Download - Solving Proportions
Solving Proportions
Warm UpLesson
Warm UpSolve each equation.
1.
Multiply.
3.
5. 7.
3.6 48
7
2. 5m = 18
Change each percent to a decimal.
4. 10
Change each fraction to a decimal.
9.
6. 112% 8. 1% 73% 0.6%
10.
0.73 0.0061.12
0.01
0.5 0.3
ratio scale rate scale modelcross productsscale drawingproportion unit ratepercent
Vocabulary
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.
Additional Example 1: Using Ratios
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.
The ratio of the number of bones in a human’s ears to the number of bones in the skull is 3:11. There are 22 bones in the skull. How many bones are in the ears?
Your Turn! Example 1
The ratio of red marbles to green marbles is 6:5. There are 18 red marbles. How many green marbles are there?
greenred
56
Write a ratio comparing green to red marbles.
15 = x
Write a proportion. Let x be the number green marbles.
Since x is divided by 18, multiply both sides by 18.
There are 15 green marbles.
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.
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.
Your Turn! Example 2a
Cory earns $52.50 in 7 hours.
Find the unit rate. Round to the nearest hundredth if necessary.
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.
Your Turn! 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.
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
Additional Example 3A: Solving Proportions
Solve the proportion.
3m = 45
3(m) = 9(5)
m = 15
Use cross products.
Divide both sides by 3.
Additional Example 3B: Solving Proportions
Solve the proportion.
Use cross products.
6(7) = 2(y – 3)
42 = 2y – 6 +6 +648 = 2y
24 = y
Add 6 to both sides.
Divide both sides by 2.
Solve the proportion. Check your answer.
Your Turn! Example 3a
–5(8) = 2(y)
–40 = 2y
–20 = y
Use cross products.
Divide both sides by 2.
Solve the proportion. Check your answer.
4g + 12 = 35
4(g + 3) = 5(7)
g = 5.75
Use cross product.
Divide both sides by 4.
–12 –12 4g = 23
Subtract 12 from both sides.
Your Turn! Example 3b
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.
Additional Example 4A: Percent Problems
Find 30% of 80.
Method 1 Use a proportion.
100x = 2400
x = 24
30% of 80 is 24.
Use the percent proportion.
Let x represent the part.
Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.
Additional Example 4B: Percent Problems
230 is what percent of 200?
Method 2 Use an equation.
230 = x 200
230 = 200x
1.15 = x
115% = x
230 is 115% of 200.
Write an equation. Let x represent the percent.
Since x is multiplied by 200, divide both sides by 200 to undo the multiplication.
The answer is a decimal.
Write the decimal as a percent. This answer is reasonable; 230 is more than 100% of 200.
Additional Example 4C: Percent Problems
20 is 0.4% of what number?
Method 1 Use a proportion.
2000 = 0.4x
Use the percent proportion.
Let x represent the whole.
5000 = x
Cross multiply.
Since x is multiplied by 0.4, divide both sides by 0.4.
20 is 0.4% of 5000.
Your Turn! Example 4a
Find 20% of 60.
Method 1 Use a proportion.
100x = 1200
x = 12
20% of 60 is 12.
Use the percent proportion.
Let x represent the part.
Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.
Your Turn! Example 4b
48 is 15% of what number?
Method 1 Use a proportion.
4800 = 15x
x = 320
48 is 15% of 320.
Use the percent proportion.
Let x represent the whole.
Find the cross product. Since x is multiplied by 15, divide both sides by 15 to undo the multiplication.
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.
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?
x 1= 3(6.5)x = 19.5
The actual length is 19.5 feet.
Write the scale as a fraction.
Let x be the actual length.
Use cross products to solve.
Additional Example 5B: Scale Drawings and Scale Models
Write the scale as a fraction.
Let x be the blueprint length.
x 3 = 1(12)x = 4
The blueprint length is 4 inches.
Use cross products to solve.
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?
A scale written without units, such as 32:1, means that 32 units of any measure corresponds to 1 unit of that same measure.
Reading Math
Your Turn! Example 5a
The actual distance between North Chicago and Waukegan is 4 mi. What is the distance between these two locations on the map?
18x = 4
x ≈ 0.2
The distance on the map is about 0.2 in.
Write the scale as a fraction.
Let x be the map distance.
Use cross products to solve.
Your Turn! 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?
32x = 16
x = 0.5
The actual heart is 0.5 feet or 6 inches.
Write the scale as a fraction.
Let x be the actual distance.
Use cross products to solve.