Bouncing ball
A ball is dropped from a height, and every time it hits the ground, it bounces to 3/5 of the height from which it fell.
A ball is dropped from a height of 125 cm. Each time it hits the ground, it bounces to $\frac{3}{5}$ of the height from which it fell.
How high does it bounce after hitting the ground the third time?
This problem is adapted from the World Mathematics Championships
Answer: 27 cm
Working out the height after each time it hits the ground
First time: $\frac{3}{5}$ of 125 cm = 125$\div$5$\times$3 = 25$\times$3 = 75 cm.
Second time: $\frac{3}{5}$ of 75 cm= 75$\div$5$\times$3 = 15$\times$3 = 45 cm.
Third time: $\frac{3}{5}$ of 45 cm= 45$\div$5$\times$3 = 9$\times$3 = 27 cm.
Working out the height after each time it hits the ground using 3s and 5s
125 = 5$\times$5$\times$5
First time: $\frac{3}{5}$ of 125 is 3$\times$5$\times$5
Second time: $\frac{3}{5}$ of 3$\times$5$\times$5 is 3$\times$3$\times$5
Third time: $\frac{3}{5}$ of 3$\times$3$\times$5 is 3$\times$3$\times$3 = 27 cm.
Using powers
Each time the ball hits the ground, it bounces to $\frac{3}{5}$ of the height from which it fell. So after each bounce, its height is multiplied by $\frac{3}{5}$.
So after 3 bounces, its height will be multiplied by $\left(\frac{3}{5}\right)^3=\frac{27}{125}$.
So after 3 bounces, its height will be $125\times\frac{27}{125}=27$ cm.