AMC 8 · 2018 · #6

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

rateratio-proportionunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

📘 View easy version →

Problem

Anh drives 50 miles on the highway and 10 miles on a coastal road. His highway speed is exactly 3 times his coastal speed. The 10-mile coastal stretch took 30 minutes. How many minutes did the entire 60-mile trip take?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

100

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a classic rate problem (distance, speed, time). Tool #8 (Analyze the Units) keeps the bookkeeping honest: miles/(minutes-per-mile) = minutes, so working with the rate "minutes per mile" lets us avoid converting to mph at all. Tool #9 (Easier Related Problem) is the elegance move: instead of computing the highway speed, first ask the smaller question "how long would 10 highway miles take?" — that gives a per-mile rate we can scale up by 5 to reach the actual 50 miles. Tool #7 (Identify Subproblems) splits the trip into the highway leg and the coastal leg, which are summed at the very end. This avoids reaching for Tool #13 (Algebra) when scaling and addition are enough.

1STEP 1

Flip "3 times as fast" into per-mile time: each highway mile takes $\frac{1}{3}$ the time of a coastal mile.

time per highway mile =

\frac{1}{3}

× time per coastal mile

2STEP 2

Solve the easier case first: 10 highway miles take $\frac{1}{3}$ of the coastal 30 minutes, or 10 minutes.

\frac{1}{3}

× 30 min = 10 min for 10 highway miles

3STEP 3

Scale up: 50 highway miles is 5 times 10 miles at the same speed, so 5 × 10 = 50 minutes.

5 × 10 min = 50 min on the highway

4STEP 4

Add the legs: highway 50 minutes plus coastal 30 minutes gives 80 minutes total.

50 + 30 = 80 min → (C)

Answer

Sanity-check the speeds. Coastal: 10 miles in 30 min = 20 mph (a slow coastal road). Highway: 3 × 20 = 60 mph (a normal highway speed). Both numbers are realistic. The highway leg, 50 miles at 60 mph, takes

\frac{50}{60}

\frac{5}{6}

hour = 50 minutes, matching our scaled answer. Total = 50 + 30 = 80 minutes, which is choice (C) and lies in the middle of the answer choices — not suspiciously extreme.

💡Key takeaway

This AMC 8 problem only needs Grade 4 multiplicative comparison ("3 times as fast means $\frac{1}{3}$ the time") that you already know!

More like this