2022 AMC 10A Problem 2

P1-10~1 minPrint

Ratios, Percents & Averages Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes?

Show solution without answering

Key insight

At a steady speed, laps go up in step with time. 27 minutes is 9/19 of the ride, so he did 9/19 of the 15 laps, a little over 7.

Solution

Because Mike's speed never changes, the fraction of the time that has passed is the same as the fraction of the laps he has done.

What part of the ride is $27$ minutes? It is $\frac{27}{57}$ of it. Both numbers divide by $3$ , so this is $\frac{9}{19}$ .
He has done $\frac{9}{19}$ of the $15$ laps: $15 \times \frac{9}{19} = \frac{135}{19}$ .
Since $19 \times 7 = 133$ , we get $\frac{135}{19} = 7\frac{2}{19}$ , just a little over $7$ .

The closest choice is $7$ .

The answer is $\boxed{\textbf{(B)}\ 7}$ .

Why this works

"Constant speed" means twice the time gives twice the distance. So a question about part of the trip is a fraction question: find what fraction of the time has passed, and take that fraction of the distance. Simplify the fraction first so the arithmetic stays small, and remember that only an approximate answer is needed.

Alternative approach

Find the time for one lap: $57 \div 15 = 3.8$ minutes. In $27$ minutes he rides $27 \div 3.8 \approx 7.1$ laps.

Or just estimate: $27$ is a bit less than half of $57$ , so he did a bit less than half of $15$ laps, which is $7.5$ . Only $7$ is close to that.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

At constant speed, the number of laps is proportional to the elapsed time. The first $27$ minutes are $\frac{27}{57} = \frac{9}{19}$ of the whole ride, so Mike completed
$15 \cdot \frac{9}{19} = \frac{135}{19} = 7\tfrac{2}{19} \approx 7.1$
laps. The closest choice is $7$ .

The answer is $\boxed{\textbf{(B)}\ 7}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)