Video: Pack 3 • Paper 1 • Question 4

Pack 3 • Paper 1 • Question 4

04:28

Video Transcript

Work out two and two-fifths multiplied by five and five-ninths. Give your answer as a mixed number in its simplest form.

To perform any arithmetic with mixed numbers, we must first ensure they are all written as improper fractions. To change a mixed number into an improper fraction, we first multiply the integer or the whole number by the denominator. In the case of two and two-fifths, that’s two multiplied by five which is 10. This tells us there are 10 fifths in two wholes.

Next, we add this number to the numerator of our fraction. For this mixed number, that’s 10 plus two which is 12. It tells us the total number of fifths that are in two and two-fifths. So we make 12 the numerator in our improper fraction and we keep the denominator the same. Two and two-fifths is the same as 12 fifths.

Let’s repeat this process for five and five-ninths. Multiplying a whole number by the denominator of the fraction gives us 45. Adding that number to the numerator of the fraction gives us 50. Remember the denominator stays the same. So five and five-ninths is equal to 50 ninths. So our sum becomes 12 fifths multiplied by 50 ninths.

There are two ways to do this: either we can 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 and then fully simplify our answer. But this does give us quite big numbers.

What we can do is make our life easier by performing a process known as cross cancelling. It involves looking diagonally at the numerators and denominators of each fraction to see if they share any common factors. In this case, 12 and nine have a highest common factor of three. We can therefore divide 12 by three to get four and nine by three to get three.

Similarly, the highest common factor of 50 and five is five. 50 divided by five is 10 and five divided by five is one. We then multiply the numerator of the first fraction by the numerator of the second and the denominator of the first fraction by the denominator of the second. Four times 10 is 40 and one times three is three. We are not quite finished though. We need to change our answer back into a mixed number.

To do this, we do the opposite of the process we did earlier. We divide 40 by three to find out how many wholes we can make from 40 thirds. 40 divided by three is 13 remainder one. That means we have 13 wholes and one-third left over. Remember the denominator stays the same. So 40 thirds is the same as 13 and one-third.

Write the numbers one, two, three, and four in the boxes to give the lowest possible total. You may write each number only once. In order to get the lowest possible total, we will need to ensure the whole numbers are as small as possible. That’s either one and some number of sevenths plus two and some number of fifths or two and some number of sevenths plus one and sum number of fifths. It doesn’t matter which way round these integers go since their size will never change. They will always sum to three.

Now, we have the numbers three and four left over. We do have to be careful which way round we choose to use these. First, let’s consider the size difference between a seventh and a fifth. Splitting a shape into seven pieces means there’s more pieces than if we were to split it into five pieces. The more pieces we have, the smaller each piece must be to fit into one whole. That means one-seventh is less than one-fifth. It also means that we can have slightly more sevenths than fifths. And we choose to put the four on the first fraction and the three on the second.

The answer is therefore two and four-sevenths plus one and three-fifths or one and four-sevenths plus two and three-fifths.

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