Unit 7 Section 1 : Equivalent Ratios

A ratio is used to describe how two quantities are related.

For example, we might say that orange squash is to be mixed with water in a ratio of 1:6.
This means that for every 1 part squash, there will need to be 6 parts of water.
If there was 100ml of squash, there would be 600ml of water.

Another common example of a ratio is a map scale. A particular map scale might be 1:50,000.
In this case it means that 1cm on the map represents 50,000cm in "real-life".
50,000cm = 500m = 0.5km, so 1cm on the map represents half a kilometre. 2cm would therefore represent 1km.

Finding equivalent ratios
The ratio of squash to water in the example above was 1:6, but this could be written as 100:600, or 20:120, or 5:30.
These ratios are equivalent because they have the same meaning - the amount of water is six times the amount of squash.

You can find equivalent ratios by multiplying or dividing both sides by the same number.
This is similar to finding equivalent fractions. Some examples of finding equivalent ratios     
are shown on the right. All the ratios in the diagram are equivalent.

Writing a ratio in its simplest form
A ratio is in its simplest form when both sides are whole numbers and there is no whole
number which both sides can be divided by. In the example opposite, 1:6 is the simplest
form of the ratio.

To write a ratio in its simplest form, keep dividing both sides by the same number until
you can't go any further without going into decimals.

Example: write 160:240 in its simplest form
160:240(divide both sides by 4)
40:60(divide both sides by 2)
20:30(divide both sides by 5)
4:6(divide both sides by 2)
2:3SIMPLEST FORM
Equivalent Ratios

Writing a ratio in the form 1 : n or n : 1
Sometimes we need to write a ratio in the form 1 : n or in the form n : 1.

To write a ratio in the form 1 : n, divide both sides by the left-hand number.
For example, with the ratio 4 : 10 you would divide both sides by 4, giving the equivalent ratio 1 : 2.5

To write a ratio in the form n : 1, divide both sides by the right-hand number.
For example, with the ratio 8 : 5 you would divide both sides by 5, giving the equivalent ratio 1.6 : 1

Practice Questions
Work out the answer to each of these questions then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

(a) Write the ratio 7 : 14 in its simplest form

(b) Write the ratio 15 : 25 in its simplest form

(c) Write the ratio 10 : 4 in its simplest form

(d) Write the ratio 5 : 13 in the form 1 : n

(e) Write the ratio 12 : 3 in the form n : 1

(f) A map has scale 1:20000. What actual distance is represented by 8cm on the map?

(g) A map has scale 1:100000. What distance on the map would represent 20km in real life?

 

Exercises

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Question 1
Write each of the ratios below in its simplest form.
(a)2:6    simplifies to     :
(b)4:20    simplifies to     :
(c)3:15    simplifies to     :
(d)6:2    simplifies to     :
(e)24:4    simplifies to     :
(f)30:25    simplifies to     :
(g)14:21    simplifies to     :
(h)15:60    simplifies to     :
(i)20:100    simplifies to     :
(j)80:100    simplifies to     :
(k)18:24    simplifies to     :
(l)22:77    simplifies to     :
Question 2
Write each of the ratios below in the form 1:n
(a)2:5    is equivalent to     1:
(b)5:3    is equivalent to     1:
(c)10:35    is equivalent to     1:
(d)2:17    is equivalent to     1:
(e)4:10    is equivalent to     1:
(f)8:20    is equivalent to     1:
(g)6:9    is equivalent to     1:
(h)15:12    is equivalent to     1:
(i)5:12    is equivalent to     1:
Question 3
Write each of the ratios below in the form n:1
(a)24:3    is equivalent to     :1
(b)4:5    is equivalent to     :1
(c)7:10    is equivalent to     :1
(d)15:2    is equivalent to     :1
(e)18:5    is equivalent to     :1
(f)6:5    is equivalent to     :1
Question 4
Jennifer mixes 600ml of orange juice with 900ml of apple juice to make a fruit drink.

Write the ratio of orange juice to apple juice in its simplest form.
:

Question 5
A builder mixes 10 shovels of cement with 25 shovels of sand.

Write the ratio of cement to sand:
(a) in its simplest form     :
(b) in the form 1:n     1:
(c) in the form n:1     :1

Question 6
In a cake recipe, 500 grams of butter are mixed with 800 grams of flour.

Write the ratio of butter to flour:
(a) in its simplest form     :
(b) in the form 1:n     1:
(c) in the form n:1     :1

Question 7
In a school there are 850 pupils and 40 teachers.

Write the ratio of teachers to pupils:
(a) in its simplest form     :
(b) in the form 1:n     1:

Question 8
A map is drawn with a scale of 1:50000.

For each of the following distances on the map, calculate the actual distance in real life:
(a) 2 cm on the map represents km
(b) 9 cm on the map represents km
(c) 30 cm on the map represents km

Question 9
A map has a scale of 1:200000. The distance between two towns is 60km.

How far apart are the towns on the map?
cm

Question 10
On a map, a distance of 5cm represents an actual distance of 15km.

Write the scale of the map in the form 1:n.
    1:

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