4 Commutative properties of multiplication and division
When you add two numbers together, the order does not matter – the same as saying that addition is commutative; so 2 + 4 is the same as 4 + 2. But 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.
On the left this shows 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 5 How much paper?
Try doing these on paper and then check your answers using a calculator.
A college bookshop buys pads of legal paper in bulk to sell to students in the law department at a cheap rate.
a) Each pack of paper contains 20 pads. If the shop wants 1500 pads for the term, how many packs should be ordered?
a) You need to work out how many times 20 goes into 1500, so need to divide.
Number of packs = 1500 ÷ 20 = 75
Therefore, the bookshop needs to order 75 packs.
b) Each pack costs £25.00 but if the college orders over 50 packs they receive a discount of £2.50 on each pack. How much will the total cost be?
b) The shop will receive a discount of £2.50 per pack since they will be buying more than 50 packs.
Discount = £2.50 × 75 = £187.50
Cost without discount = 75 × £25 = £1875
Cost with discount = £1875 − £187.50 = £1687.50
The bill for the pads should be £1687.50.
c) How much will the shop need to sell each pad for to cover these costs?
c) You know that there are 1500 pads, and the total cost is £1687.50. So you need to share this cost across all the pads, meaning you need to divide.
Cost per pad = £1687.50 ÷ 1500 = £1.125
= £1.13 (to the nearest penny)
Hence the shop will need to sell the pads for £1.13 to cover their costs (although they will be making 0.5p profit on each pad!).
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 6 The Great Malvern Priory
The Great Malvern Priory in England is a church dating back more than 900 years. In 2011, there was a notice posted at the entry to the priory, reading: ‘This Priory Church costs £3 every five minutes.’ Visitors are encouraged to leave a donation of £2.50.
(a) To maintain the priory costs £3 every five minutes. Use a calculator to find out how much it costs in one year (365 days, as this is not a leap year).
How many times do you have ‘five minutes’ in one hour? Use this to find the cost per hour. Now, how many hours are in a day, and how many days are in a year?
(a) There are 12 periods of five minutes in each hour and 24 hours in each day. Therefore, the cost for a year of 365 days will be £3 × 12 × 24 × 365 = £315 360.
The cost of running the priory for a year is £315 360.
(b) In 2013, £1 was equivalent to about 1.55 US dollars. How much will it cost an American visitor (in US dollars) when they donate £2.50 to the Priory?
To buy one British pound, an American had to pay £1.55.
(b) Since 2.5 × $1.55 = $3.875, it will cost the American visitor about $3.88 (rounded to the nearest cent).
(c) Calculate how much the priory would cost to run for a year in US dollars, using the same exchange rate as before (£1 costs $1.55).
If you find this calculation difficult, then think of a simpler version first. £1 is equivalent to 1.55 US dollars, so £2 would give twice as many dollars – multiply by 2. So, how many dollars would you get for £315 360?
(c) £315 360 = $1.55 × 315 360 = $488 808. So the cost in US dollars is $488 808.
Now you’ve covered the four basic operators (addition, subtraction, multiplication and division) you’ll turn your attention to repeated multiplication of the same number, which is exponents or powers.