Olympic College - Topic 5 Ratio and Proportions
Topic 5 Ratio and Proportions
1. Ratio
Definition: A ratio is a comparison of two numbers it can be written in three major forms.
5
For example, 5 to 7 also as 5:7 and as 7.
A ratio can be simplified in a similar to simplifying fractions – you find the Greatest Common
Divisor (GCD) of the two numbers and then divide both numbers by the GCD.
Example 1: Simplify the ratio 40 to 55
Solution:
The GCD of 40 and 45 is 5 dividing both numbers by 5 we get
40 to 45 = 8 to 9
Example 2: Simplify the ratio 33 to 63
Solution:
The GCD of 33 and 63 is 3 dividing both numbers by 3 we get 33 to 63= 11 to 21
Example 3: Simplify the ratio 36 : 27
Solution:
The GCD of 36 and 27 is 9 dividing both numbers by 9 we get
36 : 27 = 4:3
Example 4: Simplify the ratio 60 : 80
Solution:
The GCD of 60 and 80 is 20 dividing both numbers by 20 we get 60 : 80 = 3:4
24
Example 5: Simplify the ratio 44
Solution:
Exercise 1A:
The GCD of 24 and 44 is 4 dividing both numbers by 4 we get
24
44
6
= 11
Simplify the following ratios
1.
50 to 40
2.
44 to 80
3.
33 to 36
4.
240 to 400
5.
33 : 9
6.
26 : 39
7.
45 : 18
8.
60 : 80
9.
20
54
10.
20
400
11.
14
21
12.
18
81
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Olympic College - Topic 5 Ratio and Proportions
When you have two ratios then they are said to be “equivalent ratios” if the simplified form of
both ratios are identical.
For example, 8 : 12 and 6 : 9 are equivalent ratios as the ratio 8 : 12 = 2 :3 and 6 : 9 = 2 :3
There is another method to test if two ratios are equivalent.
It is known that the ratios a : b and c: d are equivalent if ad = bc
For example 3 : 17 and 9 : 51 are equivalent ratios as 3(51) = 9(17) both are equal to 153.
Example 1: Which of the following pairs of ratios are equivalent?
Solution:
Exercise 1B:
18
(b) 5 to 15 and 3 to 10
(c)
(a) 14(6) = 21(4)
Both = 84
Equivalent Ratio
(b) 5(10) ≠ 15(3)
50 and 45
Not Equivalent Ratio
(c) 18(18) = 4(81)
Both are 324
Equivalent Ratio
81
and
4
(a) 14 :21 and 4 : 6
18
Which of the following pairs of ratios are equivalent?
1. 4 : 20 and 6 : 30
2. 15 to 10 and 30 to 20
3.
4. 25 :2 and 50 : 4
5. 6 to 81 and 4 to 18
6.
10
25
8
6
and
and
15
20
16
12
2. Solving Ratio Equations
𝑥
7
A “Ratio Equation” is an equation of the form 3 = 2 in order to solve this type of equation we use
the method of “Cross Multiplication”.
𝑥
7
Example 1: Solve the equation 3 = 2
𝑥
Solution:
3
=
7
2
2x = 21
x =
21
2
Cross multiplication
Divide both sides by 2
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Olympic College - Topic 5 Ratio and Proportions
6
Example 2: Solve the equation
6
Solution:
𝑥
𝑥
4
=
2
4x =
12
4𝑥
12
4
x
4
=2
=
Cross multiplication
Divide both sides by 4.
4
3
=
3
Example 3: Solve the equation 5 =
3
Solution:
5
𝑥
60
3𝑥
60
x
𝑥
12
=
3x =
3
12
=
Cross multiplication
Divide both sides by 3.
3
20
=
3
𝑥
Example 4: Solve the equation 4 = 7
3
Solution:
4
21
4𝑥
21
x
1.
5.
𝑥
45
8
𝑥
=
=
4
9
12
5
7
4x =
4
Exercise 2A:
𝑥
=
=
Cross multiplication
Divide both sides by 4.
3
21
=
4
Solve the following equations.
2.
6.
6
=
𝑡
𝑥
3
4
8
5
=7
7.
7.
8
12
𝑥
7
=
12
8
=4
𝑥
8.
8.
3
𝑥
9
𝑥
4
=3
=
33
2
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Olympic College - Topic 5 Ratio and Proportions
3. Solving Ratio Word Problems.
Example 1: The ratio of boys to girls in a class is 5 to 4. If there are 25 boys in the class, how
many girls are there?
Solution:
Identify the two quantities in this case its boys and girls
5
The first ratio of boys to girls is to 4 gives us the fraction 4 the second piece of
information is 25 boys to x girls gives us the fraction
5
Set up the ratio equations.
4
5
Solve the ratio equation
4
=
=
𝑥
25
𝑥
60
5𝑥
60
x
=
=
𝑥
25
5x =
5
25
5
Cross multiply
Divide both sides by 3.
12 girls in the class
Example 2: Two bags of grass seed will cover 2500 sq feet, how many bags of grass seed will
you need in order to cover 7500 sq ft?
Solution:
Identify the two quantities in this case its amount of grass seed and the area it
covers.
