Question

chelsea cricket begins hopping on a number line at 0. she wants to get the point at 1, but she can only hop 1/4 of the distance. each hop tires her out so that she continues to hop 1/4 of the remaining distance. how far has she hopped after five hops? express your answer as a common fraction.

Answer 1

The distance between 0 and 1 is 1.
First hop: 1/4
Total distance traveled 1/4
Remaining distance to 1 = 1 - 1/4 = 4/4 - 1/4 = 3/4

Second hop: 1/4 * 3/4 = 3/16
Total distance traveled = 1/4 + 3/16 = 4/16 + 3/16 = 7/16
Remaining distance to 1: 1 - 7/16 = 16/16 - 7/16 = 9/16

Third hop: 1/4 * 9/16 = 9/64
Total distance traveled: 7/16 + 9/64 = 28/64 + 9/64 = 37/64
Remaining distance: 1 - 37/64 = 64/64 - 37/64 = 27/64

Fourth hop: 1/4 * 27/64 = 27/256
Total distance traveled = 37/64 + 27/256 = 148/256 + 27/256 = 175/256
Remaining distance: 1 - 175/256 = 256/256 - 175/256 = 81/256

Fifth hop: 1/4 * 81/256 = 81/1024
Total distance traveled: 175/256 + 81/1024 = 700/1024 + 81/10254 = 781/1024
Remaining distance: 1 - 781/1024 = 1024/1024 - 781/1024 = 243/1024

After 5 hops, it has gone a distance of 781/1024.