7-1 using proportions recognize and use ratios and proportions.recognize and use ratios and...
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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.