AMC 10 · 2021 · #3

Grade 6 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateunit-conversionfraction-arithmeticdimensional-analysis identify-subproblemsdimensional-analysis ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Route A is 6 miles driven at 30 miles per hour. Route B is 5 miles long: a half-mile school zone driven at 20 miles per hour, and the rest driven at 40 miles per hour. Find how many minutes less Route B takes than Route A.

Pick an answer.

(A)
$2 \frac{3}{4}$
(B)
$3 \frac{3}{4}$
(C)
$4 \frac{1}{2}$
(D)
$5 \frac{1}{2}$
(E)
$6 \frac{3}{4}$
How to solve
Strategy Analyze the Units

Every number here is a distance in miles or a speed in miles per hour, so Tool #8 (Analyze the Units) is the spine: dividing miles by miles per hour leaves hours, and multiplying by 60 leaves minutes. Route B is not one uniform trip, so Tool #7 (Identify Subproblems) splits it into the school-zone stretch and the faster stretch, each with its own speed, and the two times are added back together. Tool #15 (Organize Information in More Ways) gives a second bookkeeping — speeds rewritten as miles per minute — to confirm the total without redoing the hour conversions.

1STEP 1

Time Route A in minutes

Time Route A in minutes.

6/30=1/5 hour=1/5 · 60=12 minutes
2STEP 2

Split Route B at the school zone

Split Route B into two legs.

5-1/2=9/2 miles at 40 mph, 1/2 mile at 20 mph
3STEP 3

Time the fast stretch

Time the fast stretch.

9/2 ÷ 40=9/80 hour=9/80 · 60=27/4 minutes
4STEP 4

Time the school zone and add

The short zone costs real time.

1/2 ÷ 20=1/40 hour=3/2 minutes; 27/4+6/4=33/4 minutes
5STEP 5

Subtract to compare the routes

Subtracting gives three and three quarters minutes.

12-33/4=48/4-33/4=15/4=3 3/4 → (B)
Answer
3 3/4
The size is sensible. Route B is shorter and mostly faster, so it should win, but only by a few minutes — the whole Route A trip is just 12 minutes long, so a saving of 3 3/4 minutes is a believable fraction of it, while an answer above 6 minutes would mean Route B took under half the time, which 40 miles per hour against 30 cannot deliver. The school zone matters: without it, 5 miles at 40 miles per hour would take 5/40 hour =7 1/2 minutes, saving 4 1/2 minutes, so the slow half mile costs back 3/4 of a minute and 4 1/2-3/4=3 3/4 matches.
💡Key takeaway

When one trip has two speeds, time each stretch separately and add — and check that the stretches still add back to the stated total distance.

  • Time Route A in minutes
  • Split Route B at the school zone
  • Time the fast stretch
  • Time the school zone and add
  • Subtract to compare the routes

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