1. 8.1 Exploring Ratio and Proportion

2. Ratio – a comparison of 2 numbers measured in the same units
The ratio of x to y can be written:
x to y
x:y
Because a ratio is a quotient, its denominator cannot be zero.
Ratios are expressed in simplified form.
6:8 is simplified to 3:4.

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

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

5. Ex. 1: Simplifying Ratios
Simplify the ratios:
b. 6 ft
18 in
6 ft 6∙12 in 72 in. 4
18 in in. 18 in. 1

6. 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

7. 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.

8. 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.

9. Ex. 3: Using Ratios
The measures of the angles in ∆JKL are in the ratio 1:2:3. Find the measures of the angles.

10. 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°.

11. 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.

12. 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

15. Using Proportions
Proportion - An equation that sets two ratios equal to one another
Means
Extremes
 = 
If using an extended proportion –
a:b:c = d:e:f
The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.

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

17. Properties of proportions
RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal.
If  = , then =  b a

18. Ex. 5: Solving Proportions4 5 = x 7 28 x = 5

19. Ex. 5: Solving Proportions3 2 =
y + 2 y y 4 =

22. 8.2 – Problem Solving With Proportions

25. Ex:

26. Determine values of x and y: