Page 1 3/16/2011 1 8.1 Ratio and Proportion Objectives 8.1 Ratio & Proportion * Find and simplify the ratio of two numbers. * Use proportions to solve real-life problems, such as computing the width of a painting. Computing Ratios ? If a and b are two quantities that are measured in the same units, then the ratio of “a” to “b” is a/b. The ratio of a to b can also be written as a:b. Because a ratio is a quotient (meaning division), its denominator cannot be zero (can’t divide by 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) 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft c. 9 in. 4cm 18 ft 18 in. 4 cm 18 ft 18 in. OR a. 12:4 b. 6:18 c. 9:18 The fraction and ratio are both ways to express ratios 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft 4 m 18 in Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible. 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm 4m 4 m 12 cm 12 cm 12 . 3 4 m 4·100cm 400 100 3:100 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: b. 6 ft 18 in 18 in 6 ft 6·12 in 72 in. 4 4 18 in 18 in. 18 in. 1 8.1 Ratio & Proportion Page 2 3/16/2011 1 8.1 Ratio and Proportion Objectives 8.1 Ratio & Proportion * Find and simplify the ratio of two numbers. * Use proportions to solve real-life problems, such as computing the width of a painting. Computing Ratios ? If a and b are two quantities that are measured in the same units, then the ratio of “a” to “b” is a/b. The ratio of a to b can also be written as a:b. Because a ratio is a quotient (meaning division), its denominator cannot be zero (can’t divide by 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) 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft c. 9 in. 4cm 18 ft 18 in. 4 cm 18 ft 18 in. OR a. 12:4 b. 6:18 c. 9:18 The fraction and ratio are both ways to express ratios 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft 4 m 18 in Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible. 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm 4m 4 m 12 cm 12 cm 12 . 3 4 m 4·100cm 400 100 3:100 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: b. 6 ft 18 in 18 in 6 ft 6·12 in 72 in. 4 4 18 in 18 in. 18 in. 1 8.1 Ratio & Proportion 3/16/2011 2 Ex. 2: Using Ratios ? The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2 Find B C of AB: BC is 3:2. Find the length and the width of the rectangle w l A D 8.1 Ratio & Proportion Ex. 2: Using Ratios ? SOLUTION: Because the ratio of AB:BC is 3:2, you can represent the length of AB as 3x B C the length of AB as 3x and the width of BC as 2x. w l A D 8.1 Ratio & Proportion Solution: Statement 2l + 2w = P 2(3x) + 2(2x) = 60 Reason Formula for perimeter of a rectangle Substitute l, w and P 6x + 4x = 60 10x = 60 x = 6 Multiply Combine like terms Divide each side by 10 So, ABCD has a length of 18 centimeters and a width of 12 cm. 8.1 Ratio & Proportion 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 l K 2x° 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°. J L x° 3x° 8.1 Ratio & Proportion Solution: Statement x°+ 2x°+ 3x° = 180° 6x = 180 Reason Triangle Sum Theorem Combine like terms x = 30 Divide each side by 6 So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°. 8.1 Ratio & Proportion Ex. 4: Using Ratios ? The ratios of the side lengths of ?DEF to the corresponding side lengths of ?ABC 3 in. F C side lengths of ?ABC are 2:1. Find the unknown lengths. 8 in. D E A B 8.1 Ratio & Proportion Page 3 3/16/2011 1 8.1 Ratio and Proportion Objectives 8.1 Ratio & Proportion * Find and simplify the ratio of two numbers. * Use proportions to solve real-life problems, such as computing the width of a painting. Computing Ratios ? If a and b are two quantities that are measured in the same units, then the ratio of “a” to “b” is a/b. The ratio of a to b can also be written as a:b. Because a ratio is a quotient (meaning division), its denominator cannot be zero (can’t divide by 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) 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft c. 9 in. 4cm 18 ft 18 in. 4 cm 18 ft 18 in. OR a. 12:4 b. 6:18 c. 9:18 The fraction and ratio are both ways to express ratios 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm b. 6 ft 4 m 18 in Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible. 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: a. 12 cm 4m 4 m 12 cm 12 cm 12 . 3 4 m 4·100cm 400 100 3:100 8.1 Ratio & Proportion Ex. 1: Simplifying Ratios ? Simplify the ratios: b. 6 ft 18 in 18 in 6 ft 6·12 in 72 in. 4 4 18 in 18 in. 18 in. 1 8.1 Ratio & Proportion 3/16/2011 2 Ex. 2: Using Ratios ? The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB: BC is 3:2 Find B C of AB: BC is 3:2. Find the length and the width of the rectangle w l A D 8.1 Ratio & Proportion Ex. 2: Using Ratios ? SOLUTION: Because the ratio of AB:BC is 3:2, you can represent the length of AB as 3x B C the length of AB as 3x and the width of BC as 2x. w l A D 8.1 Ratio & Proportion Solution: Statement 2l + 2w = P 2(3x) + 2(2x) = 60 Reason Formula for perimeter of a rectangle Substitute l, w and P 6x + 4x = 60 10x = 60 x = 6 Multiply Combine like terms Divide each side by 10 So, ABCD has a length of 18 centimeters and a width of 12 cm. 8.1 Ratio & Proportion 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 l K 2x° 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°. J L x° 3x° 8.1 Ratio & Proportion Solution: Statement x°+ 2x°+ 3x° = 180° 6x = 180 Reason Triangle Sum Theorem Combine like terms x = 30 Divide each side by 6 So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°. 8.1 Ratio & Proportion Ex. 4: Using Ratios ? The ratios of the side lengths of ?DEF to the corresponding side lengths of ?ABC 3 in. F C side lengths of ?ABC are 2:1. Find the unknown lengths. 8 in. D E A B 8.1 Ratio & Proportion 3/16/2011 3 Ex. 4: Using Ratios ? SOLUTION: ? DE is twice AB and DE = 8, so AB = ½(8) = 4 ? Use the Pythagorean Theorem to determine 3 in. F C 5in 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 8 in. D E A B 4 in a 2 + b 2 = c 2 3 2 + 4 2 = c 2 9 + 16 = c 2 25 = c 2 5 = c 8.1 Ratio & Proportion 6in Using Proportions ? An equation that equates two ratios is called a proportion. For instance if the ? = ? Means Extremes For instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written: ? = ? The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion. 8.1 Ratio & Proportion Properties of proportions 1. CROSS PRODUCT PROPERTY. The product of the extremes equals the product of the means. If ? = ?, then ad = bc 8.1 Ratio & Proportion Properties of proportions 2. RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal. If ? = ?, then = ? b a To solve the proportion, you find the value of the variable. 8.1 Ratio & Proportion Ex. 5: Solving Proportions 4 x 5 7 = Write the original proportion. Reciprocal prop. Multiply each side by 4 Simplify. x 4 7 5 = 44 x = 28 5 8.1 Ratio & Proportion Ex. 5: Solving Proportions 3 y + 2 2 y = Write the original proportion. y y Cross Product prop. Distributive Property Subtract 2y from each side. 3y = 2(y+2) y = 4 3y = 2y+4 8.1 Ratio & ProportionRead More

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