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Amount of Amount of Orange Squash Water
( cm^3 ) (cm 3 )
1 6 20 120 5 30
7 Ratio and Proportion
7.1 Equivalent Ratios
Orange squash is to be mixed with water in a ratio of 1 : 6; this means that for every unit of orange squash, 6 units of water will be used. The table gives some examples:
The ratios 1 : 6 and 20 : 120 and 5 : 30 are all equivalent ratios, but 1 : 6 is the simplest form. Ratios can be simplified by dividing both sides by the same number: note the similarity to fractions. An alternative method for some purposes, is to reduce to the form 1 : n or n : 1 by dividing both numbers by either the left-hand-side (LHS) or the right-hand-side (RHS). For example: the ratio 4 :10 may be simplified to 44 : 104 ⇒ 1 : 2.
the ratio 8 : 5 may be simplified to 85 : 55 ⇒ 1.6 : 1
Example 1
Write each of these ratios in its simplest form: (a) 7 : 14 (b) 15 : 25 (c) 10 : 4
Solution
(a) Divide both sides by 7, giving 7 : 14 = 77 :^147 = 1 : 2 (b) Divide both sides by 5, giving 15 : 25 = 155 :^255 = 3 : 5 (c) Divide both sides by 2, giving 10 : 4 = 102 :^42 = 5 : 2
Example 2
Write these ratios in the form 1 : n. (a) 3 : 12 (b) 5 : 6 (c) 10 : 42
Solution
(a) Divide both sides by 3, giving 3 : 12 = 1 : 4
(b) Divide both sides by 5, giving
5 : 6 = 1 : (^65) = 1 : 1.
(c) Divide both sides by 10, giving
10 : 42 = 1 : 1042 = 1 : 4.
Example 3
The scale on a map is 1 : 20 000. What actual distance does a length of 8 cm on the map represent?
Solution
Actual distance = 8 ×20 000 = 160 000 cm = 1600 m = 1.6 km
Exercises
- Write each of these ratios in its simplest form: (a) 2 : 6 (b) 4 : 20 (c) 3 : 15 (d) 6 : 2 (e) 24 : 4 (f) 30 : 25 (g) 14 : 21 (h) 15 : 60 (i) 20 : 100 (j) 80 : 100 (k) 18 : 24 (l) 22 : 77
- Write in the form 1 : n, each of the following ratios: (a) 2 : 5 (b) 5 : 3 (c) 10 : 35 (d) 2 : 17 (e) 4 : 10 (f) 8 : 20 (g) 6 : 9 (h) 15 : 12 (i) 5 : 12
Example 1
If 6 copies of a book cost £9, calculate the cost of 8 books.
Solution
If 6 copies cost £9,
then 1 copy costs £^9 6 = £1.
and 8 copies cost £ .1 50 × 8 = £
Example 2
If 25 floppy discs cost £5.50, calculate the cost of 11 floppy discs.
Solution
If 25 discs cost £5.50 = 550p
then 1 disc costs 55025 = 22p
so 11 discs cost 11 × 22 p = 242p
= £2.
Exercises
- If 5 tickets for a play cost £40, calculate the cost of: (a) 6 tickets (b) 9 tickets (c) 20 tickets.
- To make 3 glasses of orange squash you need 600 ml of water. How much water do you need to make: (a) 5 glasses of orange squash, (b) 7 glasses of orange squash?
- If 10 litres of petrol cost £8.20, calculate the cost of: (a) 4 litres (b) 12 litres (c) 30 litres.
- A baker uses 1800 grams of flour to make 3 loaves of bread. How much flour will he need to make: (a) 2 loaves (b) 7 loaves (c) 24 loaves?
- Ben buys 21 football stickers for 84p. Calculate the cost of: (a) 7 stickers (b) 12 stickers (c) 50 stickers.
- A 20 m length of rope costs £14.40. (a) Calculate the cost of 12 m of rope. (b) What is the cost of the rope, per metre?
- A window cleaner charges n pence to clean each window, and for a house with 9 windows he charges £4.95. (a) What is n? (b) Calculate the window cleaner's charge for a house with 13 windows.
- 16 teams, each with the same number of people, enter a quiz. At the semifinal stage there are 12 people left in the competition. How many people entered the quiz?
- Three identical coaches can carry a total of 162 passengers. How many passengers in total can be carried on seven of these coaches?
- The total mass of 200 concrete blocks is 1460 kg. Calculate the mass of 900 concrete blocks.
7.3 Proportional Division
Sometimes we need to divide something in a given ratio. Malcolm and Alison share the profits from their business in the ratio 2 : 3. This means that, out of every £5 profit, Malcolm gets £2 and Alison gets £3.
Example 1
Julie and Jack run a stall at a car boot sale and take a total of £90. They share the money in the ratio 4 : 5. How much money does each receive?
Solution
As the ratio is 4 : 5, first add these numbers together to see by how many parts the £90 is to be divided. 4 + 5 = 9 , so 9 parts are needed. Now divide the total by 9. 90 9 =^10 , so each part is £10.
- Simon, Sarah and Matthew are given a total of £300. They share it in the ratio 10 : 11 : 9. How much does each receive?
