## Half Fractions

#### Problem

By concatenating all of the digits 1, 2, 3, and 4 to form two numbers, it is impossible to find a pair that divide to make one-half. However, using the digits 1, 2, 2, and 4, it is possible to make one-half in exactly two different ways.

12 24 | = | 21 42 | = | 0.5 |

By concatenating all of the digits 1, 2, 3, 4, and 5 to form the ratio of two numbers, how many ways can you make one-half?

#### Solution

The solution must be of the form:

2-digits 3-digits | = | 0.5 |

As the denominator must be exactly twice the numerator, the only 2-digit numbers we can form that double to make a 3-digit number are 51, 52, 53 and 54. However, their doubles are 102, 104, 106 and 108, respectively.

Hence there exist no solutions using the digits 1 to 5.

Investigate using the digits 1 to `n`.

Problem ID: 151 (Feb 2004) Difficulty: 2 Star