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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

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