AMC 10 · 2010 · #13

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

Angelina drove at 80 km/h, then stopped 20 minutes for gas, and after the stop drove at 100 km/h. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Let t be the number of hours she drove before the stop. Which equation could be used to find t?

Pick an answer.

(A)

$80t+100\left(\frac{8}{3}-t\right)=250$

(B)

80t=250

(C)

100t=250

(D)

90t=250

(E)

$80\left(\frac{8}{3}-t\right)+100t=250$

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

How to solve

Strategy Convert to Algebra

The question does not ask for a number — it asks which equation models the trip. That is exactly Tool #13 (Convert to Algebra): translate each English phrase into a symbol and glue them into one equation. Tool #8 (Analyze the Units) is essential first, because the stop is given in minutes while everything else is in hours — the 20 minutes must become 1/3 hour before any time can be subtracted, and each distance is a rate (km/h) times a time (h). Tool #4 (Introduce a Variable) names the two driving times: t before the stop and 8/3-t after it, since the two driving legs must share the 8/3 hours of actual driving. Guessing (Tool #6) is unhelpful here because no leg time is a whole number and the answer is an equation, not a value.

1STEP 1

Turn the stop into hours

Only the stop is measured in minutes: 20 minutes is 1/3 hour, so taking it out of the 3-hour total leaves 8/3 hours of driving.

20 min = 20/60 h = 1/3 h, 3 - 1/3 = 8/3 h driving

2STEP 2

Name the two driving times

Call the hours before the stop t; the two legs share the 8/3 driving hours, so after the stop she drives 8/3 - t hours.

before = t, after = 8/3 - t

3STEP 3

Write each leg's distance

Distance is speed times time, so the first leg covers 80t km and the second covers 100(8/3 - t) km.

first leg = 80t, second leg = 100(8/3 - t)

4STEP 4

Add the legs to 250

The two legs together cover the whole 250 km, so adding the two expressions and setting the sum to 250 gives choice (A).

80t + 100(8/3 - t) = 250 → (A)

Answer

80t+100(8/3-t)=250

Solve the chosen equation to see it behaves sensibly: 80t + 800/3 - 100t = 250 → -20t = 250 - 800/3 = -50/3 → t = 5/6 h. That is 50 minutes before the stop, leaving 8/3 - 5/6 = 11/6 h ≈ 110 min after — both positive and inside the 3-hour window, so the equation is physically meaningful. Choice (E) swaps the speeds onto the wrong legs, and (B), (C), (D) ignore the stop and the two-speed split, so only (A) fits.

💡Key takeaway

Subtract the stop time first, split the leftover driving time into before and after, then add the two speed-times-time distances to 250.

Turn the stop into hours
Name the two driving times
Write each leg's distance
Add the legs to 250

Turn the stop into hours

Name the two driving times

Write each leg's distance

Add the legs to 250

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