E
QUIVALENT
R
ATIOS
1
To make an orange drink you need 1 part juice and 5 parts water. This can be written as a RATIO
juice : water
1 : 5
The sign in the middle shows it is a ratio It doesn’t matter whether we are making
Glasses of orange drink It is still 1 part
Or juice to
jugs of orange drink 5 parts water
The ratio is 1:5
We write it in the same order as the ingredients are listed.
What if we were to make 2 glasses of the drink?
The ratio of juice to water is 1:5
But we are making 2 glasses x2 x2
We have 2 parts juice : 10 parts water
The quantities used have been doubled, but the ratio we use them in stays the same.
REMEMBER:
Whatever you do to increase one part of the ratio you also do to the other part of
the ratio.
A. Now try these:
1) How much juice and water will you need for 3 glasses of the drink
juice water 1 : 5
___ parts juice : ___ parts water
2) How much will you need for 7 glasses of orange drink?
1 : 5
X3 X3
:
X7 X7
3) How many parts of juice and water will you need for (a) 5 glasses of the drink ___ : ___
(b) 8 glasses of the drink ___ : ___ (c) 10 glasses of the drink ___ : ___
B. Now try increasing these ratios
1) Each house built in a new estate has 7 windows The ratio of houses to windows is 1 : 7
How many windows will you need for (a) 2 houses 2 : ___
(b) 4 houses 4 : ___ (c) 10 houses 10 : ___ (d) 15 houses 15 : ___
2) In a drop-in centre there are 6 chairs around each table.
The ratio is 1 : 6
How many chairs are there around (a) 5 tables 5 : ___
(b) 6 tables 6 : ___ (c) 10 tables 10 : ___ (d) 12 tables 12 : ___
3) Each day a grocer sells carrots, apples and potatoes in the ratio 1 : 2 : 5
What will he sell in (a) 2 days
A. 1) 3 glasses 3 : 15
2) 7 glasses 7 : 35
3) 5 glasses 5 : 25
4) 8 glasses 8 : 40
5) 10 glasses 10 : 50
B. 1) (a) 2 : 14
(b) 4 : 28
(c) 10 : 70
(d) 15 : 105
2) (a) 5 : 30
(b) 6 : 36
(c) 10 : 60
(d) 12 : 72
3) (a) 2 : 4 : 10
(b) 3 : 6 : 15
(c) 5 : 10 : 25
E
QUIVALENT
R
ATIOS
2
We use ratios to show how we share things or mix things. To make pastry we need
1 part butter : 2 parts flour
This ratio works whether you want to make a small amount of pastry
or a large amount
The ratio of butter to flour is 1 : 2
We might need different quantities or weights of pastry, but we keep using 1 part butter to 2 parts flour.
So . . . if we make 5 times the amount of pastry we increase both sides of the ratio by x5
This is an EQUIVALENT RATIO.
We are putting in more of each ingredient, but the mixture is the same.
(Remember that we write the ratio in the same order as the ingredients)
X5 X5
5 : 10 butter flour
1 : 2
That to make pastry we know the ratio of butter to flour is 1 : 2
A. Now complete these ratios
1) (a) How much flour would you need for 2 ounces of butter?
1 : 2
(b) 4oz of butter?
1 : 2
(c) 8 oz of butter? 1 : 2 = 50 :
2) How much butter would you need for (a) 6 ounces of flour?
1 : 2
X2 X2
2 :
X X
4 :
X3 X3
6
(b) 12 ounces of flour? 1 : 2 = ___ : 12
(c) 200 grams of flour? 1 : 2 = ___ : 200
(d) 350 grams of flour? 1 : 2 = ___ : 350
Now try these
3) A plant needs 1 part plant food for every 10 parts of water.
The ratio is 1 : 10
How much water will be mixed with (a) 1 fluid ounce of plant food? (b) 1 litre of plant food?
(c) 10 ml of plant food? (d) 2 ml of plant food? (e) 100 ml of plant food?
4) A car travels 9 miles on a litre of petrol The ratio is 9 : 1
How far will it travel on
(a) 3 litres of petrol ___ : 3? (b) 6 litres of petrol ___ : 6? (c) 9 litres of petrol ___ :9? (d) 11 litres of petrol ___ : 11? (e) 25 litres of petrol ___ : 25?
