1. Ratios, Proportions, and the Geometric Mean

2. Ratios The ratio of a to b can be written 3 ways: a:b a to b
A ratio is a comparison of two numbers expressed by a fraction.
The ratio of a to b can be written 3 ways:
a:b
a to b

3. Equivalent Ratios Can you come up with your own?
Equivalent ratios are ratios that have the same value.
Examples:
1:2 and 3:6
5:15 and 1:3
6:36 and 1:6
2:18 and 1:9
4:16 and 1:4
7:35 and 1:5
Can you come up with your own?

4. Simplify the ratios to determine an equivalent ratio.
3 ft = 1 yard
Convert 3 yd to ft
1 km = 1000 m
Convert 5 km to m

5. Simplify the ratio
Convert 2 ft to in

6. Use the number line to find the ratio of the distances

7. Using an Extended Ratio
An extended ratio compares three (or more) numbers.
In the extended ratio a : b : c, the ratios a : b, b : c, and a : c are all equivalent.

8. The lengths of the sides of a triangle are in the extended ratio 4 : 7 : 9. The perimeter is 60 cm. What are the lengths of the sides?

9. Triangles and ratios: finding interior angles
The ratio of the 3 angles in a triangle are represented by 1:2:3.
The 1st angle is a multiple of 1, the 2nd a multiple of 2 and the 3rd a multiple of 3.
Angle 1 = 1x
Angle 2 = 2x
Angle 3 = 3x
What do we know about the sum of the interior angles?
=30
=2(30) = 60
= 3(30) = 90
1x + 2x + 3x = 180
6x = 180
X = 30

10. Triangles and ratios: finding interior angles
The ratio of the angles in a triangle are represented by 1:1:2.
Angle 1 = 1x
Angle 2 = 1x
Angle 3 = 2x
1x + 1x + 2x = 180
4x = 180
x = 45
Angle 1 = 1x = 1(45) = 45
Angle 2 = 1x = 1(45) = 45
Angle 3 = 2x = 2(45) = 90

11. Proportions, extremes, means
Proportion: a mathematical statement that states that 2 ratios are equal to each other.
means
extremes

12. Solving Proportions 1y = 3(3) y = 9
When you have 2 proportions or fractions that are set equal to each other, you can use cross multiplication.
1y = 3(3)
y = 9

13. Solving Proportions
1(8) = 2x
4(15) = 12z
8 = 2x
60 = 12z
5 = z
4 = x

14. A little trickier
3(8) = 6(x – 3)
24 = 6x – 18
42 = 6x
7 = x

15. X’s on both sides?
3(x + 8) = 6x
3x + 24 = 6x
24 = 3x
8 = x

16. Now you try!

17. Now you try!
z = 3
x = 18
d = 5
x = 9
m = 7

18. Geometric Mean
When given 2 positive numbers, a and b the geometric mean satisfies:

19. Find the geometric mean
x = 2
x = 3

20. Find the geometric mean
x = 9