Ratios & Proportions

Tutorial 6e

Ratios

• In mathematics, a ratio compares two numbers using division.

• For example, the ratio of 12 dog owners in class to 3 cat owners in class would be the quotient obtained by dividing 12 by 3.• The result would be 4.

• This tells us that there is a 4 to 1 ratio of dog owners to cat owners in class. -- Or, for every 4 people who own a dog in class, 1 person owns a cat.

Ratios cont . . .

• A ratio can be written a few different ways. The ratio of the length to the width of the poster to the right is 42 to 24. The ratio can be written in any of the following ways:

42

2442 to 24 42 : 24 24

4242 24

• Note that the ratio 42 to 24 is not the same as the ratio 24 to 42.

• A ratio that is expressed in lowest terms is said to be in simplest form. For example, the ratio 6 to 8 or in simplest form is 3 to 4 or .

4

3 8

6

Ratios cont . . .

• Example 1• Write the ratio of heartbeats to seconds.• 8 heartbeats in 6 seconds.

6

8 heartbeats

seconds

• The ratio can be reduced to and is read “4 to 3”.6

8

3

4

Proportions

• A proportion is an equation stating that two ratios are equal.

• In the proportions to the right, a is the first term, b is the second term, c is the third term and d is the fourth term.

• The blue first and fourth terms are called the extremes (a & d), and the red second and third terms are called the means (b & c) of the proportion.

• This proportion is read: “a is to b as c is to d”. The next property describes the relationship between the means and extremes.

adbc

a : b = c : d

Proportions cont . . .

Means-Extremes Property of Proportions (cross-multiplication)

• In a proportion, the product of the means equals the product of the extremes.

• If then ad = bc. ,d

c

b

a

Example: Solve ,4228

12 c

12•42 = 28c

504 = 28c

28

28

28

504 c

18 = c

Proportions cont . . .

• Example: Solve ,2

9

4

3

x

(x + 3)2 = 4•9

2x + 6 = 36

2x = 30

x = 15

Proportions cont . . .

Your turn: On a separate sheet of paper, find the value of x in each proportion below.

15

561

x .

Click here to check your answers!

5

322

x

x .

2

73

3

323

xx .

Proportions cont . . .

Your turn: On a separate sheet of paper, find the value of x in each proportion below.

15

561

x .

5

322

x

x .

2

73

3

323

xx .

6•15 = 5•x

90 = 5x

18 = x

(x – 2)5 = 3•x

5x – 10 = 3x

2x = 10

x = 5

(2x – 3)2 = 3(3x – 7)

4x – 6 = 9x – 21-5x = -15

x = 3