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6.1 Ratios, Proportions, and the Geometric Mean
Objective: Solve problems by writing and solving proportions
EXAMPLE 1 Simplify ratios
SOLUTION
64 m : 6 ma.
Then divide out the units and simplify.
a. Write 64 m : 6 m as 64 m6 m
.
= 323
= 32 : 3
Simplify the ratio.
64 m6 m
Simplifying Ratios with Different Units
• What to do:
Use unit analysis to cancel out units
*note: simplifying the numbers can be done before or after canceling units
Example: Simplify 36in : 9ft
Using Ratios and to Solve ProblemsExample
• You are planning to paint a mural on a rectangular wall. You know the perimeter of the wall is 484 feet. The ratio of its length to its width is 9:2. Find the area of the wall.
Guided Practice #3 page 357
The perimeter of a room is 48 feet and the ratio of its length to its width is 7:5. Find the length and width of the room.
Example
• The area of a rectangular garden is 108 square feet, and the ratio of the length to the width is 4:3. Find the length and width of fence needed to enclose the garden.
EXAMPLE 3 Use extended ratios
Combine like terms.
SOLUTION
Triangle Sum Theorem
Divide each side by 6.= 30x=6x 180= 180
ox + 2x + 3x o oo
ALGEBRA The measures of the angles in CDE are in the extended ratio of 1 : 2 : 3. Find the measures of the angles.
Begin by sketching the triangle. Then use the extended ratio of 1 : 2 : 3 to label the measures as x° , 2x° , and 3x° .
The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90.o o o o o
ANSWER
GUIDED PRACTICE for Examples 2 and 3
4. A triangle’s angle measures are in the extended ratio of 1 : 3 : 5. Find the measures of the angles.
ANSWER 20°, 60°, 100°
EXAMPLE 4 Solve proportions
SOLUTION
a. 510
x16=
Multiply.
Divide each side by 10.
a. 510
x16=
= 10 x5 16
= 10 x80
= x8
Write original proportion.
Cross Products Property
Solve the proportion.ALGEBRA
EXAMPLE 4 Solve proportions
Subtract 2y from each side.
1y + 1
23y
b. =
= 2 (y + 1)1 3y
= 2y + 23y
=y 2
Distributive Property
SOLUTION
b. 1y + 1 = 2
3y
Write original proportion.
Cross Products Property
GUIDED PRACTICE for Example 4
5. 2 x
5 8=
6. 1x – 3
43x=
7.
y – 3 7
y14=
16 5
ANSWER
ANSWER 12
ANSWER 6
Solve the proportion.
EXAMPLE 5 Solve a real-world problem
SOLUTION
SCIENCE
As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees.
Write and solve a proportion involving two ratios that compare the number of trees with the area of the land.
EXAMPLE 5 Solve a real-world problem
Simplify.
Cross Products Property
There are about 20,250 trees in the 150 acre area.
=2702
n150
=270 150 2 n
n=20,250
number of treesarea in acres
Write proportion.
GUIDED PRACTICE for Examples 5 and 6
WHAT IF ? In Example 5, suppose you count 390 trees in a 3 acre area of the 150 acre area. Make a new estimate of the total number of trees.
8.
ANSWER 19,500 trees
Geometric Mean
• A number x that satisfies the proportiona/x = x/b
• To find the geometric mean of two numbers, multiply them together and take the square root– Then simplify the square root (no decimal
answers)
EXAMPLE 6 Find a geometric mean
Simplify.
Definition of geometric mean
Substitute 24 for a and 48 for b.
Factor.
Find the geometric mean of 24 and 48.
SOLUTION
=x ab
= 24 48
= 24 24 2
= 224
The geometric mean of 24 and 48 is 24 33.9.2
GUIDED PRACTICE for Examples 5 and 6
Find the geometric mean of the two numbers.
9. 12 and 27
ANSWER 18
10. 18 and 54
ANSWER 18 3
11. 16 and 18
ANSWER 12 2
Daily Homework Quiz
For use after Lesson 6.1
1. Simplify the ratio 1200 cm : 1.8 m.
ANSWER 2 : 3
2. The perimeter of a rectangle is 528 millimeters . The ratio of length of the width is 8 : 3. Find the length and the width.
ANSWER 192 mm, 72 mm
Daily Homework Quiz
For use after Lesson 6.1
4. Find the geometric mean of 42 and 12
ANSWER 6 14
1s + 8 = 3
363. Solve
ANSWER 4
Daily Homework Quiz
For use after Lesson 6.1
5. The extended ratio of the angles of a triangle is 5: 12: 13. Find the angle measures of the triangle.
ANSWER 30 ,72 , 78 o o o
6. The area of a rectangle is 720 square inches. If the ratio of the length to the width is 5 : 4, find the perimeter of the rectangle.
ANSWER 108 in.