RATIOS INVOLVING PERCENTAGES
Ratios
A ratio is a comparison of two quantities with the same units. For example a dilution of 1 mL
to 10 mL is 1:10. A ratio has no units.
Simplifying ratios
 Divide each number by any common factor
 3:6:9 simplifies to 1:2:3
 14:35:49 simplifies to 2:5:7
 Quantities must be in the same units before they can be expressed as a ratio.
 A cake requires 1 hour of cooking time compared with cookies which require 15
minutes of cooking time. What is the ratio of cooking time for the cake to that of
the cookies?
Cake : cookies
1 hour : 15 minutes which becomes 60 minutes : 15 minutes
without units this is 60 : 15 which simplifies to
4:1
 1 cm : 1 km becomes 1 cm : 100 000 cm which simplifies to 1 : 100 000
 A two stroke motor has oil added to the petrol in the ratio
300 mL oil to 4 L petrol.
This can be written as
Oil : Petrol = 300 mL : 4000 mL
Once the two quantities have been converted to the same unit, the unit notation is
dispensed with, as a ratio does not include any units.
300 : 4000
Now simplify the unit by dividing by 100, Oil : Petrol = 3 : 40
When only two quantities are involved this can be written as:
However, often three or more quantities are dealt with so the format 3:40 is more
common.
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Updated Feb 2013
Dividing a quantity in a given ratio
 Divide $36 in the ratio 1:3:8
The total number of parts is 1+3+8=12
$36 divided by 12 is $3 which is one part
Multiply each number in the ratio by 3 to get $3 : $9 : $24
Check the answer by adding up the parts of the ratio to get $36.
 Aroma Brand coffee is a mixture of Grade A and Grade B coffee powder in the ratio
2 : 3. If 1 kg of Grade A costs $12 and 1 kg of Grade B costs $8, what is the cost of 1
kg of the mixture?
Grade A : Grade B = 2 : 3 = 2 kg : 3 kg
5 kg of the mixture contains 2 kg Grade A and 3 kg Grade B
Cost of 2 kg Grade A = 2  $12 = $24
Cost of 3 kg Grade B = 3  $8 = $24
Cost of 5 kg of mixture = $48
Cost of 1 kg of mixture = $48  5 = $9.60
Dilution ratio
Solutions are often diluted, water being added to the concentrate
 Hydrochloric Acid (1:1) means one volume of acid to one volume of water.
 Sulphuric Acid (1:4) means one volume of concentrated acid to four volumes water.
 What volumes of acetic acid and water would be used to prepare 300 mL of 1 : 5
solution?
Ratio is 1 : 5, so we have 1 + 5 = 6 volumes
300 mL divided by 6 is 50 mL which is the one volume of acid
5  50 mL = 250 mL = volume of water
So add 50 mL of acid to 250 mL of water
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Ratios involving percentages
It is useful to use units to help.
 A certain alloy contains 70% copper, 16% zinc and 14% tin by weight. Calculate the
weight of tin in 50 kg alloy.
14% tin by weight means
14 kg tin
14 g tin
or
100 g alloy
100 kg alloy
So
70 kg copper
 50 kg alloy  35 kg copper
100 kg alloy
So
16 kg zinc
 50 kg alloy  8 kg zinc
100 kg alloy
So
14 kg tin
 50 kg alloy  7 kg tin
100 kg alloy
Parts per million (PPM)
This gives the ratio of one part of one substance to one million parts of another substance.
 The concentration of an air pollutant may be quoted as 10 ppm. This means:
10 L pollutant to 1 000 000 L air.
Pollutant : air = 10 : 1 000 000
or 1 : 100 000
 How much pollutant is there in a room of capacity 300 000 L?
10 L pollutant
 300 000 L air  3 L pollutant
1 000 000 L air
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Direct proportion
When we use the term proportion, we are still applying ratios. Quantities which increase in
the same ratio are in direct proportion.
 The following table shows how the cost of notebooks increases.
Number of notebooks bought
Cost
1
$2
2
$4
3
$6
4
$8
5
$10
Notice that the cost increases by $2 for each additional notebook bought. Therefore
if the number of notebooks is doubled, the cost is also doubled. i.e. the cost
increases proportionally.