2
The first ratio of gives us the fraction 2500 the second piece of
𝑥
information is x bags to 6250 sq ft gives us the fraction 6250
Set up the ratio equations.
Solve the ratio equation
2
2500
2
2500
=
=
2500x =
2500𝑥
2500
x
=
=
𝑥
7500
𝑥
7500
1500
1500
2500
Cross multiply
Divide both sides by 2500.
6 bags of grass seed
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Olympic College - Topic 5 Ratio and Proportions
Example 3: The exchange rate of 1 British pound is worth $1.67, how many pounds (to the
nearest whole number) will you get for $500?
Solution:
Identify the two quantities in this case its pounds and dollars
1
The first ratio of pounds to dollars is 1 to 1.67 gives us the fraction 1.67 the second
𝑥
piece of information is x pounds to $500 gives us the fraction 500
1
Set up the ratio equations.
1.67
1
Solve the ratio equation
1.67
1.67x
=
=
=
1.67𝑥
=
1.67
𝑥
500
𝑥
500
500
500
1.67
Cross multiply
Divide both sides by 1.67
x
=
299.4011 pounds
x
=
299 pounds
Example 4: Judi runs 4 laps of a track in 25 minutes, how long will it take to run 7 laps at the
same speed.
Solution:
Identify the two quantities in this case its laps completed and time.
4
The first ratio of laps to time is 4 to 25 gives us the fraction 25 the second
7
piece of information is 7 laps to x minutes gives us the fraction 𝑥
4
Set up the ratio equations.
Solve the ratio equation
25
4
25
=
=
7
𝑥
7
𝑥
4x =
175
4𝑥
175
4
x
=
=
4
Cross multiply
Divide both sides by 4
43.75 minutes
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Olympic College - Topic 5 Ratio and Proportions
Exercise 3A:
Solve the following equations.
1.
The ratio of boys to girls in a class is 3 : 2. If there are 24 boys in the class, how many
girls are there?
2.
The ratio of boys to girls in a class is 5 to 6. If there are 30 girls in the class, how many boys
are there?
3.
In a piece of jewellery the ratio of gold to silver is 5 : 2.
If the jewellery contains 40 grammes of gold, what weight of silver does it contain?
4.
An lottery win was shared between two brothers, Frank and Pat, in the ratio 3 : 4.
If Pat received $720, how much did Frank receive ?
5.
Two farmers, Bill and Dan, decided to split a herd of cows in the ratio 5 : 7.
If Dan's share was 42 cows, how many cows did Bill get ?
6.
An airplane ascends 200 feet as it flies a horizontal distance of 1,000 feet. How much
altitude will it gain as it flies a horizontal distance of 1 mile? [Use 5,280 feet = 1 mile.]
7. In a scale drawing, a building 60 feet tall is drawn 4 inches high. The building next to it is
5 feet tall. How high will it be drawn in the scale drawing?
8.
In a scale drawing, a flagpole 40 feet tall is drawn 2 inches high.
The building next to it is drawn 3. 5 inches high. How tall is the building?
9.
Donald is making a model car to the scale 1 : 20
(a) His model is 16cm long. How long is the real car?
(b) The real car is 160cm wide. How wide is the model?
10. A watch loses 3 minutes in 6 hours. At this rate, how much would it lose in 24 hours?
11. A contractor finds that 5 gal. of stain will cover 2000 ft.² of floor space. How much stain
would be needed for a job of 3600 ft.² ?
12. A 10 lb. turkey breast contains 25 servings.
How many pounds of turkey breast would be needed for 45 servings?
13. A factory manufacturing computer circuits found 35 defective circuits in a lot of 500
circuits. At this rate, how many defective circuits can be expected in a lot of 1200 circuits?
14. To determine the number of rabbits in a park, a ranger catches 270 rabbits, tags them, and
then releases them. Later, 500 rabbits are caught, and it is found that 108 of them are
tagged. Estimate how many rabbits are in the park.
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Olympic College - Topic 5 Ratio and Proportions
Solutions.
Exercise 1A:
Simplify the following ratios
1.
5 to 4
2.
11 to 20
3.
7.
5:2
8.
3:4
9.
Exercise 1B:
11 to 12
10
4.
10.
27
3 to 5
1
20
5.
11.
11 : 3
2
6.
12.
3
2:3
2
9
#1,2,4,6 are Equivalent Ratios
#3 and 5 are not Equivalent Ratios.
Exercise 2A:
1. x =
5.
Exercise 3A:
160
x=
9
10
3
2. t = 12
6.
x=
15
7
3. x = 16
4.
x=
7. x = 164
8.
x=
1. 16 girls.
2. 25 boys.
3. 16 grams of silver.
4. Frank gets $540.
5. Bill gets 30 cows.
6. Altitude of 21,120 ft.
7. model is 3.2 inches tall.
8. Building is 70ft tall.
9.(a) real car is 320 cm long.
9.(b) model is 8 cm wide.
10. watch loses 12 minutes.
11. 9 gallons of stain.
12. 18 lb turkey
13. 84 defects
9
4
6
11
14. 1250 rabbits are in the park
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