AMC 10 · 2006 · #5

Grade 6 rate-ratio

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

ratelinear-equations-one-varunit-conversion convert-to-algebra ↑ Prerequisites: rate

📏 Short solution 💡 1 insight

Problem

Two people walk the same direction at 3 and 5 miles per hour, the faster one starting 1 mile behind. Find how many minutes until the faster reaches the other's starting spot.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

120

How to solve

Strategy Introduce a Variable

Two things move at once, so tracking each walker separately invites confusion about who is where. Name the elapsed time with a single letter, write each walker's position as an expression in that letter, and then subtract the two expressions to get the gap. The gap is the quantity the question really cares about: catching up means the gap is zero. Turning the words into one expression for the gap also makes it visible why the two speeds may be subtracted, and forces the units to be checked at the end, because the speeds are per hour while the answer is wanted in minutes.

1STEP 1

Put both walkers on one line

Both walk one way, and the faster starts behind.

gap now = 1 mile, 5 ＞ 3 (same direction)

2STEP 2

Name the time, write the positions

Naming the time writes both positions.

B(t) = 5t, J(t) = 1 + 3t

3STEP 3

Track the gap, not the walkers

Tracking the gap is simpler than tracking either walker.

g(t) = J(t) - B(t) = (1 + 3t) - 5t = 1 - 2t

4STEP 4

Check the gap really closes

The gap shrinks at the speed difference, so it really closes.

g(0) = 1 ＞ 0, slope = -(5-3) = -2 ＜ 0

5STEP 5

Set the gap to zero and solve

Setting the gap to zero gives half an hour.

1 - 2t = 0 → 2t = 1 → t = 1/2

6STEP 6

Turn hours into minutes

Converting gives 30 minutes, choice (D).

t = 1/2 hour × (60 minutes)/(1 hour) = 30 minutes

Answer

Put the time back into the original picture. In half an hour Bob walks 5 x 0.5 = 2.5 miles and John walks 3 x 0.5 = 1.5 miles. Measured from Bob's starting spot, Bob is at 2.5 and John is at 1 + 1.5 = 2.5, the same point, so 30 minutes is right and (A) matches. A second check bounds the answer without solving: if John stood still, Bob would need 1 mile at 5 miles per hour, which is 12 minutes, and John walking away can only make it take longer, so any answer under 12 minutes would be impossible and 30 sits comfortably above it. The wrong choices correspond to gaps closing at the wrong rate: 120 minutes would mean the gap shrinks only 0.5 miles per hour, and 60 minutes would mean 1 mile per hour, but the true closing rate is 5 - 3 = 2.

💡Key takeaway

When two people move the same way on the same line, stop watching the people and watch the gap: here it shrinks by 5 - 3 = 2 miles every hour, so a 1-mile gap is gone in half an hour.

Put both walkers on one line
Name the time, write the positions
Track the gap, not the walkers
Check the gap really closes
Set the gap to zero and solve
Turn hours into minutes