Ratios and Proportion
Introduction
Ratios occur in mixing things - such as concrete which is made of cement, sand and gravel in a definite ratio. For example, a ratio of 1:3:4 would mean that no matter what volume of concrete you have, 1 part is cement, 3 parts is sand and 4 parts is gravel. An alternative way of stating this is to say that \(\frac{1}{8}\) is cement, \(\frac{3}{8}\) is sand and \(\frac{4}{8}\) \(\left(\mbox{i.e. }\frac{1}{2}\right)\) is gravel.
The aspect ratio which is commonly used in describing the width to length ratios of aircraft wings, is also commonly seen nowadays to describe the ratio of width to height of a TV screen. "Old-style" TV screens feature a 4:3 (1.33:1) aspect ratio, but newer widescreen TVs have a 16:9 (1.78:1) ratio; and most feature films are shot in at least a 1.85:1 ratio.
Simplification of Ratios
Conventionally ratios are stated as whole numbers, or at least decimal numbers (as used for some examples in the Introduction).
Whole number ratios would normally be stated in their lowest terms. A ratio of the form
can be reduced to
by dividing every number by 3 (note the similarity with reducing fractions to their lowest terms).
On the other hand, if a ratio was stated as
(which is much less common) then it would be usual to multiply every term by an appropriate number to
achieve a ratio stated in whole numers only. Here we could multiply every term by 6 to get
or
To go back to the decimal representation mentioned in the
first paragraph, you can see from the examples in the Introduction that when this is used
then at least of one of the ratios is 1 .
This is the crux of the representation, the ratios have been divided through so that
the lowest ratio is 1, and the others are either 1 or more than 1.
For example, the ratio
by dividing the original ratios through by 2.
You do come across ratios of this type where the numbers are quoted as decimals less than 1,
contravening the rule I have just given. Mathematically, there is nothing wrong with this, but
hopefully you can see that this type of represenation would not be as easy to understand.
Example
Of 200 customers of a cafe, 80 ordered and 120 ordered coffee. What is the ratio of
tea drinkers to coffee drinkers ?
State the given figures as a ratio (in the right order)
and then reduce this to its lowest terms if applicable. Here we can see
that both sides can be divided by 40, to give
which is the final answer.
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Calculation
Ratios are closely related to fractions, for example if two items ( A and B ) are connected in the ratio
then
Likewise, if three quantities are related in the ratio
then the quantities will constitute
Example
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If a line 10 cms long is to be divided in the ratio then stated as fractions of the whole, the two lengths will be The two required lengths are then \[ \frac{5}{8} \times 10 = \frac{50}{8} = 6.25\ \mbox{cm} \] |
Example
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If £ 4.50 is to be divided between three people in the ratio
then stated as fractions of the whole, the three amounts will be So the required monetary amounts are then \[ \frac{1}{3} \times 4.50 = \frac{4.5}{3} = £ 1.50 \] \[ \frac{6}{15} \times 4.50 = \frac{27}{15} = £ 1.80\] |
Proportions
Consider a question like
A car travels for 300 kilometres on 35 litres
of petrol. How far will it travel on 54 litres ?
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To explain how to do this in words -
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If needs be, you can start off doing calculations of this type in these two stages, but once you get more practised, you can start to do it in one step. For example, the calculations for the above question would be
= 462.9 (to 1 d.p.)
If a similar (but different) question had been asked
A car travels for 300 kilometres on 35 litres of petrol. How much petrol would be needed for a journey of 369 kilometres ?
Similar logic would be needed, but applied differently.
To explain how to do this in words -
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As before, you can start off doing calculations of this type in these two stages, but once you get more practised, you can start to do it in one step. For example, the calculations for this question would be
= 43.05 litres
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Inverse Proportion
Some problems have an inverse proportion.
Very common ones involve workers doing a particular job - the more workers you have, the less time the job will take (assuming an ideal situation where all workers produce exactly the same, at the same rate).
For problems like these, the technique would be the opposite to that used for 'proportions' previously. There you carried out a two-stage operation, first dividing and then multiplying. For inverse proportion, a two-stage operation is involved, first multiplying and then dividing.
Example
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To explain specifically how to do this in words -
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As explained in other sections, with practise you can conflate these two steps into one line of calculation
Example
Note that the figure of 3 000 articles does not enter into the calculation
(previously the details about the wall that the workers were building did not enter into
the calculation,
apart from knowing that it had been finished).
To explain specifically how to do this in words -
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Conflating these two steps into one line of calculation
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Past Exam Questions
Jake is making 55 biscuits for the playgroup Christmas party.
He has a recipe for 20 biscuits, which requires:
- 150g margarine
- 150g sugar
- 1 egg
- 300g self-raising flour
- 50g ground almonds
- 150g sugar
1. How much flour will he need to make exactly 55 biscuits?
- A 413g
- B 825g
- C 900g
- D 1 650g
2. What is the ratio of ground almonds to sugar to self-raising flour in the recipe?
- A 3 : 6 : 1
- B 6 : 1 : 3
- C 1 : 3 : 6
- D 1 : 6 : 3
This question is about decorating a room
1. The border for the top of the walls costs £ 3.97 per metre. Which estimate is most accurate for the total cost of the border?
- A (8 + 3) x 4 = £ 44
- B (8 + 3 + 8 + 3) = £ 22
- C (8 + 3) x 2 x 4 = £ 88
- D (8 x 3) x 4 = £ 96
2. 3.2 litres of paint are needed to decorate the bedroom. (1 litre = 1 000cm3). The decorator mixes 3 paint colours together. The amounts of paint are in the ratio: 15 parts of Honey Yellow to 12 parts of Aztec Orange to 5 parts of Ravishing Red. How much Ravishing Red will she need?
- A 500cm3
- B 720cm3
- C 1 067cm3
- D 1 200cm3
Wayfleet Hotel had 15 329 guests in 1998
1. 5 748 of the guest used the swimming pool. What is the approximate ratio of those using the pool to those not using the pool ?
- A 3:8
- B 3:5
- C 5:8
- D 5:3
2. 1 904 guests occupied single rooms. About what percentage of guests occupied single rooms ?
- A 6%
- B 12%
- C 25%
- D 125%
3. Another hotel has 56 single rooms and 58 double rooms. What is the best approximation to the maximum number of possible occupancies in one year ? ?
- A 70 000
- B 80 000
- C 90 000
- D 10 000
Here are the scores that 20 people get for a test
| 8, 7, 5, 6, 9 |
| 4, 5, 7, 2, 1 |
| 6, 1, 9, 2, 1 |
| 5, 9, 7, 3, 8 |
To pass the test a person needs to score 8 or more
1. What percentage of people fail to pass by 1 mark ?
- A 3%
- B 15%
- C 25%
- D 75%
2. What is the ratio of the number of people who pass to the number of people who do not ?
- A 1:4
- B 2:5
- C 3:2
- D 1:3
Stage 3
A store has a cafeteria that bakes its own cakes. The recipe for a fruitcake mixture is :
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a. Find the actual weight in grams of flour and baking powder used for 3kg. of the cake mixture.
b. The baking time for a cake depends on its weight and this is shown in a graph.
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Estimate the baking time for the 3kg. cake
c. The original recipe gave the baking temperature as 350° F. Modern ovens use °
C and the conversion formula is
\[ F = \frac{9}{5} C + 32 \]
where F is temperature in Fahrenheit and C is temperature in Centigrade (Celsius)
Calculate the equivalent temperature in °C to appropriate accuracy.