# Penny Farthing

Boris' bicycle has a smaller back wheel than front wheel. Can you work out how many revolutions the front wheel made if the back wheel did 120,000?

## Problem

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Boris Biker entered the Tour de Transylvania with an unusual bicycle whose front wheel is larger than the back.

The radius of the large wheel is 40cm, and the radius of the small wheel is 30cm.

On the first stage of the race the smaller wheel made 120,000 revolutions. How many revolutions did the larger wheel make?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

## Student Solutions

**Answer**: 90 000

**Using a small number of revolutions**

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Wheels travel the same total distance

Small wheel:

1 revolution: 60$\pi$ cm

2 revolutions: 120$\pi$ cm

4 revolutions: 240$\pi$ cm

Large wheel:

1 revolution: 80$\pi$ cm

2 revolutions: 160$\pi$ cm

3 revolutions: 240$\pi$ cm

3 revolutions on the large wheel = 4 revolutions on the small wheel

Small wheel: 120 000 revolutions = (4 revolutions)$\times$30 000

= (3 revolutions of large wheel)$\times$ 30 000

= 90 000 revolutions of the large wheel

**Using ratio**

Ratio of the radii is $3\colon 4$

Ratio of the number of revolutions (to travel the same distance) is $4\colon3$

$\therefore$ larger wheel makes $120000\times\frac{3}{4}= 90000$ revolutions