# Sunday Times Teaser 2635 – Three Sisters

by Nick MacKinnon

A father shared his estate among his three daughters with each getting a fraction of it.

In their simplest form, these fractions had numerators less than 100 and shared a common denominator.

The largest fraction was less than three times the smallest and each fraction recurred after three places when expressed in decimal form (e.g. 0.abcabc…) .

Each of the digits 1 to 9 occurred in one of these three decimals.

What were the three fractions?

1. Here is my version.

2. Only one trio of this set is the answer obeying the given constraints:
5/37=0.135.. , 6/37=0.162.. , 7/37=0.189.. , 8/37=0.216.. , 9/37=0.243.. , 11/37=0.297.. , 12/37=0.324.. , 13/37=0.351.. , 14/37=0.378.. , 16/37=0.432.. , 17/37=0.459.. , 18/37=0.486.. , 19/37=0.513.. , 21/37=0.567.. , 22/37=0.594..

• HI Ahmet,

Your list is longer than it needs to be.

22 is not possible because this only leaves 15 for the sum of the other two. So the smaller of these has to be 7 or less and the maximum hence has to be less than 21, which it isn’t.

The same is true for 21 so the maximum value is 19/ 37 (as you suggest, 20 isn’t possible).

Similarly if the smallest is 5/37, the other two sum to 32/37 and this makes the maximum at least 16/37. But this is not possible because it is larger than 15 – three times the smallest.

So the minimum is 6/37 and the maximum is 19/37.

3. 