Ratios, Proportions, and the Geometric Mean

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

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?

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

Simplify the ratio Convert 2 ft to in

Use the number line to find the ratio of the distances

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.

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?

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

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

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

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

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

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

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

Now you try!

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

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

Find the geometric mean x = 2 x = 3

Find the geometric mean x = 9