Transcript 2-5 Solving Proportions
2-5 Solving Proportions
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2-5 Solving Proportions
Warm Up Solve each equation.
1. Multiply.
48 2. 5m = 18 3.6
3.
7
4.
Change each percent to a decimal.
10
5.
73% 0.73
6. 112% 1.12
7.
0.6% 0.006
Change each fraction to a decimal.
9.
0.5
10.
0.3
8. 1% 0.01
2-5 Solving Proportions California Standards
15.0 Students apply algebraic
techniques to solve rate problems, work problems, and percent mixture problems.
2-5 Solving Proportions
Vocabulary
ratio scale rate scale model cross products scale drawing proportion unit rate percent
2-5 Solving Proportions
A ratio is a comparison of two quantities. The ratio of a to b can be written as a:b or , where b ≠ 0.
A statement that two ratios are equal, such as is called a proportion.
2-5 Solving Proportions
Additional Example 1: Using Ratios The ratio of the number of bones in a human There are 22 bones in the skull. How many bones are in the ears? ’ s ears to the number of bones in the skull is 3:11.
Write a ratio comparing bones in ears to bones in skull.
Write a proportion. Let x be the number of bones in ears.
Since x is divided by 22, multiply both sides of the equation by 22.
There are 6 bones in the ears.
2-5 Solving Proportions
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Example 1 The ratio of red marbles to green marbles is 6:5. There are 18 red marbles. How many green marbles are there? green red
5 6
Write a ratio comparing green to red marbles. Write a proportion. Let x be the number green marbles. Since x is divided by 18, multiply both sides by 18.
15 = x There are 15 green marbles.
2-5 Solving Proportions
A common application of proportions is rates. A rate is a ratio of two quantities with different units, such as Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as or 17 mi/gal. You can convert any rate to a unit rate.
2-5 Solving Proportions
Additional Example 2: Finding Unit Rates Ralf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth.
Write a proportion to find an equivalent ratio with a second quantity of 1.
3.47 ≈ x
Divide on the left side to find x.
The unit rate is approximately 3.47 pancake flips per second.
2-5 Solving Proportions
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Example 2a Find the unit rate. Round to the nearest hundredth if necessary. Cory earns $52.50 in 7 hours.
7.50 = x
Write a proportion to find an equivalent ratio with a second quantity of 1.
Divide on the left side to find x.
The unit rate is $7.50 per hour.
2-5 Solving Proportions
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Example 2b Find the unit rate. Round to the nearest hundredth if necessary. A machine seals 138 envelopes in 23 minutes.
6 = x
Write a proportion to find an equivalent ratio with a second quantity of 1.
Divide on the left side to find x.
The unit rate is 6 envelopes seals per minute.
2-5 Solving Proportions
In the proportion the products a
d
and b c are called cross products. You can solve a proportion for a missing value by using the Cross Products Property
2-5 Solving Proportions
Additional Example 3A: Solving Proportions Solve the proportion.
Use cross products.
3(m) = 9(5) 3m = 45 m = 15
Divide both sides by 3.
2-5 Solving Proportions
Additional Example 3B: Solving Proportions Solve the proportion.
Use cross products.
6(7) = 2(y – 3) 42 = 2y – 6 +6 +6 48 = 2y
Add 6 to both sides.
Divide both sides by 2.
24 = y
2-5 Solving Proportions
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Example 3a Solve the proportion. Check your answer.
Use cross products.
–5 ( 8 ) = 2(y) –40 = 2y –20 = y
Divide both sides by 2.
2-5 Solving Proportions
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Example 3a Continued Solve the proportion. Check your answer.
Check
Substitute –20 for y.
–2.5 –2.5
2-5 Solving Proportions
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Example 3b Solve the proportion. Check your answer.
Use cross product.
4 ( g + 3 ) = 5(7) 4g + 12 = 35 –12 –12 4g = 23 g = 5.75
Subtract 12 from both sides.
Divide both sides by 4.
2-5 Solving Proportions
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Example 3b Continued Solve the proportion. Check your answer.
Check
Substitute 5.75 for b.
