# 8 Does order matter for multiplication and division?

When you add two numbers together, the order does not matter – addition is said to be ‘commutative’ as 2 + 4 is the same as 4 + 2. However, subtraction isn’t commutative as 6 − 2 isn’t the same as 2 − 6. So what about multiplication and division? Is 3 × 2 the same as 2 × 3? Is 4 ÷ 2 the same as 2 ÷ 4? To help with this, we’ll use a diagram in Figure 17.

Figure 17 shows on the left three rows of two dots (3 × 2), and on the right two rows of three dots (2 × 3).

The number of dots in both arrangements is the same, 6, and hence you can see that 3 × 2 = 2 × 3.

However, you can’t say the same for division, where order does matter. For example, if you divide £4.00 between two people, each person gets £2.00. If instead you need to divide £2.00 among four people, each person only gets £0.50. Division is not commutative.

Now you’ve looked at the fundamentals of multiplication and division, you are going to get a chance to apply these to a more everyday problem.

The next activity uses both multiplication and division to solve a problem.

## Activity 7 How many teabags?

Try doing these on paper and then check your answers using a calculator.

A café buys boxes of teabags in bulk to cater for their customers. Each box contains 20 packs of teabags. Each pack contains 100 teabags.

a) How many teabags are there in one box?

### Answer

Since each box contains 20 packs, and each pack contains 100 teabags, there are 20 × 100 = 2000 teabags in each box.

b) The café estimates they use 500 teabags a day. How many days will a box last them?

### Answer

2000 ÷ 500 = 4 days.

This looks like a difficult division, but it is possible to work it out like this:

You know that 2 × 500 gives 1000, so each lot of 1000 contains two 500s. Therefore, in 2000 there are 2 × 2 = 4 lots of 500.

You can also think about it as 500 + 500 + 500 + 500 = 2000, so they last 4 days.

c) If the café is open 300 days a year, how many boxes of teabags do they need for the year?

### Answer

300 ÷ 4 = 75 boxes.

From Part b of this activity, you know that a box lasts 4 days. So to work out how many boxes are needed for 300 days, you need to find out many lots of 4 days there are in 300 days.

To do that, you find 300 ÷ 4 = 75.

d) Each box normally costs £80 but if the shop orders more than 50, they get a discount of £10 on each box. How much will their teabags for the year cost them?

### Answer

Cost is £70 per box. £70 × 75 = £5250.

You know the shop is going to order 75 boxes from what you have worked out already in this activity, which means they will get the discount.

Therefore they pay £70 per box. You need to work out 75 boxes at £70 per box, so you need to do £70 × 75.

One way to do that is to work out £7 × 75, and then multiply the result by 10 to find the answer to £70 × 75. Using the multiplication strategies from earlier £7 × 75 = £525 and then multiplied by 10 is £5250.

The next activity is a slightly more complex problem, or puzzle, than you’ve encountered so far in this week. You can use your calculator to help solve it and remember to use the hints, if you need to, by clicking on reveal.

## Activity 8 Running a hospital

The local hospital is keen to encourage people not to miss appointments. It has a sign up saying ‘This hospital costs £4000 every three minutes to run.’

a) Use a calculator to find out how much it costs to run the hospital in a 365-day year.

### Answer

An hour is 60 minutes which is 20 lots of 3 minutes.

So it costs £4000 × 20 = £80000 per hour.

There are 24 hours in a day so 24 × £80 000 = £1 920 000 per day.

There are 365 days in a year, so £1920000 × 365 = £700 800 000 per year.

b) An American visitor who comes to the hospital seeking treatment is charged £400 for this. For how long (in seconds) does his payment fund the running of the hospital?

### Answer

There are 180 seconds in 3 minutes. 180 seconds costs £4000.

£400 is a tenth of £4000. So it pays for 18 seconds.

c) A pound is currently worth $1.39. How much does the American pay in dollars for their treatment?

### Comment

400 × 1.39 = $556.

In Week 1, you have learnt how to carry out the four arithmetic operations – adding, subtracting, multiplying and dividing – on paper. You’ve also learnt that maths has a lot in common with doing puzzles – both need a logical systematic approach, and persistence.