Fractured Fractions

By Mike Turnbull

The Pizza Problem

I recently saw on You Tube a “trick” mathematics problem that went something like the following.

Sam ate 1/3 of his pizza and Fred ate 1/5 of his pizza – but Fred ate more pizza than Sam!
How is that so?

As the astute reader will realise, the “answer” to that question is the obvious one – Fred’s pizza was larger than Sam’s pizza; at least, it was sufficiently larger than Sam’s for Fred to have consumed more in the absolute sense (as usually opposed to the relative sense; but, in this case both relatively and absolutely).

So, how much bigger than Sam’s pizza does Fred’s need to be if Fred is to consume more pizza, based on the fractions stated in the original question?

We proceed as follows.

We first assume that Sam’s pizza is the standard unity pizza, and that Fred’s is a non-standard larger pizza that is $x$ times bigger than Sam’s.

We can then express the problem statement as follows.

We then multiply both sides of this inequation by $5$, and eliminate the unity value on the right-hand side, to get the following.

and, by converting the improper fraction to a mixed fraction:

Therefore, for Fred to have eaten more pizza than Sam, Fred’s pizza has to have been at least $1\frac{2}{3}$ times the size of Sam’s pizza.

Fractions are Relative Quantities

The main take-home fact pointed out in the Pizza Problem is that a fraction, taken in isolation with no point of contextual reference cannot be compared with any other fraction that similarly has no contextual point of reference.

It is meaningless to say that someone has 1/8th and some other person has 1/10th unless the two fractions are referenced to whatever whole they are each fractions of. For instance, it would be meaningful to say that someone has 1/8th of a 100ml soda drink and someone else has 1/10th of 200ml soda drink.

Fractions in isolation are meaningless!

Back to Reality

Despite the above truth we often see mathematical statements containing something like the following.

Typically, in reality, when we come across that sort of statement, we just do the multiplication and come up with the following.

So, how come we can do that when we know that when two fractions are stated with no common reference, we simply cannot use them together? What are we missing?

The answer to that question has the technical name elision. There is something missing from the common fractional statements that we all just simply assume is there – but it is not explicitly written into the mathematical statement. It has been elided!

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