7-1 Using Proportions Recognize and use ratios and proportions.Recognize and use ratios and proportions. Apply the properties of proportions.Apply the.

7-1 Using Proportions7-1 Using Proportions7-1 Using Proportions7-1 Using Proportions•Recognize and use ratios and proportions.Recognize and use ratios and proportions.•Apply the properties of proportions.Apply the properties of proportions.

Ratio• A ratio is a comparison of two

quantities using division.• Example: The number of boys to

girls in a class.

Ways to express a ratio• a/b• a to b• a:b

Example 1

• A baseball player’s batting average is the ratio of the number of base hits to the number of at-bats, not including walks. Minnesota Twins’ Joe Mauer had the highest batting average in Major League Baseball in 2006. If he had 521 official at-bats and 181 hits, find his batting average.

• Number of hits = 181 = 0.347• Number of at-bats 521 1

• Joe Mauer’s batting average was 0.347

Extended ratios

• An extended ratio can be used to compare three or more quantities.

• The expression a:b:c means that the ratio of the first two quantities is a:b, the ratio of the last two quantities is b:c, and the ratio of the first and last quantities is a:c.

Ex. 2 The ratio of the measures of the angles in a triangle is 3:4:5. Find the

measures of the angles.

• Just as ratio ¾ can be written as 3x/4x or 3x:4x, the extended ratio 3:4:5 can be written as 3x:4x:5x.

• 3x + 4x + 5x = 180• 12x = 180• X = 15 • So the measures of the angles are 3(15)

or 45, 4(15) or 60, and 5(15) or 75.• 45 + 60 + 75 = 180

Proportion

• An equation stating that two ratios are equal is a proportion.

• Extreme Means• extreme a = c mean• mean b d extreme• The product of the extremes ad and

the product of the means bc are called cross products.

The cross products of a proportion are equal.

Equality of Cross Product

• For any numbers a and c and any nonzero numbers b and d,

• a = c • b d• If and only if ad = bc.

Example 2

• Solve• 3t – 1 = 7 • 4 8

• Nikki can word process 7 words in 6 seconds. At that rate, how many words can she word process in 3 minutes?

• Words 7 words = x words • Time 6 seconds 180

seconds

Example 3

• In a triangle, the ratio of the measures of three sides is 8:7:5. and its perimeter is 240 centimeters. Find the measure of each side of the triangle.

• 8x + 7x + 5x = 240• 20x = 240• x = 12

• Side 1 = 8x = 8(12) = 96cm• Side 2 = 7x = 7(12) = 84cm• Side 3 = 5x = 5(12) = 60cm

4. Determine which proportions are equivalent. Explain your reasoning.

• 7 = x y = 8 y = x 7 = 8

• 8 y x 7 7 8 x y

6. 2 inches on a map represent 150 miles. Find a ratio involving 1 inch.

• 2 inches = 1 inch • 150 miles 75 miles

7. The perimeter of a rectangle is 84 feet. The ratio of the width to the length is 2:5, Find the length and width.

• P = 2l + 2w• 84= 2(5x) + 2(2x)• 84 = 10x + 4x• 84 = 14x• 6 = x• Length 5(6) is 30, width 2(6) is 12

Ex. 8 The area of a rectangle is 108cm2. The ratio of the width to the length is 3:4.

Find the length and width.

• A = lw• 108 = (3x)(4x)• 108 = 12x2

• 9 = x2

• 3 = x• Length 4(3) is 12cm,• width 3(3) is 9cm

Solve each proportion by using cross products.

• x = 11 • 5 35• 35x = 55• x = 1.57

• 13 = 26 • 49 7x• 91x = 1274• x = 14

• X – 2 = 3 • x 8 • 8x – 16 = 3x• - 16 = -5x• 3.2 = x

• If a 6-foot post casts a shadow that is 8 feet long, how tall is an antenna that casts a 60-foot shadow at the same time?

Class work on page 464, problems 1-16

Homework on page 465, problems 17-36 even numbers.