Ratio and Proportions

Ratio of a to b

• The quotient a/b if a and b are 2 quantities that are measured in the same units can also be written as a:b.

* b cannot = 0, because the denominator cannot be 0.

• Always write ratios in simplified form! (reduce the fraction!)

3

Ratios

• A ratio is a comparison of numbers that can be expressed as a fraction.

• If there were 18 boys and 12 girls in a class, you could compare the number of boys to girls by saying there is a ratio of 18 boys to 12 girls. You could represent that comparison in three different ways:– 18 to 12– 18 : 12– 18

12

4

Ratios

• The ratio of 18 to 12 is another way to represent the fraction

• All three representations are equal.– 18 to 12 = 18:12 =

• The first operation to perform on a ratio is to reduce it to lowest terms– 18:12 =

– 18:12 = 3:2

1812

1812

1812 3 2

3 2

÷ 6

÷ 6

Example: simplify.

ft

ft

24

12

ft

yd

6

3

in

ft

18

6

2

1

yd

ft

1

36

9

2

3

ft

in

1

12

18

724

1

4

6

Ratios

• A basketball team wins 16 games and loses 14 games. Find the reduced ratio of:– Wins to losses – 16:14 = =

– Losses to wins – 14:16 = =

– Wins to total games played –

16:30 = =

• The order of the numbers is critical

1614

8 7

1416

7 8

1630

815

Example: the perimeter of an isosceles is 56in. The ratio of LM:MN is 5:4. find

the lengths of the sides of the .

L

MN

5x

4x

5x +5x+ 4x=56

x=4

) (

14x=56

2 equal sides Because it’s an isoscelestriangle

Ex: the measure of the s in a are in the extended ratio 3:4:8. Find the

measures of the s of the .

3x+4x+8x=180

15x=180

x=12

Substitute to find the angles:

3(12)=36, 4(12)=48, 8(12)= 96

Angle measures: 36o, 48o, 96o

Proportion

• An equation stating 2 ratios are =

• b and c are the means • a and d are the extremes

d

c

b

a

10

Proportions

• A proportion is a statement that one ratio is equal to another ratio.– Ex: a ratio of 4:8 = a ratio of 3:6– 4:8 = = and 3:6 = =– 4:8 = 3:6– =

– These ratios form a proportion since they are equal to other. =

4 8

1 2

3 6

1 2

4 8

3 6

1 2

1 2

Properties of Proportions

• Cross product property- means=extremes

1. If then, ad=bc

• Reciprocal Property- (both ratios must be flipped)

2. If , then

d

c

b

a

d

c

b

a

c

d

a

b

12

Proportions

• In a proportion, you will notice that if you cross multiply the terms of a proportion, those cross-products are equal.

4 8

3 6

3 2

18 12

4 x 6 = 8 x 3 (both equal 24)

3 x 12 = 2 x 18 (both equal 36)

=

=

13

Proportions N 12

3 4

=

4 x N = 12 x 3

4N = 36

4 N 36 4 4

1N = 9

N = 9

=

Cross multiply the proportion

Divide the terms on both sides of the equal sign by the number next to the unknown letter. (4)

That will leave the N on the left side and the answer (9) on the right side

Proportions

• Solve for N • Solve for N

14

2 5

N 35

=

5 x N = 2 x 35

5 n = 70

5 N 70 5 5

1N = 14

N = 14

=

15 N

3 4

=

6 7

102 N

=

4 N

6 27

=

15

Proportions

• At 2 p.m. on a sunny day, a 5 ft woman had a 2 ft shadow, while a church steeple had a 27 ft shadow. Use this information to find the height of the steeple.

• 2 x H = 5 x 27• 2H = 135• H = 67.5 ft.

5 2

H 27

= heightshadow

= heightshadow

You must be careful to place the same quantities in corresponding positions in the proportion

Ratios

• The ratio of freshman to sophomores in a drama club

is 5:6.

There are 18 sophomores in the drama club.

How many freshmen are there?

Freshman = 5 = x Sophomore 6 18

15 freshmen

5 * 186 * x

6x 90=

=

X = 15Divide both sides by 6

Example

10(s-5)=4s

10s-50=4s

-50= -6s

104

5 ss

s3

25

Butterfly effect cross-multiply

distribute

The ratios of the side lengths of QRS to the corresponding side lengths of

VTU are 3:2. Find the unknown length.

Q

RS

Xy

V

uT

18cm

2cmz

w

Example cont

333y 148z

2

318

w2

3

2x

x= 3cm

a2+b2=c 2

32+182=y2

9+324=y2

333=y2

y ≈ 18.25cm

22+122=z2

4+144=z2

148=z2

z ≈ 12.17cm

w= 12cm