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Last week, I travelled from Birmingham to Glasgow (via bicycle).

On the first day I travelled one quarter of the distance.

On day two, I travelled one half of the remaining distance.

On day three, I travelled three quarters of the remaining distance.

Yesterday I travelled one third of the remaining distance.

I now have 21 miles left to travel.

How far is it from Birmingham to Glasgow in total?

On the first day I travelled one quarter of the distance (84, leaving 252). On day two, I travelled one half of the remaining distance (126, leaving 126). On day three, I travelled three quarters of the remaining distance (94.5, leaving 31.5). Yesterday I travelled one third of the remaining distance (10.5). I now have 21 miles left to travel.

To work these numbers out, it is easiest to start at the end and work backwards.

Yesterday I travelled 1÷3 of the distance to leave 21 miles, therefore there were 31.5 miles (=21 x 3÷2) at the start of yesterday.

On day 3 I travelled 3÷4 of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 x 4) at the start of day 3.

On day 2 I travelled 1÷2 of the distance to leave 126 miles, therefore there were 252 miles (=126 x 2) at the start of day 2.

On the first day I travelled 1÷4 of the distance to leave 252 miles, therefore there were 336 miles (=252 x 4÷3) at the start of the first day. QED.

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