Notice that 4 notebooks : 2 notebooks
= 4:2 = 2:1
Cost of 4 notebooks : Cost of 2 notebooks = $8 : $4 = 2 : 1
Similarly:
3 notebooks : 1 notebook = 3 : 1
Cost of 3 notebooks : Cost of 1 notebook = $6 : $2 = 3 : 1
 Find the value of x if
7 : 13 = 21 : x
7 : 13
21 : x
( 21 = 7  3 )
73 : x
( so x = 13  3 = 39)
 A piece of string is cut into two parts in the ratio 2 : 3. If the length of the shorter
piece is 12 cm, find the length of the longer piece.
2 parts
3 parts
? cm
12 cm
2 parts = 12 cm so 1 part = 6 cm
Hence the longer piece of string is 3  6cm = 18 cm
 10 writing pads cost $11. How much will 2 pads cost?
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10 pads cost $11
1 pad costs
$11
= $1.10
10
2 pads cost $1.10  2 = $2.20
 A car requires 21 litres of petrol to travel 320 km. How much petrol is needed to
travel 80 km?
320 km requires
1 km requires
80 km requires
21 L
21
L
320
21
21
L = 5.25 L
 80 L =
320
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Inverse proportion
 A man has to pay income tax of $1200. He is given the following instalment options:
Amount paid per month
Time taken in months
$100
12
$200
6
$300
4
$400
3
$600
2
Increasing
amount
Decreasing
number of
months
Notice that the time taken for payment decreases as the amount paid per
month is increased.
Therefore if the amount paid per month is doubled, the time taken for payment
is halved.
Similarly if the amount paid per month is tripled, the time taken is
1
of the
3
original time.
Also when you multiply the time taken by the amount repaid you always get the
total sum repaid which is $1200.
Therefore one quantity increases at the same rate of decrease of the other
quantity.
 If an emergency pack of food can last 10 men for 6 days, how long would the food
last if there were 15 men?
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The food will last a shorter time for more men, so the amount are inversely
proportional.
10 men can last 6 days
so
1 man can last 6  10 days = 60 days
and 15 men will last
60
days = 4 days.
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 At a speed of 60 km/h, a car takes 20 minutes to travel a certain stretch of road. At
what speed must it travel the same stretch of road for it to take only 15 minutes?
20 minutes require a speed of 60 km/h
1 minute requires a speed of 60  20 km/h
15 minutes require a speed of
60  20
km/h = 80 km/h
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Ratio and proportion exercises
1. Write the following ratios in their simplest form:
a) 33:44
b) 4:16
c) 3:120
d) 63:18
e) 10 mm:1 m
f) 27:3
g) 1 cm:1 km
h) 36:45:81
i) 50 mL:1 L
2. Divide the following numbers in the given ratios:
a) 200 into 2:2:4
b) 140 into 2:5:3:4
c) 49 into 2:5
d) 3570 into 3:5:2
e) 150 into 3:5:7
f) 275 into 2:4:5
3. Divide a resistance of 1500 ohms into 2 resistors in the ratio 2:3.
4. The centre of gravity of a tapering beam 24 m long divides the beam in the ratio1:3.
How far from the nearer end is the centre of gravity?
5. On a job the ratio labour costs : materials is 3 : 4. Find the labour costs if the total cost of
the job is $1645.
6. What volumes of hydrochloric acid and water would be used to prepare 800 mL of 1 : 24
solution?
7. Four colours are to be mixed in the ratio of 4 : 5 : 8 : 3 to produce 25 litres of paint. How
many litres of each are needed for the mixture?
8. A drawing of a machine is made to a scale 1 : 24.
a) What length would you draw for a machine part measuring 260 mm?
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Updated Feb 2013
b) What is the actual diameter of a flywheel drawn 22.5 cm long?
9. A brass contains 7 parts of weight of copper to 3 of zinc. Find the weight of copper in
500 kg of the brass.
10. Find the quantities in cubic metres of cement, sand and aggregate respectively to cover
a rectangular area of 5 m by 4 m to a depth of 25 cm if the concrete mixture is in the
ratio 1 : 3 : 6.
11. An alloy contains a mixture of 55% copper, 27% nickel and 18% zinc. How much of each
to make 2 kg of the alloy?
12. The ratio of water to flesh and bone in humans is 47 : 3. How much of a 70 kg person is
water?