In a meeting the ratio of women to men is
4 : 3
This means if there are 8 women there will be 6 men
B. Now try these
1) How many men would there be at this meeting if there were
(a) 12 women? (b) 20 women? (c) 32 women?
2) How many women would there be at the meeting if there were
(a) 9 men? (b) 12 men? (c) 18 men? (d) 30 men?
X2 X2
8 6: 4 : 3
3) 5 miles is the same distance as 8 km The ratio is 5 : 8
How many km will you have travelled if you go for (a) 10 miles?
(b) 25 miles? (c) 45 miles? (d) 60 miles? (e) 7.5 miles? (f) 12.5 miles?
How many miles would you have travelled if you go (g) 24 km?
(h) 48 km? (i) 32 km? (j) 28 km? (k) 84 km?
4) A grocer sells carrots, apples and potatoes in the ratio of 1 : 2 : 5
How many tonnes of apples will he sell if he sells (a) 3 tonnes of carrots?
Answers: Equivalent Ratios 2
A. 1) (a) 2 : 4
(b) 4 : 8
(c) 8 : 16
(d) 50 : 100
2) (a) 3 : 6
(b) 6 : 12
(c) 100 : 200
(d) 175 : 350
3) (a) 10 fluid ounces of water 1 : 10
(b) 10 litres of water 1 : 10
(c) 100 ml of water 10 : 100
(d) 20 ml of water 2 : 20
(e) 1000 ml (1 litre) water 100 : 1000
4) (a) 27 : 3
(b) 54 : 6
(c) 81 : 9
(d) 99 : 11
B. 1) (a) 9
(b) 15
(c) 24
2) (a) 12 women
(b) 16 women
(c) 24 women
(d) 40 women
3) (a) 16 km
(b) 40 km
(c) 72 km
(d) 96 km
(e) 12 km
(f) 20 km
(g) 15 miles
(h) 30 miles
(i) 20 miles
(j) 17.5 miles
(k) 52.5 miles
4) (a) 6 tonnes
(b) 20 tonnes
(c) 4 tonnes
(d) 6 tonnes
C
ANCELLING
R
ATIOS
In a college the number of male to female students is
Male female
1000 : 2000
The most useful way of working with ratios is when they are in their simplest form, which is
Male female
1 : 2
When writing a ratio in its simplest form we follow the same rules as for cancelling down fractions.
Divide both sides of the ratio by the same number.
Example 1 Cancel the ratio to its simplest form
5 : 15
5 : 15
÷5 ÷5
1 : 3
Example 2 Cancel the ratio to its simplest form
6 : 9
6 : 9
Example 3 In a hospital there are 10 nurses for every 100 patients. How many patients does each nurse look after?
10 : 100
÷10 ÷10
1 : 10
Each nurse looks after 10 patients.
Make sure you understand the examples
Now have a go at these
Cancel these ratios down to their simplest form.
1) 6 : 12 2) 8 : 24 3) 3 : 21
4) 4 : 10 5) 9 : 12 6) 15 : 25
7) 6 : 30 8) 25 : 150 9) 11 : 33
10) 10 : 45
11) A farmer uses 20 bags of fertilizer for 4 fields. How many bags for each field?
12) A recipe for shortbread uses 200g butter, 100 grams sugar and 300 grams flour. Write this as a ratio, then cancel it to its simplest form.
13) A car’s petrol tank holds 9 gallons. On a full tank the car can travel 270 miles.
14) Sonya tiles her bathroom using 96 plain tiles and 12 patterned tiles. What is the ratio of plain to patterned tiles in its simplest form?
Answers: Cancelling Ratios
1) 1 : 2 2) 1 : 3 3) 1 : 7
4) 2 : 5 5) 3 : 4 6) 3 : 5
7) 1 : 5 8) 1 : 6 9) 1 : 3
10) 2 : 9
11) 5 bags
12) 200 : 100 : 300, 2 : 1 : 3
13) 9 : 270, 1:30
14) 8 : 1
S
PLITTING
R
ATIO
1
Splitting a quantity in a ratio means sharing it into unequal parts.
