Transcript No Slide Title

Solving Proportions
Warm Up
Lesson
Warm Up
Solve each equation.
1.
48
Multiply.
2. 5m = 18 3.6
3.
10
4.
7
Change each percent to a decimal.
5.
7.
0.6% 0.006 8. 1% 0.01
73% 0.73 6. 112%
1.12
Change each fraction to a decimal.
9.
0.5
10.
0.3
Vocabulary
ratio
rate
cross products
scale drawing
proportion
unit rate
percent
scale
scale model
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
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?
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.
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?
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.
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
Find the unit rate. Round to the nearest
hundredth if necessary.
Cory earns $52.50 in 7 hours.
Write a proportion to find an equivalent
ratio with a second quantity of 1.
7.50 = x
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.
Write a proportion to find an equivalent
ratio with a second quantity of 1.
6=x
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.
Use cross
products.
3(m) = 9(5)
3m = 45
Divide both sides by 3.
m = 15
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
Your Turn! Example 3a
Solve the proportion. Check your answer.
Use cross products.
–5(8) = 2(y)
–40 = 2y
Divide both sides by 2.
–20 = y
Your Turn! Example 3b
Solve the proportion. Check your answer.
Use cross product.
4(g + 3) = 5(7)
4g + 12 = 35
–12 –12
4g = 23
Subtract 12 from both sides.
Divide both sides by 4.
g = 5.75
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
find unknown values.
to
Additional Example 4A: Percent Problems
Find 30% of 80.
Method 1 Use a proportion.
Use the percent proportion.
Let x represent the part.
100x = 2400
x = 24
30% of 80 is 24.
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
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.
230 is 115% of 200.
115% = x
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
Cross multiply.
Since x is multiplied by 0.4,
divide both sides by 0.4.
5000 = x
20 is 0.4% of 5000.
Your Turn! Example 4a
Find 20% of 60.
Method 1 Use a proportion.
Use the percent proportion.
Let x represent the part.
100x = 1200
x = 12
20% of 60 is 12.
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.
Use the percent proportion.
Let x represent the whole.
4800 = 15x
x = 320
48 is 15% of 320.
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?
Write the scale as a fraction.
Let x be the actual length.
Use cross products to solve.
x  1= 3(6.5)
x = 19.5
The actual length is 19.5 feet.
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.
Use cross products to solve.
x  3 = 1(12)
x=4
The blueprint length is 4 inches.
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.
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?
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.
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?
Write the scale as a fraction.
Let x be the actual distance.
32x = 16
Use cross products to solve.
x = 0.5
The actual heart is 0.5 feet or 6 inches.