13. Two men form a partnership with $2000 and $3000 respectively. How should a profit
of $850 be shared?
14. Tom, Dick and Jeff invest $7000, $5000 and $20 000 respectively in a business and the
profits for the year are $5000. How should the profits be shared?
15. The ratio of lead, tin and antimony in a solder has to be in the ratio 6 : 5 : 1. If 85 kg of
solder is required, what is the mass of each component?
16. A welder and her apprentice earn $85.70 in a weekly bonus. If they share it in the ratio 7
: 4, how much will each receive?
17. If a 300 mL bottle of vinegar contains 5.4% acetic acid by volume and the rest is water,
calculate the volume of water present in the bottle.
18. A fertiliser contains 6.8% nitrogen, 30% phosphoric acid, 5% soluble potash and 0.05%
copper. How much of each is there in 3 kg fertiliser? Is anything else in the fertiliser?
How can you tell?
19. A compound consists of calcium and oxygen in the ratio 40 : 16 by weight. Convert this
into percentages by weight of each element in the compound.
20. Two cog wheels A and B are geared together. If A has 60 teeth and B has 24, calculate:
a) the ratio of B’s speed to A’s;
b) the number of revolutions B makes for each one of A.
21. Three metals A B and C are mixed in an alloy in the ratio by weight of 6 : 3 : 1.
a) What weight of metal B is there in 1 kg of the alloy?
b) Metal A costs 10 c/g, metal B 15 c/g and metal C 75 c/g. What weight of metal C is
contained in $15 worth of the alloy?
22. An additive in a solution is found to be at a concentration 0.006 g additive in 2700 g
solution. How many parts per million is this?
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23. The concentration of chlorine in a water treatment plant is 60 g chlorine in a 10 000 kg
tank of water. What is this in ppm?
24. A compound is 0.06 % selenium by weight. Convert this to ppm.
25. A mixture is 8 parts alcohol and 6 parts water. How many litres of each are in 140 L of
the mixture?
26. A room is 6 m long by 4 m wide. What is the ratio of length to width?
27. Two gears have 90 teeth and 20 teeth respectively. What is the ratio of the number of
teeth?
28. One town has a population of 8000 and another has a population of 20 000. What is the
ratio of their populations?
29. Divide 80 apples between two people in the ratio 5 : 3.
30. If people can fly from Chicago to New York in 1 hour 50 min and can go by train in 16
hours, what is the ratio of flying time to train time?
31. 51 students enrolled in a class and 33 of them finished the course.
a) What percentage finished?
b) What is the ratio of the number that finished to the whole number?
32. Four residents owning adjoining properties on the same street with frontages 25 m, 25
m, 50 m and 75 m respectively are assessed a total of $15 000 for paving. Find the
amount each should pay.
33. What is c in the ratio 2 : 3 = c : 12 ?
34. What is b in the ratio 3 : b = 6 : 20 ?
35. If 15 tonnes of coal cost $3600, what will 27 tonnes cost?
36. If 25 workers complete a piece of work in 30 days, in how many days can 40 workers
complete it?
37. Bell metal is 25 parts copper to 12 parts tin. Find the weight of each in a bell weighing
1850 kg?
38. If 8 m of cloth costs $40, what will 24 m cost?
39. If 6 people can dig a ditch in 8 days, in what time can 8 people dig it?
40. Find x in the following:
a) 8 : 4 = 26 : x
b) 8 : x = 24 : 4
c) x : 80 = ¼ : 1
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41. How long will it take $1000 to produce $1000 interest, if it produces $40 in one year?
42. If 15 people can complete a task in 36 days, in how many days can they perform the
same task with the assistance of 9 more people?
43. Two gears, one having 26 teeth and the other 20 teeth, run together. In how many
revolutions of the large wheel will the smaller wheel gain 12 revolutions?
44. If sound travels 1877.7 m in 5½ seconds, how far does it travel in 1 minute?
45. If 16 L of petrol will drive a car 228 km, how many litres will be required to drive the
same car from Auckland to Wellington, a distance of 485 km?
46. A mixture for a casting has 4 parts copper, 3 parts lead and 2 parts tin. How many
kilograms of each metal are in a casting weighing 96kg?
47. On a certain map, 1½ cm represents 50 km. How many kilometres are there between
two cities 5 cm apart on the map?
