# 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.