2015 AMC 10B Problem 2

P1-10~1 minPrint

Clocks, Calendars & Time Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

Show solution without answering

Key insight

From 1:00 to 2:40 is 100 minutes, and that covers two tasks, not three. So each task takes 50 minutes.

Solution

Find how long the first two tasks took. From 1:00 PM to 2:40 PM is $1$ hour and $40$ minutes, which is $60 + 40 = 100$ minutes.
Count the tasks in that time. She started the first task at 1:00 and finished the second at 2:40, so that time holds exactly two tasks.
Each task takes $100 \div 2 = 50$ minutes.
The third task starts at 2:40 PM. Adding $20$ minutes gets to 3:00 PM, and the other $30$ minutes get to 3:30 PM.

The answer is $\boxed{\textbf{(B)}\ \text{3:30 PM}}$ .

Why this works

Change clock times to minutes before dividing: an hour is $60$ minutes, not $100$ , so "1 hour 40 minutes" is $100$ minutes. Then count carefully how many tasks fit in the time you measured. Here the finish of the second task marks two tasks.

Concise solution

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

From 1:00 PM to 2:40 PM is $1$ hour $40$ minutes, which is $100$ minutes. During that time Marie completed exactly two tasks (the first and the second), so each task takes
$\frac{100}{2} = 50 \text{ minutes}.$

The third task starts at 2:40 PM and lasts $50$ minutes: $20$ minutes bring her to 3:00 PM, and the remaining $30$ minutes bring her to 3:30 PM.

The answer is $\boxed{\textbf{(B)}\ \text{3:30 PM}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)