48. An iron casting weighs 142 kg and costs $72.50. At the same rate, what is the cost of a
casting weighing 255 kg?
49. A steel rail in a railway track expands 0.000 006 36 cm for every centimetre of length, for
each degree increase in temperature. If a steel rail is 32 m long, what is the change in
length due to a change from a winter temperature of 20oC below zero to a summer
temperature of 110oC above zero?
50. A man who is 1.83 m tall casts a shadow 1.52 m long. If a building at the same time has
a shadow 68.58 m long, how high is the building?
51. If 36inches = 91.44 cm, how many centimetres are there in 12 inches?
52. A map scale shows 5 cm represents 150 km. What actual distance is represented by a
map distance of 3.2 cm?
53. Determine the map scale if 1 km = 1000 m.
54. Analysis of carbon dioxide shows that for every 12 g of carbon there are present 32 g of
oxygen.
a) How many grams of carbon are combined with 100 g of oxygen in a sample of
carbon dioxide?
b) If 25 g of carbon dioxide is decomposed into carbon and oxygen, how much of
each is present?
55. A stake 10 m high casts a shadow 8 m long at the same time that a tree casts a shadow
60 m long. What is the height of the tree?
56. The ratio of a propeller rotation rate to that of an engine is 2 : 3. If the engine is turning
at 4200 rpm, what is the propeller rate?
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Updated Feb 2013
1
57. A photographic negative is
what should its length be?
64
1
cm by
38
3
cm. If the width of an enlarged print is
98
cm,
58. An aeroplane travels 550 km in 1 hour and 50 minutes. At the same speed, what
distance would the plane cover in 2 hours 10 minutes?
59. The volume of a gas at a constant temperature is directly proportional to its absolute
temperature. The volume is 23.5 mL at 300oK. Find the temperature of the gas when the
volume is 19.5 mL?
60. An aeroplane flying at 450 km/h covers a distance in 3 hours 15 minutes. At what speed
would it have to fly to cover the same distance in 2 hours 30 minutes?
61. If 10 men can complete a project in 12 days, how many days will it take 15 to complete
it?
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Ratio and proportion exercises solutions
1.
2.
a) 3:4
b) 1:4
c) 1:40
d) 7:2
e) 1:100
f) 9:1
g) 1:100 000
h) 4:5:9
i) 1 :20
a) 50 50 100
b) 20 50 30 40
c) 14 35
d) 1071 1785 714
e) 30 50 70
f) 50 100 125
3. 600 ohms
900 ohms
17. 280 mL
4. 6 m
18. 0.2 kg
5. $705
19. 71.4% 28.6%
6. 32 mL hydrochloric acid
20. a) 60 : 24 b) 2 : 5
768 mL water
0.9 kg
0.2 kg
21. a) 300 g b) 8.3 g
7. 5 L 6.25 L 10 L 3.75 L
22. 2 ppm
8.
23. 6 ppm
a) 10.8 b) 540 cm
24. 600ppm
25. 80 L alcohol, 60L Water
9. 350 kg
10. 0.5 m3
1.5 m3
3 m3
26. 3: 2
11. 1.1 kg
0.54 kg
0.36 kg
27. 9 : 2
12. 66 kg
28. 2 : 5
13. $340 $510
29. 50 apples, 30 apples
14. $1094
15. 43 kg
16. $54.54
$781
35 kg
30. 11 : 96
$3125
31. a) 64.7% b) 33 : 51
7.1 kg
$31.16
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0.002 kg
32. $2142.86, $2142.86, $4285.71,
$6428.57
54. a) 37.5 g b) 6.8 g carbon, 18.2g
oxygen
33. 8
55. 75 m
34. 10
56. 2800 rpm
35. $6480
57. 18.75 cm
36. 18.75 days
58. 650 km
37. 1250 kg copper, 600 kg tin
59. 248.9oK
38. $120
60. 585 km/h
39. 6 days
61. 8 days
40. a) 13
b) 1
1
3
c) 20
41. 25 years
42. 22.5 days
43. 15.6 revolutions
44. 9574.91 m
45. 34.04 L
46. 42.67 kg copper, 32 kg lead, 31.33 kg
tin
47. 166.67 km
48. $130.19
49. 1.83 cm
50. 82.57 m
51. 30.48 cm
52. 96 km
53. 1 : 1
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Updated Feb 2013