On Monday, I posed this question:

A hiker walks from one town to another. On the first day she covers one fifth of the total distance. The next day she covers a quarter of what is left. The following day she covers two fifths of the remainder and on the fourth day half of the remaining distance. She now has 25 miles left. How many miles has she travelled?

And now all shall be revealed! Well, after the break.

The best way I’ve found to do this fairly simply is to take it one day at a time and use some algebra. On which note, let the total distance between the two towns be ‘x’, and let’s continue.

Day 1

Here the hiker walks 1/5 of x.

**Running total: 1/5 of x.**

Day 2

Since we’ve already covered 1/5, there is 4/5 of the distance left. So, on this day the hiker covers 1/4 of 4/5, which is 1/5.

**Running total: 1/5 + 1/5 = 2/5 of x.**

Day 3

We’ve covered 2/5, so there are 3/5 left. Therefore on this day she covers 2/5 of 3/5, which is 6/25 as a fraction.

**Running total: 2/5 + 6/25 = 16/25 of x.**

Day 4

Having covered 16/25, we have 9/25 left, and the hiker walks half of this, which is 9/50.

**Running total: 16/25 + 9/50 = 41/50 of x.**

Finding x

So, we know that we now have 9/50 of x left to go, and this is equal to 25 miles. Therefore x = 25 / (9/50) = 1250/9 = 138.89 miles.

However, the questions asks how far the hiker has gone, not just the total distance, so we need to subtract the 25 miles which are left, giving us **113.89 miles**.

How’d you do this week?

Vel.

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