Applications Proportions Section 1.9. Proportions What are proportions? - If two ratios are equal, they form a proportion. Proportions can be used in.

Applications Proportions

Section 1.9

Proportions

What are proportions?

- If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures.

What do we mean by similar?

- Similar describes things which have the same shape but are not the same size.

1

248

= 1:3 = 3:9

ExamplesThese two stick figures are similar. As you can see both are the same shape. However, the bigger stick figure’s dimensions are exactly twice the smaller.

So the ratio of the smaller figure to the larger figure is 1:2 (said “one to two”). This can also be written as a fraction of ½.

A proportion can be made relating the height and the width of the smaller figure to the larger figure:

2 feet

4 feet

8 feet

4 feet4 ft

2 ft=

8 ft

4 ft

Solving Proportion Problems

First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent?

Then solve for x by cross multiplying:

2 feet

4 feet

8 feet

? feet

4 ft

2 ft=

8 ft

x ft

4x = 16

X = 4

Try One Yourself

4 feet

8 feet 12 feet

x feet

x = 6 feet

OR

Similar Shapes

In geometry similar shapes are very important. This is because if we know the dimensions of one shape and one of the dimensions of another shape similar to it, we can figure out the unknown dimensions.

Proportions and Triangles

What are the unknown values on these triangles?

16 m

20 m

4 m

3 m

x m

y m

First, write proportions relating the two triangles.

4 m

16 m=

3 m

x m

4 m

16 m=

y m

20 m

Solve for the unknown by cross multiplying.

4x = 48

x = 12

16y = 80

y = 5

Solving for the Building’s Height

Here is a sample calculation for the height of a building:

48 feet

4 feet

3 feet

yardstick

building

x feetx ft

3 ft=

48 ft

4 ft

4x = 144

x = 36

The height of the building is

36 feet.

Ex: The dosage of a certain medication is 2 mg for every 80 lbs of body weight. How many milligrams

of this medication are required for a person who weighs 220 lbs?

Use this rate to determine the dosage for 220-lbs by setting up a proportion (match units)

lbs80mg2

Let x = required dosage

=220 lbs x mg 2(220) = 80x

440 = 80x x = 5.5 mg

Ex: To determine the number of deer in a game preserve, a forest ranger catches 318 deer, tags

them, and release them. Later, 168 deer are caught, and it is found that 56 of them are tagged. Estimate how many deer are in the game preserve.

Set up a proportion comparing the initial tag rate to the later catch tag rate

Initial tag rate = later catch tag rate

sizecatchlatertagged#catchlater

sizepopulationtaggedinitially#

deer168tagged56

deerdtagged318

(318)(168) = 56d

53,424 = 56d 56 56

d = 954 deer in the reserve

Ex: An investment of $1500 earns $120 each year. At the same rate, how much additional

money must be invested to earn $300 each year?

What do we need to find?

Let m = additional money to be invested

What is the annual return rate of the investment?

$120 for $1500 investment

What is the desired return?

$300

Set up a proportion comparing the current return rate and the desired return rate

Initial return rate = desired return rate

investmentnewreturndesired

investmentinitialreturninitial

invested)m1500($returndesired300$

invested1500$return120$

120(1500 + m) = (1500)(300)

180,000 + 120m = 450,000

120m = 270,000 m = $2250 additional needs to be invest new investment = $1500 + $2250 = $3750

Divide by 120

Ex: A nurse is to transfuse 900 cc of blood over a period of 6 hours. What rate would the nurse

infuse 300 cc of blood?

What do we need to find?The rate of infusion for 300 cc of blood

What is the rate of transfusion?

900 cc of blood in 6 hoursSet up a proportion comparing the rate of tranfusion

to the desired rate of infusion

But to set up the proportion we need to know how long it takes to insfuse 300 cc of

blood Let h = hours required

hourshcc300

hours6cc900

proportion comparing the rate of tranfusion to the desired rate of infusion

900h = (6)(300) 900h = 1800 h = 2 hours

Therefore, it will take 2 hours to insfuse 300 cc of blood

New insfusion rate = 300 cc / 2 hours

hours2cc300

hours1cc150

150 cc/hour is the insfusion rate

For Polygons to be Similar

corresponding angles must be congruent,

andcorresponding sides must

be proportional

(in other words the sides must have lengths that form equivalent ratios)

Congruent figures have the same size and shape. Similar figures have the same

shape but not necessarily the same size. The two figures below are similar. They have the same shape but not the same

size.

Let’s look at the two triangles we looked at earlier to see if they

are similar.

Are the corresponding angles in the two

triangles congruent?

Are the corresponding sides proportional?

(Do they form equivalent ratios)

Just as we solved for variables in earlier

proportions, we can solve for variables to find

unknown sides in similar figures.

Set up the corresponding sides as a proportion and

then solve for x.

Ratios x/12and5/10

x 5

12 10

10x = 60

x = 6

Determine if the two triangles are similar.

In the diagram we can use proportions

to determine the height of the tree.

5/x = 8/288x = 140

x = 17.5 ft

The two windows below are similar. Find the unknown width of the larger

window.

These two buildings are similar. Find the height of

the large building.