Question

Kevin Kangaroo begins hopping on a number line at 0. He wants to get to 1, but he can hop only 1/3 of the distance. Each hop tires him out so that he continues to hop 1/3 of the remaining distance. How far has he hopped after five hops? Express your answer as a common fraction.

Answer 1

Answer:

$\frac{211}{243}$

Explanation:

This is called a series, to solve it you need to give the first hop which is going to move you 1/3 of the way, the you hop another time, this will move you 1/3 of the 2/3 missing, this means you have moved now:

$\frac{1}{3}+ \frac{\frac{2}{3}}{3} =\frac{1}{3}+ \frac{2}{9}=\frac{5}{9}$

and you are missing 4/9 of the way.

Next hope will move you 1/3 of the 4/9 missing, which is $\frac{\frac{4}{9} }{3} = \frac{4}{27}$, adding this to the path you have already moved is:

$\frac{5}{9}+ \frac{4}{27} =\frac{19}{27}$

and you are missing 8/27 of the way.

The fourth hop is the same, one third of the missing path:  $\frac{\frac{8}{27} }{3} = \frac{8}{81}$, and adding this to the traveled path:

$\frac{19}{27}+ \frac{8}{81} =\frac{65}{81}$

and you are missing 16/81 of the way.

The last and fifth hop is again one third of the missing path: $\frac{\frac{16}{81} }{3} = \frac{16}{243}$, and adding this to the already moved way:

$\frac{65}{81}+ \frac{16}{243} =\frac{211}{243}$

And you end here.