AMC 8 · 2018 · #12

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateratio-proportionfraction-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Medium solution 💡 3 insights

Problem

Sri's car clock runs fast at a steady rate. At noon, both his accurate wristwatch and the car clock read 12{:}00. By the time his watch reads 12{:}30, the car clock already reads 12{:}35. Later that day his watch is gone, and the car clock reads 7{:}00. What is the true time?

Pick an answer.

(A)

5:50

(B)

6:00

(C)

6:30

(D)

6:55

(E)

8:10

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

How to solve

Strategy Analyze the Units

This is a rate problem with two different time "units" — car-clock minutes and real minutes. Tool #8 (Analyze the Units) keeps them straight: the data point (35 car-min per 30 real-min) defines an exchange rate between the two. Tool #5 (Look for a Pattern) turns that data point into the cleaner repeating block "7 car-min = 6 real-min," which is easy to scale by counting how many blocks fit in 420 car-minutes. Tool #6 (Guess and Check) gives a quick sanity test against the multiple-choice answers at the end.

1STEP 1

The car gains 5 extra minutes per 30 real minutes, so car-time : real-time = 35{:}30, which simplifies to 7{:}6.

\frac{car min}{real min}

=

\frac{35}{30}

=

\frac{7}{6}

2STEP 2

As a repeatable block: every 6 real minutes the car clock advances 7 minutes, and this repeats forever at the constant rate.

6 real min_one block ⟷ 7 car min_one block

3STEP 3

The car clock went from 12{:}00 to 7{:}00, i.e. 7 hours, so in minutes that is 7 × 60 = 420 car-minutes.

7 hr × 60 min/hr = 420 car-min

4STEP 4

How many 7-minute blocks fit in 420? 420 ÷ 7 = 60 blocks, and each is 6 real minutes, so 60 × 6 = 360 real-minutes.

\frac{420 car-min}{7 \frac{car-min}{block}}

= 60 blocks → 60 × 6 = 360 real-min

5STEP 5

Turn 360 real-minutes back into hours: 360 ÷ 60 = 6 hours, and 12{:}00 + 6 hours = 6{:}00 — choice (B).

360 min ÷ 60 = 6 hr, 12{:}00 + 6{:}00 = 6{:}00 → (B)

Answer

6:00

The car clock runs

\frac{7}{6}

as fast as real time, so it should always read a bigger number than the actual elapsed time. The car shows 7 hours elapsed; the real elapsed time should be smaller, and 6 hours is indeed smaller. Also, 7 ×

\frac{6}{7}

= 6 exactly, so the answer comes out as a whole number of hours with no leftover minutes — a clean match to choice (B) 6{:}00. Choices (A) 5{:}50 and (C) 6{:}30 would require the ratio to be different from 7{:}6, and (D) and (E) are clearly too far off.

💡Key takeaway

This AMC 8 problem only needs Grade 6 ratio reasoning — turning "35 car-min : 30 real-min" into the clean 7{:}6 ratio — that you already know!