, 2016

Every 2 seconds, Esther takes 5 steps and each of her steps is 0.4 m long.

Therefore, in 2 seconds Esther travels a distance of $5\times0.4=2$ m. Solution 1 Every 2 seconds, Paul takes 5 steps and each of his steps is 1.2 m long. Therefore, every 2 seconds Paul travels a distance of $5\times1.2=6$ m. Since Paul travels 6 m every 2 seconds, his speed is $6\div2=3$ m/s. Solution 2 Paul takes 5 steps every 2 seconds and so Paul takes 2.5 steps every second. The length of each of Paul’s steps is 1.2 m, and so every second Paul travels a distance of $2.5\times1.2=3$ m. Thus, Paul travels at a speed of 3 m/s. Solution 1 Paul travels at a speed of 3 m/s, and so in 120 seconds (2 minutes), Paul travels a distance of $3\times120=360$ m. Esther travels 2 m every 2 seconds and so she travels at a speed of 1 m/s. In 120 seconds, Esther travels a distance of $1\times120=120$ m. If Paul and Esther start a race at the same time, then after 2 minutes, Paul will be $360-120=240$ m ahead of Esther. Solution 2 Each of Paul’s steps is $1.2-0.4=0.8$ m longer than each of Esther’s steps. Every 2 seconds, Paul and Esther each take 5 steps, and so in $2\times60=120$ seconds (2 minutes), Paul and Esther each take $5\times60=300$ steps. If Paul and Esther start a race at the same time, then after 2 minutes Paul will be $300\times0.8=240$ m ahead of Esther. Solution 3 Paul travels at a speed of 3 m/s, and Esther travels 2 m every 2 seconds, so she travels at a speed of 1 m/s. This means that in 1 second, Paul travels 3 m and Esther travels 1 m. Every second, Paul travels $3-1=2$ m farther than Esther. If Paul and Esther start a race at the same time, then after 120 seconds (2 minutes), Paul will be $120\times2=240$ m ahead of Esther. Solution 1 Esther travels 2 m every 2 seconds or 1 m/s, and so in 180 seconds (3 minutes), Esther travels $180\times1=180$ m. Paul travels at a speed of 3 m/s, which is 2 m/s faster than the speed at which Esther travels. Therefore, every second, Paul travels 2 m farther than Esther travels. Since Esther begins the race 180 m ahead of Paul, it will take Paul $180\div2=90$ s to catch Esther. Solution 2 Esther travels 2 m every 2 seconds or 1 m/s, and so in 180 seconds (3 minutes), Esther travels $180\times1=180$ m. Each of Paul’s steps is $1.2-0.4=0.8$ m longer than each of Esther’s steps. Since Paul and Esther each step at the same rate (5 steps every 2 seconds), then it will take Paul $180\div0.8=225$ steps to catch Esther. Paul takes 5 steps every 2 seconds, and so it will take Paul $\frac{225}{5}\times2=45\times2=90$ s to catch Esther.