on an old bicycle.

Question

On an old bicycle, the front wheel has a radius of 2.5 feet and the back wheel has a radius of 4 inches. How many complete revolutions will the back wheel make to cover the same distance as the front wheel covers in 100 complete revolutions?

Answer 1

The distance covered by the front wheel and the back wheel is the same.

We calculate the distance covered by front wheel in first step and write an equation to find the unknown number of revolutions in the second step.

 

The front wheel will travel 100 (rev) times one rotation:

Distance = D = 100 x (2x2.5 ft) x pi

                    =100 x 15.7 ft =1570 ft

                    =18,840 in.

 

The same distance will be traveled by the back wheel:

Distance = D = X (rev) x (2x4 in) x pi.

                18,840 in= X • 25.12 in

                X = 18,840 in / (25.12 in)

                X = 750 rev.

 

The back wheel will make 750 revolutions, when the front wheel makes 100 revolutions.

 

Any questions?

 

Investigate:

How the ratio of front wheel radius and back wheel radius and the number of revolutions are related?

Answer 2

 

C = 2(pi)r 

 

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The ratio of the circumferences (front/rear) will give the number of revolutions of the rear wheel for each revolution of the front wheel:

C_f = 2.5(2)(pi) = 5pi ft

C_r= (0.3333)(2)(pi) = 0.6667pi ft

 

Ratio = (5pi) / (0.6667pi) = 7.5

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Notice we can just take the ratio of the radii, since 2pi's will cancel:    2.5ft / (0.33333ft) = 7.5  

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So in 100 revolutions of the front wheel, the rear wheel will make  750 revolutions.