1.75 1.75
2-5 Solving Proportions
Another common application of proportions is percents. A percent is a ratio that compares a number to 100. For example, 25% = You can use the proportion to find unknown values.
2-5 Solving Proportions
Additional Example 4A: Percent Problems Find 30% of 80.
Method 1 Use a proportion.
Use the percent proportion.
100x = 2400 x = 24 30% of 80 is 24.
Let x represent the part.
Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.
2-5 Solving Proportions
Additional Example 4B: Percent Problems 230 is what percent of 200?
Method 2 Use an equation.
230 = x 200 230 = 200x
Write an equation. Let x represent the percent.
1.15 = x
Since x is multiplied by 200, divide both sides by 200 to undo the multiplication. The answer is a decimal.
115% = x 230 is 115% of 200.
Write the decimal as a percent. This answer is reasonable; 230 is more than 100% of 200.
2-5 Solving Proportions
Additional Example 4C: Percent Problems 20 is 0.4% of what number?
Method 1 Use a proportion.
Use the percent proportion.
Let x represent the whole.
2000 = 0.4x 5000 = x 20 is 0.4% of 5000.
Cross multiply.
Since x is multiplied by 0.4, divide both sides by 0.4.
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Example 4a Find 20% of 60.
Method 1 Use a proportion.
Use the percent proportion.
100x = 1200
Let x represent the part.
Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.
x = 12 20% of 60 is 12.
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Example 4b 48 is 15% of what number?
Method 1 Use a proportion.
Use the percent proportion.
4800 = 15x x = 320 48 is 15% of 320.
Let x represent the whole.
Find the cross product. Since x is multiplied by 15, divide both sides by 15 to undo the multiplication.
2-5 Solving Proportions
Proportions are used to create scale drawings and scale models. A scale is a ratio between two sets of measurements, such as 1 in.:5 mi. A scale drawing, or scale model, uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing.
2-5 Solving Proportions
Additional Example 5A: Scale Drawings and Scale Models A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.
A wall on the blueprint is 6.5 inches long. How long is the actual wall?
Write the scale as a fraction.
Let x be the actual length.
x
1= 3(6.5) x = 19.5
Use cross products to solve.
The actual length is 19.5 feet.
2-5 Solving Proportions
Additional Example 5B: Scale Drawings and Scale Models A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.
A wall in the house is 12 feet long. How long is the wall on the blueprint?
Write the scale as a fraction.
Let x be the blueprint length.
x
3 = 1(12)
Use cross products to solve.
x = 4 The blueprint length is 4 inches.
2-5 Solving Proportions
Reading Math
A scale written without units, such as 32:1, means that 32 units of any measure corresponds to 1 unit of that same measure.
2-5 Solving Proportions
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Example 5a The actual distance between North Chicago and Waukegan is 4 mi. What is the distance between these two locations on the map?
Write the scale as a fraction.
Let x be the map distance.
18x = 4 x ≈ 0.2
Use cross products to solve.
The distance on the map is about 0.2 in.
2-5 Solving Proportions
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Example 5b A scale model of a human heart is 16 ft long. The scale is 32:1 How many inches long is the actual heart that the model represents?
Write the scale as a fraction.
Let x be the actual distance.
32x = 16 x = 0.5
Use cross products to solve.
The actual heart is 0.5 feet or 6 inches.
2-5 Solving Proportions
Lesson Quiz: Part l
1. In a school, the ratio of boys to girls is 4:3. There are 216 boys. How many girls are there? 162
Find each unit rate. Round to the nearest hundredth if necessary.
2. Nuts cost $10.75 for 3 pounds.
$3.58/lb 3. Sue washes 25 cars in 5 hours.
Solve each proportion.
5 cars/h
4.
6
5.
16
2-5 Solving Proportions
Lesson Quiz: Part ll
6. Find 20% of 80.
16 7. What percent of 160 is 20? 12.5% 8. 35% of what number is 40?
114.3
9. A scale model of a car is 9 in. long. The scale is 1:18. How many inches long is the actual car the model represents?
162 in.