Comparing fractions with different denominators

Leon and Lina are shooting penalty kicks against each other. Leon scored four out of five shots. Lina scored seven out of nine. - I shot four goals. - I got seven - the best! - What? I scored four goals from five penalty kicks, I only missed one. I'm the best. - Yes.

Who is the best? Lina shot more goals, but might Leon have a bigger proportion of goals? It's not easy to figure out because they took different numbers of penalty kicks. How can we do this? It’s not easy to solve with mental arithmetic, because the fractions have different denominators.

To answer this question, we need the same denominator in both fractions: a common denominator. Leon scores four out of five shots. We can make the denominator twice as big, if we make the numerator twice as big. Four out of five shots is the same ratio as eight out of ten shots. And even if the numerator and the denominator are three times as big, the fraction still has the same ratio.

The proportion is the same. This is called expanding the fraction. Look at the denominator: five, ten, fifteen ... That’s the five times table! So when we expand the fraction “four fifths”, the denominator is always in the five times table!

We can expand the other fraction too: Seven ninths is the same ratio as fourteen eighteenths Or twenty-one twenty-sevenths Nine, eighteen, twenty-seven ... ..that’s the nine times table. Now, we need to expand both fractions so that they get a common denominator. Can you find any number that appears in both tables? Oh yes! The number “forty-five” appears in both tables.

So “forty-five” is a common denominator for the two fractions. We rewrite the fractions - expand them - so that both get the denominator “forty-five”. Let’s start with four fifths ... To expand this fraction so its denominator is forty-five, we need to multiply it by nine. Then we also have to multiply the numerator by nine, so that the fraction has the same ratio as before.

Four times nine equals.. thirty-six and five times nine equals.. forty-five. The new fraction will be “thirty-six forty-fifths”. The second fraction has nine as denominator.

This has to be expanded by five, for the denominator to become forty-five. When the numerator is multiplied by five, it becomes .. seven times five equals thirty-five. The new fraction will be “thirty-five forty-fifths”. Now that the fractions have a common denominator - forty-five - it's easy to compare them: thirty-six over forty-five is more than thirty-five over forty-five.

Thus, four fifths is more than seven ninths. It is Leon who scored the biggest proportion of goals. - I won, I won, my ratio was higher! - Not anymore! - Is Lina right? Who is winning now?