AMC 10 · 2010 · #10

Grade 7 rate-ratio

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance

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

📏 Medium solution 💡 2 insights

Problem

Shelby rides her scooter at 30 miles per hour when it is not raining and at 20 miles per hour when it is raining. Today she rode in the sun first and in the rain afterwards, covering 16 miles in a total of 40 minutes. How many of those minutes did she ride in the rain?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Two unknown time chunks share one total time and one total distance, so Tool #4 (Introduce a Variable) is the natural fit: name the rain time, write the sun time as "what's left," and turn the distance fact into one equation. Tool #8 (Analyze the Units) keeps the arithmetic honest — speeds are per hour but the answer is wanted in minutes, so units must be converted at the start and again at the end.

1STEP 1

Match the units

Speeds are per hour, so the total time must be in hours too: dividing 40 minutes by 60 gives $\frac{2}{3}$ hour.

40 min = 40/60 hr = 2/3 hr

2STEP 2

Name the unknown and set up

Let r be the rain time in hours; the sun time is then $\frac{2}{3}$ - r. Each distance is speed times time, and the two add to 16 miles.

30(2/3 - r) + 20r = 16

3STEP 3

Solve for the rain time

Distribute the 30, combine the r terms, then isolate: 20 - 10r = 16, so 10r = 4 and r = $\frac{2}{5}$ hour.

20 - 30r + 20r = 16 → 20 - 10r = 16 → 10r = 4 → r = 2/5 hr

4STEP 4

Convert back to minutes

The question wants minutes, so multiply by 60 again: $\frac{2}{5}$ × 60 = 24 minutes, choice (C).

2/5 hr × 60 = 24 min → (C)

Answer

Check both totals. Rain time = 2/5 hr, so sun time = 2/3 - 2/5 = 10/15 - 6/15 = 4/15 hr. Distances: sun = 30 × 4/15 = 8 miles, rain = 20 × 2/5 = 8 miles, total 8 + 8 = 16 miles. Times in minutes: sun = 4/15 × 60 = 16 min, rain = 24 min, total 40 min. Both the distance and the time come out exactly right, and 24 minutes is choice (C).

💡Key takeaway

Name the rain time as one variable, make the sun time the leftover, and add up the distances — the whole race problem shrinks to a single Grade 7 equation.

Match the units
Name the unknown and set up
Solve for the rain time
Convert back to minutes

Match the units

Name the unknown and set up

Solve for the rain time

Convert back to minutes

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