistent. Earlier choices fail: at 4{:}00 Andy is 8 × 2.5 = 20 miles out but Betsy only 12 × 1.5 = 18 miles, so Betsy is still behind. The gap first hits zero at 4{:}30.

💡Key takeaway

When two people ride straight away from the same spot, don't juggle both distances — watch the gap between them and how fast it closes. Andy's 8-mile head start vanishing at 4 miles per hour means they tie 2 hours after Betsy starts, at 4{:}30.

Pick the model from the units
Andy's one-hour head start
Track the gap, not both distances
When does the gap reach zero?
Convert to clock time

Pick the model from the units

Andy's one-hour head start

Track the gap, not both distances

When does the gap reach zero?

Convert to clock time

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