### Multiplying fractions

Interestingly, multiplying fractions is, in many ways, simpler than adding or subtracting them, though it is important to learn the method correctly and precisely at otherwise you will end up with a mess.

It is vital to make sure that you work in the correct order.

- Cancel where possible.
- Multiply the numerators and denominators together.
- Check that the fraction can be simplified (if you missed a cancellation in point 1).
- Write down the answer.

### Worked example 1: Multiplying fractions with no cancelling

- Notes
- $$ {2 \over 3} \times {4 \over 5} $$
- $${2 \over 3} \times {4 \over 5} = {{2 \times 4} \over {3 \times 5}}={8 \over 15}$$To multiply fractions like this together, where there is no cancelling to be done, you can multiply the numerators together and then multiply the denominators together.
- $${8 \over 15}$$

### Worked example 2: Multiplying fractions where cancelling is necessary.

- Notes
- $$ {2 \over 5} \times {5 \over 8}$$
- $${2 \over 5} \times {5 \over 8}$$
Start by writing down the question. You cancel (divide) wherever you have any numerator with a common factor with any denominator.

Starting at the top left, note that 2 and 8 (on the diagonal) have a common factor of 2. So we divide both by 2. We can then replace the diagonal 2/8 with 1/4.Likewise, the other diagonal is 5/5 which can be replaced with 1/1, as both can be divided by 5.

It is important that the cancelling must be done before the multiplication, otherwise you run the risk of losing marks, but also a much higher risk of going wrong. The next example explains clearly, why this is.

- $${1 \over 4}$$

### Worked example 3: Multiplying many fractions together.

- Notes
- $$ {1 \over 2} \times {2 \over 3} \times {3 \over 4} \times {4 \over 5} \times {5 \over 6} \times {6 \over 7} \times {7 \over 8} \times {8 \over 9} \times {9 \over 10}$$
- $$={1 \over 10}$$
This example is an old favourite to make the point about cancelling first. It is, of course, possible to get the answer by multiplying all of the numerators and denominators together, and then cancel the resulting fraction down. However, the time taken to do it this way, is time wasted, pointlessly.

Do please try it both ways if you don't believe me. But don't forget to time both methods!

- $${1 \over 10}$$

### Worked example 4: A fraction of a whole number.

- Notes
- $${2 \over 3} \ of \ {27 \over 1}$$
- $$\begin{aligned}& \ \ \ \ \ {2 \over 3} \times {27 \over 1} \\&= {2 \over 1} \times {9 \over 1} \\&= {18 \over 1} \\&= 18 \end{aligned}$$
When combining a whole number by a fraction, it is a good idea, while you are learning, to write the whole number(s) as fractions by placing them over a denominator of 1. This has the advantage of meaning that you are only having to think in terms of fractions and not both fractions and whole numbers at the same time.

There is a fascinating piece of research by a pair of scientists called Dunning and Kruger, which speaks very directly to the "I prefer to do it my way" brigade, which is about how when our competence is quite low in a discipline, we tend to be rather more confident than we ought to be. I have written a short essay on this phenomenon, here.

- $$18$$

### Worked example 5: A fraction of a fraction

- Notes
- $$\frac{4}{5} \ of \ \frac{15}{16}$$
- $$ \begin{aligned} &\frac{4}{5} \ of \ \frac{15}{16}\\ &=\frac{\cancelto{1}{4}}{\cancelto{1}{5}} \times \frac{\cancelto{3}{15}}{\cancelto{4}{16}}\\ &= \frac{3}{4} \end{aligned} $$
Rewrite the problem as a multiplication.

Look for possible cancellations: the 4 and 16 have a factor of 4 in common and the 5 and 15 have a common factor of 5.

- $$\frac{3}{4}$$

### Exercise 1: Multiplying: interpreting 'of'

**Instructions:**Calculate the following products (multiplications).

- Q1$${1 \over 5} \ of \ 10$$2
- Q2$${2 \over 3} \ of \ 6$$4
- Q3$${3 \over 4} \ of \ 18$$$$12{1 \over 2}$$
- Q4$${5 \over 6} \ of \ 27$$$$22 \frac{1}{2}$$
- Q5$${2 \over 3} \ of \ {1 \over 4}$$$$\frac{{1}{6}$$
- Q6$${5 \over 6} \ of \ {3 \over 10}$$$$\frac{{1}{4}$$
- Q7$${7 \over 6} \ of \ {4 \over 5}$$$$\frac{14}{15}$$
- Q8$${8 \over 9} \ of \ {3 \over 4}$$$$\frac{2}{3}$$
- Q9$${4 \over 5} \ of \ {5 \over 4}$$$1$
- Q10$${2 \over 3} \ of \ {4 \over 5}$$$$\frac{8}{15}$$

### Exercise 2: Multiplying: standard notation.

**Instructions:**Multiply the following fractions, giving your answers as simply as possible.

- Q1$$\frac{8}{15} \times \frac{5}{12}$$$${2 \over 9}$$
- Q2$$\frac{9}{10} \times \frac{5}{18}$$$${1 \over 4}$$
- Q3$$\frac{9}{10} \times \frac{20}{27}$$$${2 \over 3}$$
- Q4$$\frac{10}{11} \times \frac{2}{15}$$$${4 \over 32}$$
- Q5$$\frac{16}{25} \times \frac{15}{32}$$$${3 \over 10}$$
- Q6$$\frac{3}{8} \times \frac{6}{11}$$$${9 \over 44}$$
- Q7$$\frac{14}{15} \times \frac{10}{21}$$$${4 \over9 }$$
- Q8$$\frac{34}{35} \times \frac{14}{17}$$$${4 \over 5}$$
- Q9$$\frac{14}{15} \times \frac{5}{8}$$$${7 \over 12}$$
- Q10$$\frac{9}{16} \times \frac{11}{27}$$$${11 \over 48}$$