Question

An old-fashioned bicycle has two differently sized wheels. The circumference of the front wheel is 9 feet larger than the circumference of the back wheel. Thomas biked for a while and, according to his equipment, the front wheel went all the way around 500 times and the smaller wheel went all the way around 1400 times. How far, in feet, did Thomas bike?

Answer 1

So, the bicycle looks like the one in the picture below.

now, the circumference of the smaller wheel, will just be C = 2πr, with a radius of "r".

we also know that the large wheel has a circumference that is 9 feet larger than the small one, so, since the small one has a circumference of 2πr, then the large one will have a circumference of 2πr + 9.

after Thomas cycled for a while, the large did 500 revolutions, or times around, whilst the small one did 1400, since it's smaller.  On that time, they covered however, the same amount of ground, since they're on the same bike.

The amount covered by the small one in 1400 cycles, is 1400(2πr), that's how much ground it covered.

The amount covered by the large one in 500 cycles, is 500(2πr + 9).

And we know that ground covered, is the same for both, therefore, we also know that   1400(2πr)  =  500(2πr + 9).

$\bf \stackrel{small~wheel}{1400(2\pi r)}~~=~~\stackrel{large~wheel}{500(2\pi r+9)}\implies 14(2\pi r)=5(2\pi r+9) \\\\\\ 28\pi r=10\pi r+45\implies 18\pi r=45\implies r=\cfrac{45}{18\pi } \\\\\\ \boxed{r=\cfrac{5}{2\pi }}$

so, how much ground did they cover anyway?

well, we could use either, since we know what the radius is, hmmmm let's use say the small wheel's circumference,

$\bf 1400(2\pi r)\implies 1400\left(\underline{2\pi }\cdot \cfrac{5}{\underline{2\pi }} \right)\implies 1400(5)\implies \stackrel{feet}{7000}$

Answer 2

1400x=500(x+9)

x=9 feet