2 Computing RatiosIf 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 ft4 m inSolution: 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 cm4 m12 cm 12 cm 12 34 m ∙100cm
6 Ex. 1: Simplifying Ratios Simplify the ratios:b. 6 ft18 in6 ft 6∙12 in 72 in18 in in. 18 in. 1
7 Ex. 2: Using RatiosThe 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 RatiosSOLUTION: 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 ReasonFormula for perimeter of a rectangleSubstitute l, w and PMultiplyCombine like termsDivide each side by 10So, 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 TheoremCombine like termsDivide each side by 6So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.
12 Ex. 4: Using RatiosThe 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) = 4Use the Pythagorean Theorem to determine what side BC is.DF is twice AC and AC = 3, so DF = 2(3) = 6EF is twice BC and BC = 5, so EF = 2(5) or 104 ina2 + b2 = c2= c2= c225 = c25 = c
14 Using ProportionsAn 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:MeansExtremes = 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 = baTo solve the proportion, you find the value of the variable.
17 Ex. 5: Solving Proportions 45Write the original proportion.Reciprocal prop.Multiply each side by 4Simplify.=x74x74=4528x=5