Mathematical Problem 3: Solution

This is the solution to the problem set on 1 April 2010. This was about the metric/imperial curiosity that 10 feet 3 inches converts to 3 metres 10 centimetres, rounded down to the nearest 5 cm. I wondered if there were other lengths like this where you just swap over the two numbers to convert from feet and inches to metres and centimetres. For reference purposes 1 metre is about 3.28084 feet.

Well, the obvious one to try next is 20 feet 6 inches – which works, because it converts to 6 m 20 cm, when rounded down to the nearest 5 cm. But 30 feet 9 inches does not work, because of the rounding error; it actually rounds down to 9 m 35 cm.

It's not too difficult to check all the possibilities. Because of the instruction to round down to the nearest 5 cm, we need check only imperial lengths that start 0 feet, 5 feet, 10 feet, 15 feet, and so on. Then because the number of inches cannot exceed 11, the number of metres cannot exceed 11. So we do not need to check lengths beyond 11 metres 95 cm, which is under 40 feet.

So, I've checked all the possibilities and all I can offer you are these two results:

10 ft 3 ins = 3 m 10 cm and 20 ft 6 ins = 6 m 20 cm

(both metric lengths rounded down to the nearest 5 cm).