1. Applications Proportions
Section 1.9
This PowerPoint was made to teach primarily 8th grade students proportions. This was in response to a DLC request (No. 228).

2. 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 2 4 8 =
1:3 = 3:9

3. Examples
These 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:
8 feet
4 feet
2 feet
4 feet
4 ft
2 ft =
8 ft

4. 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:
8 feet
4 ft
2 ft =
8 ft
x ft
4 feet
Due to the math it does not make a difference whether the smaller side is the numerator or denominator. The only thing which matters is that it is consistent on both sides of the equation.
2 feet
4x = 16
X = 4
? feet

5. Try One Yourself OR 8 feet 12 feet x = 6 feet 4 feet x feet
Knowing the two figures are similar the proportion between the two stick figures is 8 feet:12 feet.
Once written as a fraction 8/12 reduces to 2/3. So the proportion between the two stick figures is 2:3.
If the proportion is 2:3 then the student should set up this equation and solve for x:
2 / 3 = 4 / x
2 * x = 3 * 4
x = 12 / 2
x = 6 feet
4 feet
x feet

6. 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.

7. Proportions and Triangles
What are the unknown values on these triangles?
First, write proportions relating the two triangles.
4 m
16 m =
3 m
x m
4 m
16 m =
y m
20 m
20 m
16 m
Solve for the unknown by cross multiplying.
x m
4x = 48
x = 12
16y = 80
y = 5
y m
4 m
3 m

8. Solving for the Building’s Height
Here is a sample calculation for the height of a building:
building
x ft
3 ft =
48 ft
4 ft
x feet
48 feet
4x = 144
x = 36
yardstick
3 feet
The height of the building is
36 feet.
4 feet

9. 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) 
Let x = required dosage
x mg =
 2(220) = 80x
220 lbs
 440 = 80x
 x = 5.5 mg

10. 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.

11. Initial tag rate = later catch tag rate
Set up a proportion comparing the initial tag rate to the later catch tag rate
Initial tag rate = later catch tag rate
 (318)(168) = 56d
 53,424 = 56d
 d = 954 deer in the reserve

12. Let m = additional money to be invested
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

13. Initial return rate = desired return rate
Set up a proportion comparing the current return rate and the desired return rate
Initial return rate = desired return rate
 120( m) = (1500)(300)
 180, m = 450,000
 120m = 270,000
Divide by 120
m = $2250 additional needs to be invest
new investment = $ $2250 = $3750

14. The rate of infusion for 300 cc of blood
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 hours
Set 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

15. Therefore, it will take 2 hours to insfuse 300 cc of blood 
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 
150 cc/hour is the insfusion rate

16. For Polygons to be Similar corresponding angles must be congruent, and corresponding sides must be proportional (in other words the sides must have lengths that form equivalent ratios)

17. Congruent figures have the same size and shape
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.

18. 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)

19. 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/12
and
5/10 x
10x = 60
x = 6

20. Determine if the two triangles are similar.

21. In the diagram we can use proportions to determine the height of the tree. 5/x = 8/28 8x = 140 x = 17.5 ft

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

23. These two buildings are similar. Find the height of the large building.