- In a fruit cocktail drink, pineapple juice, orange juice and apple juice are mixed in the ratio 7 : 5 : 4. How much of each type of juice is needed to make: (a) 80 ml of the cocktail, (b) 1 litre of the cocktail?
- Blue, red and yellow paints are mixed to produce 200 ml of another colour. How much of each colour is needed if they are mixed in the ratio: (a) 1 : 1 : 2, (b) 3 : 3 : 2, (c) 9 : 4 : 3?
- To start up a small business, it is necessary to spend £800. Paul, Margaret and Denise agree to contribute in the ratio 8 : 1 : 7. How much does each need to spend?
- Hannah, Grace and Jordan share out 10 biscuits so that Hannah has 2, Grace has 6 and Jordan has the remainder. Later they share out 25 biscuits in the same ratio. How many does each have this time?
7.4 Linear Conversion
The ideas used in this unit can be used for converting masses, lengths and currencies.
Example 1
If £1 is worth 9 French francs, convert: (a) £22 to Ff, (b) 45 Ff to £, (c) 100 Ff to £.
Solution
(a) £22 = 22 × 9 = 198 Ff
(b) 1 Ff = £^19
so 45 Ff = 45 ×^19
= (^459) = £
(c) 100 Ff = 100 ×^19
= (^1009)
= £11^19 = £11.11 to the nearest pence
Example 2
Use the fact that 1 foot is approximately 30 cm to convert:
(a) 8 feet to cm, (b) 50 cm to feet, (c) 195 cm to feet.
Solution
(a) 8 feet = 8 × 30 = 240 cm
(b) 1 cm = 30 1 feet
so 50 cm = 50 × 301
= (^53)
= 1 23 feet
(c) 195 cm = 195 × 301
= (^19530)
= (^132)
= 6 12 feet
Example 3
If £1 is worth $1.60, convert: (a) £15 to dollars (b) $8 to pounds.
- On a certain day, the exchange rate was such that £1 was worth $1.63. Use a calculator to convert the following amounts to £, giving each answer correct to the nearest pence. (a) $100 (b) $250 (c) $75.
- The Japanese currency is the Yen (Y). The exchange rate gives 197 Yen for every £1. Using a calculator, convert the following amounts to pounds, giving each answer correct to the nearest pence. (a) 1000 Y (b) 200 Y (c) 50 000 Y.
- A weight of 1 lb is approximately equivalent to 450 grams. There are 16 ounces in 1 lb. Give answers to the following questions correct to 1 decimal place. (a) Convert 14 oz to lb. (b) Convert 200 grams to lb. (c) Convert 300 grams to ounces.
- If £1 is worth 2.8 German Marks (DM), and 1 DM is worth 2800 Italian Lira (L), use a calculator to convert: (a) 800 DM to £, (b) 10 000 L to DM, (c) 50 000 L to £.
- There are 8 pints in one gallon. One gallon is equivalent to approximately 4.55 litres. Use a calculator to convert: (a) 12 pints to litres, (b) 20 litres to pints. Give your answers correct to 1 decimal place.
7.5 Inverse Proportion
Inverse proportion is when an increase in one quantity causes a decrease in another. The relationship between speed and time is an example of inverse proportionality: as the speed increases, the journey time decreases, so the time for a journey can be found by dividing the distance by the speed.
Example 1
(a) Ben rides his bike at a speed of 10 mph. How long does it take him to cycle 40 miles? (b) On another day he cycles the same route at a speed of 16 mph. How much time does this journey take?
Solution
(a) Time = 1040 (b) Time = 4016 = (^2 )
= 4 hours = 2 12 hours Note: Faster speed ⇒ shorter time.
Example 2
Jai has to travel 280 miles. How long does it take if he travels at: (a) 50 mph, (b) 60 mph? (c) How much time does he save when he travels at the faster speed?
Solution
(a) Time = (^28050) = 5.6 hours = 5 hours 36 minutes
(b) Time = 280 60 = 4 23 hours = 4 hours 40 minutes (c) Time saved = 5 hours 36 mins – 4 hours 40 mins = 56 minutes
Example 3
In a factory, each employee can make 40 chicken pies in one hour. How long will it take: (a) 6 people to make 40 pies, (b) 3 people to make 240 pies, (c) 10 people to make 600 pies?
- A person can make 20 badges in one hour using a machine. How long would it take: (a) 4 people with machines to make 20 badges, (b) 10 people with machines to make 300 badges, (c) 12 people with machines to make 400 badges?
- A train normally complete a 270-mile journey in 4 1 2 hours. How much faster would it have to travel to complete the journey in 4 hours?
- On Monday Tom takes 15 minutes to walk one mile to school. On Tuesday he takes 20 minutes to walk the same distance. Calculate his speed in mph for each day's walk.
- Joshua shares a 2 kg tin of sweets between himself and three friends. (a) How many kg of sweets do they each receive? (b) How much less would they each have received if there were four friends instead of three?
- Nadina and her friends can each make 15 Christmas cards in one hour. How long would it take Nadina and four friends to make: (a) 300 cards, (b) 1000 cards?
