Eva is practising a two and half-minute gymnastics floor routine. The dance elements of her routine last one and three-sevenths of a minute. What fraction of her routine do the dance elements represent? Give your answer in its simplest form.
One way of answering this question would be to divide the total time spent on the dance elements by the total time spent on the routine in seconds. However, whilst we can turn two and a half minutes into seconds fairly easily, one and three-sevenths isn’t so simple.
Instead, we’ll divide the numbers as they are in fraction form. Before we can do any division though, we need to turn them into improper fractions. To change a mixed number into an improper fraction, the first thing we do is we multiply the integer by the denominator.
Here, the integer, the whole number value, is one and the denominator of our fraction is seven. So we multiply one by seven to get seven. Next, we add the numerator to this number. The numerator of the fraction part of our mixed number is three. So we add seven and three to get 10.
This number is now the numerator of our improper or top heavy fraction. The denominator stays the same. And this means that one and three-sevenths is equal to ten-sevenths.
Let’s repeat this process for two and a half. We multiply the integer by the denominator. That’s two multiplied by two which is four. We then add the numerator from the fraction part of our mixed number to this. Four plus one is five.
This is the numerator of our new fraction and the denominator remains unchanged. So two and a half is equal to five over two. And our division problem becomes 10 over seven divided by five over two.
When dividing fractions, you may have seen the letters KCF to help you decide what to do. K stands for keep: we keep the first fraction as it is; here that’s 10 over seven. C stands for change: we change the division symbol to a multiplication symbol. And F stands for flip: we flip or we invert the second fraction. This is called finding its reciprocal.
The reciprocal of five over two is two over five. And our problem becomes 10 over seven multiplied by two over five. Now, at this point, we could multiply the numerator of the first fraction by the numerator of the second and then multiply the denominator of the first fraction by the denominator of the second or we can choose to cross cancel.
To do this, we look diagonally across and see if each of these numbers share any common factors. 10 and five have a highest common factor of five. So we can divide both of these numbers by five. 10 divided by five is two and five divided by five is one.
Seven and two are coprime. That means they share no factors apart from one. And now, we can multiply the two numerators. Two multiplied by two is four. And then, we multiply the denominators. Seven multiplied by one is seven.
And we can see that the fraction of the routine that the dance elements represent is four-sevenths.