Ratio and proportionRatio and proportion

Definition of a Definition of a ratioratio

If If a a and and b b are two quantities that are are two quantities that are measured in the measured in the same same units, then the units, then the ratio of ratio of aa to to b b is is a/b.a/b.

This ratio is written as This ratio is written as a a : : bb. . Ratios are preferably expressed in Ratios are preferably expressed in

simplified form.simplified form.

A ratio is a comparison of two A ratio is a comparison of two numbers and can be written numbers and can be written several ways. Each of the several ways. Each of the following are ways to write the following are ways to write the ratio of 3 to 4.ratio of 3 to 4.

3

43/ 4

3: 4

3 4

3 to 4

If the quantities to be If the quantities to be compared include units, then compared include units, then the units should be the same.the units should be the same. To find the ratio of 1 ft to 15 To find the ratio of 1 ft to 15 in, express both quantities in in, express both quantities in inches and then find the ratio.inches and then find the ratio.

1 12 12 4

15 15 15 5

ft in

in in

For PracticeFor Practice

Simplify the following ratios:Simplify the following ratios:

Definition of Definition of proportionproportion

An equation that equates two ratios An equation that equates two ratios is a is a proportionproportion. For instance, if the . For instance, if the ratio ratio a/b a/b is equal to the ratio is equal to the ratio c/dc/d, , then the following proportion can be then the following proportion can be written:written:

A proportion has four terms. A proportion has four terms. In the proportionIn the proportion

the first term is 2, the second term the first term is 2, the second term is 5 the third term is 4, and the is 5 the third term is 4, and the fourth term is 10. The first and fourth term is 10. The first and fourth terms are called the fourth terms are called the extremesextremes and the second and and the second and third terms are called the third terms are called the meansmeans..

2 4

5 10

2 42 : 5 4 :10

5 10

In a proportion the product of the In a proportion the product of the means is equal the product of the means is equal the product of the extremes.extremes.

2 4

5 102(10) 5(4)

20 20

A proportion is often used to A proportion is often used to solve such applications in solve such applications in which we are given one of the which we are given one of the ratios and then asked to find ratios and then asked to find one part of the other ratio one part of the other ratio equal to the first one.equal to the first one.

To solve a proportion means to To solve a proportion means to find the value of the missing find the value of the missing

term. To do this use cross term. To do this use cross products to write an equation products to write an equation

then solve the equation.then solve the equation.Solve:Solve:

8

3 1212 24

2

x

x

x

Practice: Solve each of the Practice: Solve each of the following proportions.following proportions.

1 4

16a

12

5 10

x

1.1 4

6 12

x

10 50

3 7x

Solve the following proportions.Solve the following proportions.

Some Related ExamplesSome Related Examples

If 125 bolts cost $7.50, how If 125 bolts cost $7.50, how much do 200 bolts cost?much do 200 bolts cost?

125 200

$7.50 $125 1500

$12

bolts bolts

xx

x

Suppose 826 bricks are used to Suppose 826 bricks are used to build a wall 14 ft long. How build a wall 14 ft long. How

many bricks are needed for a many bricks are needed for a similar wall 35 ft long?similar wall 35 ft long?

14 35

82614 28910

2065

ft ft

br xbrx

x

The perimeter of rectangle The perimeter of rectangle ABCD ABCD is is 60 centimeters. The ratio of 60 centimeters. The ratio of ABAB::BC BC is is 3:2. Find the length and width of the 3:2. Find the length and width of the rectangle.rectangle.

The measure of the angles in triangle The measure of the angles in triangle JKL JKL are in the are in the extended ratio extended ratio of 1:2:3. of 1:2:3. Find the measures of the angles.Find the measures of the angles.

11 If 32 rulers cost £3.52 what would If 32 rulers cost £3.52 what would 25 rulers cost ?25 rulers cost ?

22 32 boxes of crayons cost 32 boxes of crayons cost £47.36 £47.36 What would 48 boxes cost ?What would 48 boxes cost ?

33 25 bottles of wine cost £ 143.7525 bottles of wine cost £ 143.75What would 15 bottles cost ?What would 15 bottles cost ?

44

If 40 set squares cost £7.60 , If 40 set squares cost £7.60 , what would 64 cost ?what would 64 cost ?

55 19 metres of curtain material cost £ 149.15 19 metres of curtain material cost £ 149.15 What would 12 metres cost ?What would 12 metres cost ?

66 A set of 8 dental mirrors cost £ 110 A set of 8 dental mirrors cost £ 110 What would a set of 12 cost ? What would a set of 12 cost ?

77 56 books cost £266 56 books cost £266 What would 40 books cost ?What would 40 books cost ?

88 80 tennis balls cost £84. What would 64 tennis balls cost ?80 tennis balls cost £84. What would 64 tennis balls cost ?

Some more Examples

99 A box of 250 toothbrushes cost £315A box of 250 toothbrushes cost £315How much should you pay for the larger How much should you pay for the larger box which contains 480 toothbrushes ?box which contains 480 toothbrushes ?

1010 A package containing 52 ties can be bought at a A package containing 52 ties can be bought at a warehouse for £621·40.warehouse for £621·40.How much should you pay if you wanted to buy How much should you pay if you wanted to buy the smaller package containing 20 ties ?the smaller package containing 20 ties ?

1111 A store buys 140 childrens shirts from A store buys 140 childrens shirts from their supplier for £539. The store wish their supplier for £539. The store wish to buy in a further 360 of the same to buy in a further 360 of the same shirts. How much would they pay for shirts. How much would they pay for them ?them ?

1212 A shoe shop buy 12 pairs of the slippers shown A shoe shop buy 12 pairs of the slippers shown from the manufacturer for £107.40from the manufacturer for £107.40If they wish to order a further 20 pairs, how If they wish to order a further 20 pairs, how much would they pay for them ?much would they pay for them ?

1313 A jeweler buys 50 bangles for his shopA jeweler buys 50 bangles for his shopfor £997·50. As they sell so well , he for £997·50. As they sell so well , he decides to buy a further 40 bangles. decides to buy a further 40 bangles. How much will these cost ?How much will these cost ?

1414 If 56 calendars cost £277.20 how much would If 56 calendars cost £277.20 how much would 44 cost ?44 cost ?