Cambridge IGCSE Mathematics
Exercise 21.2
1 Find the unknown values in the following equivalent ratios. Use whichever method you
find easiest.
a 2:3 = 6:x
e 3:8 = 66:x
b 6:5 = y:20
f 1:5 = 13:y
c 12:8 = 3:y
g x:25 = 7:5
a x:20 = 3:4
e 1:10 = x:6
b 12:21 = x:14
f 8:13 = 2:y
c 2:5 = 8:y
g 4:5 = x:7
d 27:x = 9:2
h 40:9 = 800:y
1 1
:
2: y
i 3:7 = 600:y
j 2:7 = 30:x
k 1.5:x = 6:5
l
6 3
2 Use the equation (cross multiplying fractions) method to find the unknown values in the
following equivalent ratios.
d 3:5 = x :4
h 5:4 = 9:y
3 Say whether these statements are true or false. If a statement is false, explain why it is false.
a
b
c
d
The ratio 1:6 is the same as the ratio 6:1.
The ratio 1:6 is equivalent to 3:18.
The ratio 20:15 can be expressed as 3:4.
If the ratio of a mother’s age to her daughter’s age is 8:1, the daughter will be nine when her
mother is 48 years old.
5
e If Mr Smith’s wages are of Mr Jones’ wages, then the ratio of their wages is 20:32.
8
Applying your skills
4 An alloy is a mixture of metals. Most of the gold used in jewellery is an alloy of pure gold and
other metals which are added to make the gold harder. Pure gold is 24 carats (ct), so 18 carat
18
gold is an alloy of gold and other metals in the ratio 18:6. In other words,
parts pure gold
24
6
and
other metals.
24
a A jeweller makes an 18 ct gold alloy using three grams of pure gold. What mass of other
metals does she add?
b An 18 ct gold chain contains four grams of pure gold. How much other metal does it contain?
c What is the ratio of gold to other metals in 14 ct gold?
d What is the ratio of gold to other metals in 9 ct gold?
5 An alloy of 9 ct gold contains gold, copper zinc and silver in the ratio 9:12.5:2.5.
a Express this ratio in simplest form.
b How much silver would you need if your alloy contained six grams of pure gold?
c How much copper zinc would you need to make a 9 ct alloy using three grams of pure gold?
6 An epoxy glue comes in two tubes (red and black), which have to be mixed in the ratio 1:4.
a If Petrus measures 5 ml from the red tube, how much does he need to measure from the black
tube?
b How much would you need from the red tube if you used 10 ml from the black tube?
7 A brand of pet food contains meat and cereal in the ratio 2:9. During one shift, the factory
making the pet food used 3500 kg of meat. What mass of cereal did they use?
Dividing a quantity in a given ratio
Ratios can be used to divide or share quantities. There are two ways of solving problems like these.
• Method 1: find the value of one part. This is the unitary method.
1 Add the values in the ratio to find the total number of parts involved.
2 Divide the quantity by the total number of parts to find the quantity per part (the value
of one part).
3 Multiply the values in the ratio by the quantity per part to find the value of each part.
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Unit 6: Number
21 Ratio, rate and proportion
• Method 2: express the shares as fractions. This is the ratio method.
1 Add the values in the ratio to find the total number of parts involved.
2 Express each part of the ratio as a fraction of the total parts.
3 Multiply the quantity by the fraction to find the value of each part.
Worked example 4
Share $24 between Jess and Anne in the ratio 3:5.
Method 1
There are 8 parts in the ratio.
This is the value of 1 part.
Jess gets 3 parts:3 × 3 = 9
Anne gets 5 parts:5 × 3 = 15
3+5=8
24 ÷ 8 = 3
Jess gets $9, Anne gets $15.
Method 2
There are 8 parts in the ratio.
3+5=8
3
3
Jess gets of $24 = × 24 $9
8
8
5
5
Anne gets of $24 = × 24 $15 .
8
8
Exercise 21.3
1
Express each part as a fraction of the total
parts and multiply by the quantity.
Divide:
a
c
e
g
200 in the ratio 1:4
50 in the ratio 3:7
600 in the ratio 3:9
300 in the ratio 11:4
b
d
f
h
1500 in the ratio 4:1
60 in the ratio 3:12
38 in the ratio 11:8
2300 in the ratio 1:2:7.
2 Fruit concentrate is mixed with water in the ratio of 1:3 to make a fruit drink.
How much concentrate would you need to make 1.2 litres of fruit drink?
3 Josh has 45 marbles. He shares them with his friend Ahmed in the ratio 3:2.
How many marbles does each boy get?
4 $200 is to be shared amongst Annie, Andrew and Amina in the ratio 3:4:5.
How much will each child receive?
5 A line 16 cm long is divided in the ratio 3:5. How long is each section?
The capital letters N:P:K on fertiliser
bags are the chemical symbols
for the elements. The ratio of
chemicals is always given on packs
of fertiliser.
6 A bag of N:P:K fertiliser contains nitrogen, phosphorus and potassium in the ratio 2:3:3.
Work out the mass of each ingredient if the bags have the following total masses:
a one kilogram
b five kilograms
c 20 kilograms
d 25 kilograms.
7 The lengths of the sides of a triangle are in the ratio 4:5:3. Work out the length of each side
if the triangle has a perimeter of 5.4 metres.
8 A rectangle has a perimeter of 120 cm. The ratio of its length to its breadth is 5:3.
Sketch the rectangle and indicate what the lengths of each side would be.
9 In a group of 3200 elderly people, the ratio of men to women is 3:5.
Calculate how many men there are in the group.
10
Show that the ratio area:circumference for any circle is r:2, where r is the radius of the
circle
b Find the ratio volume:surface area for a sphere of radius r, giving your answer as simply as
possible
a
Unit 6: Number
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