Example
Paul is 3 and John is 2
A gift of £30 is to be split between them in the ratio of their ages.
So Paul gets 3 parts and John gets 2 parts.
This means the money has to be divided into 5 parts (3 + 2)
6 5 ) 3 0
So 1 part is £6
Paul gets 3 parts 3 x 6 = £ 18
John gets 2 parts 2 x 6 = £ 12
(Check: £ 18 + £ 12 = £ 30) Paul : John
Try these questions
1) Jam is made in a ratio of 2:1 fruit to sugar. How much fruit and sugar are needed to make 18kg of jam?
2) How much sand and cement are needed to make 40kg of concrete in a ratio of 3:1?
3) David and Surita receive a gift of £65 in the ratio 3:2. How much will they get each?
4) To make pastry, flour and margarine are used in a ratio of 2:1. How much of each is needed to make 750g of pastry?
5) Jamie makes 1500mls of lemon squash. The juice is mixed with water in the ratio 3:7. How much of each is needed.
6) Chris and Les share a lottery win of £ 400 in the ratio of their ticket prices. Chris paid £ 5 and Les paid £ 3. How much will they each win?
7) The ratio of male to female students in a college is 4:5. There are 2070 students altogether. How many men and women are there?
9) A roadside is planted with trees in the ratio of 1 oak to every 4 birch. Altogether the farmer wants 65 trees. How many of each will there be?
Answers: Splitting Ratio
1) Fruit 12kg
Sugar 6kg
2) Sand 30kg
Cement 10kg
3) David £39
Surita £26
4) Flour 500g
Margarine 250g
5) Juice 450mls
Water 1050mls
6) Chris £250
Les £150
7) Men 920
Women 1150
8) Grey 315
Terracotta 225
9) Oak 13
Birch 52
10) Copper 1450g
S
PLITTING
IN
A
R
ATIO
2
A quantity may be split in more than 2 parts.
Example 1
A 2000ml bottle of lemonade is shared between 3 children in the ratio 2:3:5. How many millilitres does each drink?
2 + 3 + 5 = 10 parts
200 10 ) 2000
So 1 part = 200mls
The first child gets 2 x 200 = 400mls The next child gets 3 x 200 = 600mls The third child gets 5 x 200 = 1000mls
Sometimes you can cancel before you start, to make the numbers easier to handle.
Example 2
A flapjack is made in the ratio of 40 grams oats to 30 grams
margarine and 30 grams fruit. How much of each is needed to make 3000 grams of mixture?
40 : 30 : 30 can all be cancelled by 10
The total number of parts is 10 ( 4 + 3 + 3 )
300 10 ) 3000
So 1 part is 300 grams
The mixture will contain: Oats 4 x 300 grams = 1200g Margarine 3 x 300 grams = 900g Fruit 3 x 300 grams = 900g
Try these questions
1) A recipe for shortbread uses butter, sugar and flour in the ratio 2:1:2. How much of each is used in 500 grams of shortbread?
2) Concrete is mixed using cement, sand and gravel in the ratio 1:3:6. If we want to make 6 kilograms of concrete, how much of each do we need?
(Hint: change kilograms to grams first.)
3) Apples picked from an orchard were sorted into large, medium and small in the ratio 2:3:10. How many of each were there if 660 apples were picked?
4) A small sample of punch is made using 45mls of lemonade to 20mls of spirit and 25mls of fruit juice. How much of each would be needed to make 18 litres of punch?
6) Prize money of £50 is shared in the ratio of ticket prices. Jack paid £2, Anna paid £1 and Stacey paid £1. How much should they get each?
7) A bonus of £ 2 000 is split between workers in the ratio of their wages. Marie earns £ 16 000, Sam earns
£ 8 000 and Joe earns £ 40 000. How much will they each get?
8) A salad dressing is mixed in the ratio of 5 parts olive oil to 3 parts lemon juice to 1 part mustard. How much of each is needed for 1.8 litres of dressing?
9) The angles in a triangle are in the ratio 2:3:10. What are the sizes of the angles? (Reminder: there are 180° in a triangle.)
