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

`(211)/(243)`

Step-by-step 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:

`(1)/(3)+ ((2)/(3))/(3) =(1)/(3)+ (2)/(9)=(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
`((4)/(9) )/(3) = (4)/(27)`, adding this to the path you have already moved is:

`(5)/(9)+ (4)/(27) =(19)/(27)`

and you are missing 8/27 of the way.

The fourth hop is the same, one third of the missing path:
`((8)/(27) )/(3) = (8)/(81)`, and adding this to the traveled path:

`(19)/(27)+ (8)/(81) =(65)/(81)`

and you are missing 16/81 of the way.

The last and fifth hop is again one third of the missing path:
`((16)/(81) )/(3) = (16)/(243)`, and adding this to the already moved way:

`(65)/(81)+ (16)/(243) =(211)/(243)`

And you end here.