8.1 Exploring Ratio and Proportion

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.

Ex. 1: Simplifying Ratios Simplify the ratios: 12 cm b. 6 ft 4 m 18 in

Ex. 1: Simplifying Ratios Simplify the ratios: 12 cm 4 m 12 cm 12 cm 12 3 4 m 4∙100cm 400 100

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

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

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.

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.

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

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°.

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.

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 32 + 42 = c2 9 + 16 = c2 25 = c2 5 = c

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.

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

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

Ex. 5: Solving Proportions 4 5 = x 7 28 x = 5

Ex. 5: Solving Proportions 3 2 = y + 2 y y 4 =

8.2 – Problem Solving With Proportions

 

Ex:

Determine values of x and y: