From: Geecee

Yesterday I went for a long bicycle ride around the local lakes. As the ride was quite long, I rode in various stages. In the first stage I rode half of the overall distance. Stage two saw half of the remaining distance plus 35 metres covered. Stage three covered three-quarters of the remaining distance. Stage four completed half of the remaining distance plus 75 metres. Stage five completed the journey with a final burst of 150 metres.
How far did I cycle in total?

Hi Geecee.

We are most clear about the length of the last stage, so let's start there and work backwards.

V. Stage five started with a remaining distance of 150 meters, because this is the last stage and the cyclist covered 150 meters in that stage.
So stage four must have ended with 150 meters to go.

IV. Stage four started with a remaining distance we'll call D. In this stage, the cyclist covered half of D, then another 75 meters and ended with 150 meters to go.
That's D - (½D + 75) = 150 or D = 450 meters.
So stage three must have ended with 450 meters to go.

III. Stage three started with a remaining distance we'll call C. In this stage, the cyclist covered three-quarters of C and ended with 450 meters to go.
That's C - (¾C) = 450 or C = 1800 meters.
So stage two must have ended with 1800 meters to go.

II. Stage two started with a remaining distance we'll call B. In this stage, the cyclist covered half of B, then another 35 meters and ended with 1800 meters to go.
That's B - (½B + 35) = 1800 or B = 3670 meters.
So stage one must have ended with 3670 meters to go.

I. Stage one started with the whole trip distance ahead of the cyclist. Let's call this A.
In this stage, the cyclist covered half of A and finished the stage with 3670 meters to go.
That's A - (½A) = 3670 or A = 7340 meters.

So the trip was 7340 meters in total.

Stephen La Rocque.