Digit Cancelling Fractions

The fraction $49/98$ is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that $49/98 = 4/8$, which is correct, is obtained by cancelling the $9$s.
We shall consider fractions like, $30/50 = 3/5$, to be trivial examples.
There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
If the product of these four fractions is given in its lowest common terms, find the value of the denominator.

The fractions you’re looking for have the following properties:

1. They are less than 1.
2. They are reduced to lowest terms.
3. They have two digits in the numerator and denominator.
4. Cancelling one of the digits gives the correct reduced form of the fraction.

Firstly, we quickly rule out any fractions where the tens digit of the denominator and numerator are the same, as any such fraction would obviously be one of the ‘trivial’ fractions you mentioned like 30/50.

So, each fraction may have one of 9 possible tens digits (1-9), and this determines the tens digit of the denominator and also the ones digit of the numerator.

Notice that for the fraction to be less than 1, the tens digit of the numerator must be less than the tens digit of the denominator, which means there are less possible choices for the tens digit of the numerator.

Given the tens digit of the numerator and denominator, there are 10 choices for the ones digit of the denominator, but one of these (being equal to the tens digit of the numerator) results in an irreducible fraction, and one of the 10 choices makes a ‘trivial’ fraction (where the ones digit of the denominator is 0), so there are 8 possible choices.

Given all of this, you can simply list all such fractions and check which ones are equal to their ‘incorrectly cancelled’ form.

After doing this, you find that there are exactly four such fractions:

16/64, 19/95, 26/65, and 49/98.

The product of these four fractions is

(16/64) x (19/95) x (26/65) x (49/98) = 1/4 x 1/5 x 2/5 x 1/2 = 1/100.

So, the denominator of the product when written in lowest terms is 100.

More Answers: