# 2005 CEMC Gauss (Grade 7) Problems/Problem 16

## Problem

Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting $42$ sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly half of the total flock has crossed the road since 2:00 p.m. How many sheep are there in the entire flock? $\text{(A)}\ 630 \qquad \text{(B)}\ 621 \qquad \text{(C)}\ 582 \qquad \text{(D)}\ 624 \qquad \text{(E)}\ 618$

## Solution

Nicholas sleeps for an hour and a half, or $90$ minutes. Since three sheep cross the road per minute, then $3\times 90 = 270$ sheep cross while he is asleep. Since $42$ sheep crossed before he fell asleep, then $42 + 270 = 312$ sleep have crossed the road in total when he wakes up. Since this is half of the total number of sheep in the flock, then the total number in the flock is $2 \cdot 312 = 624$. Therefore, the answer is $D$.