Answers: Splitting in a Ratio 2
1) Butter 200g
Sugar 100g
Flour 200g
2) Cement 600g (0.6kg)
Sand 1800g (1.8kg)
Gravel 3600g (3.6kg)
3) Large 88
Medium 132
Small 440
4) 9 : 4 : 5
Lemonade 9L
Spirit 4L
Fruit juice 5L
5) 2 : 1 : 5 : 6
Bacon 30
Onion 15
Chicken 75
Cheese 90
6) Jack £ 25.00
Anna £ 12.50
Stacey £ 12.50
7) 2 : 1 : 5
Marie £500
Sam £250
Joe £1250
8) Oil 1000ml (1L)
Lemon 600ml (0.6L)
Mustard 200ml (0.2L)
9) 24°
36° 120°
10) Bus 180°
Tram 80°
D
IRECT
P
ROPORTION
If we know what 1 lemon costs then we can work out what 4 lemons cost.
1 lemon costs 20p
4 lemons cost 20p x 4 = 80p
Can you see that the quantity and price are directly related? As the quantity goes up so does the price.
This is direct proportion.
1) What will be the cost of: (a) 7 lemons?
(b) 9 lemons? (c) 20 lemons?
2) 1 kilo of carrots costs 37p What will be the cost of: (a) 2 kilos of carrots (b) 5 kilos of carrots (c) 8 kilos of carrots
3) 1 deckchair costs £12.98 What will be the cost of: (a) 3 deckchairs
Sometimes we have to use another step to work it out.
4 ounces of toffee cost 80p How much will 6 ounces cost?
If we start off by finding out what 1 ounce of toffee costs
1 oz = 80p ÷ 4
1 oz = 20p
Now we can find out what 6 ounces cost.
1 oz for 20p 6 oz for 20p x 6 6 oz for £1.20
4) So if 4 ounces of toffee cost 80p what will these cost? (a) 3oz
(b) 2oz (c) 8oz (d) 10oz (e) 16oz
5) Oranges are 5 for £1.20 (a) How much will 7 cost? (b) How much will 3 cost?
6) Mary jogs 3 miles in 36 minutes.
7) Flour costs 35p for 2.5 kg. How much will these cost? (a) 0.5 kg
(b) 2 kg (c) 5 kg (d) 6.5 kg
8) Souhad pays £2.10 for 3 cauliflowers. How much will 8 cost?
9) Gail buys biros. They cost £2.60 for 20. (a) How much will 50 cost?
(b) How much will 15 cost? (c) How much will 1 cost? (d) How much will 123 cost?
10) Jane can type 36 words in 30 seconds. What can she type in
(a) 15 seconds? (b) 1 minute?
Answers: Direct Proportion
1) (a) £1.40 7) (a) 7p
(b) £1.80 (b) 28p
(c) £4.00 (c) 70p
(d) 91p
2) (a) 74p
(b) £1.85 8) £5.60
(c) £2.96
9) (a) £6.50
3) (a) £38.94 (b) £1.95
(b) £90.86 (c) 13p
(c) £129.80 (d) £15.99
4) (a) 60p 10) (a) 18 words
(b) 40p (b) 72 words
(c) £1.60 (c) 162 words
(d) £2.00 (d) 12 words
(e) £3.20
5) (a) £1.68
(b) 72p
6) (a) 96 minutes or 1 hour 36 minutes
I
NDIRECT
P
ROPORTION
Have a look at this example:
2 builders roof a house in 6 hours so it would take 1 builder twice as long
6 hours x 2 = 12 hours
How long would it take 3 people? 1 builder takes 12 hours
3 builders take 12 ÷ 3 = 4 hours
You can see that as one quantity increases, another decreases. This is indirect proportion.
Look at this one:
Jo, Jill and Jack can paint a room in 3 hours 20 minutes. How long will it take them if Jenny and John help too?
It will take 1 person
3 hours 20 minutes x 3 = 10 hours
It will take 5 people 10 hours ÷ 5
= 2 hours
A. Try these
The first 2 are set out for you.
1) 7 typists take 8 hours to copy type a book. How long will it take 4 typists?
