In Chapter 1, Section 1.4 of the Calculator Book, you saw that multiplying a price by, say, 1.30 is equivalent to increasing it by 30%. Similarly, multiplying a price by 0.94 is equivalent to decreasing it by 6%. The figures 1.30 and 0.94 are called price ratios. In Table 6, the price of a loaf of bread went up from 50p to 65p. The price ratio for bread is just the new price divided by the old price, so in this case it is:
or 1.30.
Suppose that a bottle of milk rose in price from 30p to 39p.
(a) Calculate the ratio of the later price to the earlier price (that is, 39p divided by 30p).
(b) Calculate the proportional price increase and the percentage price increase.
(c) Compare the answers to parts (a) and (b). How is the proportional price increase related to the price ratio?
(d) How could you convert a proportional or a percentage price rise into price ratio and vice versa? (Try a few examples to get a sense of the underlying method, then try to write down a rule.)
(a) The price ratio is
(b) The proportional price increase is given by
The percentage price increase is the proportional price increase times 100%, so it is 0.30 x 100% = 30%.
(c) The price ratio is the proportional price increase plus 1.
(d) The rules for converting a proportional price rise into a price ratio and vice versa may be written as follows:
The rules for converting a percentage price rise into a price ratio and vice versa are:
This activity shows that information about a price change can be given either as a proportional or a percentage price change, or as a price ratio: the methods are equivalent. You may care to think about whether the same is true for weighted means. Is a weighted mean of percentage price increases equivalent to the weighted mean of the corresponding price ratios? In the next activity, you will do a calculation similar to the one you did in Activity 16, but this time using price ratios instead of percentage price increases, so you will be able to test whether or not the two methods produce equivalent answers. The weights involved are those used in Activity 16.
(a) Convert the percentage price increases in Table 15 into price ratios.
(b) Use your calculator to find the weighted mean of the price ratios for 2004 relative to 1990, based on these five items. This time the price ratios are the values being considered and the 1990 average weekly expenditures are to be used as weights.
(a) The price ratios are given in the table below.
| Item | Price ratio (2004 price 4÷1990 price) | Average 1990 weekly bill (pence) |
|---|---|---|
| Large loaf (white) | 1.30 | 288 |
| Milk | 1.17 | 443 |
| Eggs | 1.40 | 52 |
| Potatoes | 3.31 | 94 |
| Sugar | 1.17 | 23 |
| Total | 900 |
(b) If the price ratios are labelled r and the weights w, then Σrw = 1303.56 and Σw = 900. So the weighted mean of the price ratios is:
(If you calculate this using ‘1-Var Stats?, you will get the result directly without working out the two sums.)
In Activity 16, you found that the weighted mean of the percentage price increases between 1990 and 2004 was 44.84% (before rounding), and this corresponds to a proportional price increase of 0.4484. Now, in Activity 19, you have found that the weighted mean of the price ratios using the same weights is 1.4484. Since a price increase of 44.84% is equivalent to a price ratio of 1.4484, this example suggests that it does not matter whether percentage (or proportional) price increases or price ratios are used to find the ‘average? percentage price increase. The two answers will always be equivalent and this is an important point: as you will see in Section 5, a weighted mean of price ratios is used by the UK Government to assess changes in prices. The mean price ratio can therefore be converted directly into a proportional or a percentage price increase using the method described in the comment on Activity 18.
So the ‘average? percentage price increase can be found by using price ratios. This illustrates a general point about mathematics, namely that there is often more than one way to perform a calculation.
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