1. Ratio and Proportion

2. Computing Ratios
If a and b are two quantities that are measured in the same units, then the ratio of a to be is a/b. The ratio of a to be can also be written as a:b. Because a ratio is a quotient, its denominator cannot be zero. Ratios are usually expressed in simplified form. For instance, the ratio of 6:8 is usually simplified to 3:4. (You divided by 2)

3. Ex. 1: Simplifying Ratios
Simplify the ratios:
12 cm b. 6 ft c. 9 in.
4 cm ft in.

4. Ex. 1: Simplifying Ratios
Simplify the ratios:
12 cm b. 6 ft
4 m in
Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible.

5. Ex. 1: Simplifying Ratios
Simplify the ratios:
12 cm
4 m
12 cm 12 cm 12 3
4 m ∙100cm

6. Ex. 1: Simplifying Ratios
Simplify the ratios:
b. 6 ft
18 in
6 ft 6∙12 in 72 in
18 in in. 18 in. 1

7. Ex. 2: Using Ratios
The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2. Find the length and the width of the rectangle

8. Ex. 2: Using Ratios
SOLUTION: Because the ratio of AB:BC is 3:2, you can represent the length of AB as 3x and the width of BC as 2x.

9. Solution: Statement 2l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60
Reason
Formula for perimeter of a rectangle
Substitute l, w and P
Multiply
Combine like terms
Divide each side by 10
So, ABCD has a length of 18 centimeters and a width of 12 cm.

10. Ex. 3: Using Extended Ratios
The measures of the angles in ∆JKL are in the extended ratio 1:2:3. Find the measures of the angles.
Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°.
2x°
3x° x°

11. Solution: Statement x°+ 2x°+ 3x° = 180° 6x = 180 x = 30 Reason
Triangle Sum Theorem
Combine like terms
Divide each side by 6
So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

12. Ex. 4: Using Ratios
The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths.

13. Ex. 4: Using Ratios SOLUTION:
DE is twice AB and DE = 8, so AB = ½(8) = 4
Use the Pythagorean Theorem to determine what side BC is.
DF is twice AC and AC = 3, so DF = 2(3) = 6
EF is twice BC and BC = 5, so EF = 2(5) or 10
4 in
a2 + b2 = c2
= c2
= c2
25 = c2
5 = c

14. Using Proportions
An equation that equates two ratios is called a proportion. For instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:
Means
Extremes
 = 
The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.

15. Properties of proportions
CROSS PRODUCT PROPERTY. The product of the extremes equals the product of the means. If
 = , then ad = bc

16. Properties of proportions
RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal.
If  = , then =  b a
To solve the proportion, you find the value of the variable.

17. Ex. 5: Solving Proportions4 5
Write the original proportion.
Reciprocal prop.
Multiply each side by 4
Simplify. = x 7 4 x 7 4 = 4 5 28 x = 5

18. Ex. 5: Solving Proportions3 2 =
y + 2 y
3y = 2(y+2)
3y = 2y+4 y 4 =