7 typists take 8 hours
1 typist takes 8 hours x 7 = ___ hours so 4 typists take ___ hours ÷ 4 = ___ hours.
2) 4 gardeners take 2 hours to plant out pansy seeds. How long would it take 6 people?
4 gardeners take 2 hours
1 gardener takes 2 hours x 4 = ___ hours so 6 gardeners take ___ hours ÷ 6 = ___ hours.
(remember to work in minutes)
3) 2 bricklayers take 6 hours to build a wall. How long would it take 4 of them?
4) It takes Arthur and Rebecca 5 hours to clean their house.
How long would it take 3 people?
5) 5 kitchen hands take 12 minutes to chop vegetables. (a) How long would it take 6 people?
(b) How long would it take 15 people?
7) It takes 3 people 7 hours to prepare a wedding buffet. How long would it take
(a) 2 people? (b) 4 people? (c) 7 people?
8) It takes 5 people 4 hours to clear up the wedding buffet.
How long would it take (a) 10 people?
Answers: Indirect Proportion
1) 1 typist takes 56 hours so 4 take 14 hours
2) 1 gardener takes 8 hours so 6 take 1 hour 20 minutes
3) 4 brick layers take 3 hours
4) 3 hours 20 minutes
5) (a) 10 minutes
(b) 4 minutes
6) 10 months
7) (a) 10½ hours
(b) 5¼ hours
(c) 3 hours
8) (a) 2 hours
(b) ½ hour
(c) 10 hours
S
CALE
We use scales and ratio to make models and draw plans. By using these we make sure each part is in proportion.
For example
A company makes children’s and adults’ furniture.
They make the children’s furniture on a scale of 1 : 2 to the adult’s furniture.
childrens adults
1 : 2
So every measurement on the children’s furniture is half of the measurement on the adult’s furniture
Childrens : adults
1 : 2
height of a chair 25cm : 50cm height of a chair
A. Try working out these measurements using the ratio of 1:2
1) A child’s wardrobe is 1m (100cm) in height and 50cm in width.
What will the measurements of an adult’s wardrobe be?
4) An adult’s desk is 150cm long, 90cm high and 80cm wide. What will the measurements of a child’s desk be?
Any scale can be used – but we always increase or decrease each measurement in proportion to the scale.
A child’s bike is made in the scale 1 : 3
compared to an adult bike.
B. Find the dimension of the adult bike if
1) (a) The child’s wheel is 20cm in diameter. (b) The child’s handlebar is 18cm in length. (c) The height of the bike is 33cm.
(d) The length of the bike is 50cm.
2) A statue is made for the city centre based on a scale of 1 : 5 (the statue is 5 times as big as a real person).
Fill in this table
person (in cm) statue
height 180cm _____
arm length _____ 4.5m
chest width _____ 2.35
nose 5cm _____
mouth 6.5cm _____
leg length _____ 4.75m
3) Brian makes a dolls house for his granddaughter. He uses a scale of 1 : 20.
(a) If the height of the dolls house is 45cm what will the height of the real house be?
(b) If the doll’s house kitchen measures 20cm by 17cm, what will this measure in the real house?
(c) The doll’s house bathroom measures 16cm by 17.5cm. What will this measure in the real house?
(d) The real house has a living room measuring 4.2m (420cm) by 3.6m (360cm).
What will this measure in the doll’s house?
(e) The real house has a staircase. Each step is 20cm high. What will this measure in the doll’s house?
(f) How many steps will fit into a 14cm staircase in the doll’s house?
(g) How tall will the staircase be in real life?
Answers: Scale
A. 1) 2m by 1m (or 200cm by 100cm)
2) 80cm
3) 1m by 70cm
4) 75cm long
45cm high 40cm wide
B. 1) (a) 60cm
(b) 54cm
(c) 99cm
(d) 150cm
2) height 900cm or 9m
arm 0.9m or 90cm
chest 0.47m or 47cm
nose 25cm
mouth 32.5cm
leg 0.95m or 95cm
neck 75cm
3) (a) 900cm or 9m
(b) 400cm or 4m and 340cm or 3.4m
(c) 320cm or 3.2m and 350cm or 3.5m
(d) 21cm by 18cm
(e) 1cm
(f) 14
