1. 6.1 Ratios, Proportions, and the Geometric Mean
6.1
Bell Thinger
3x2 .
1. Simplify
9x
ANSWER
x
3
2. Solve 2n = 18 • 32. ANSWER
288
3. Solve 4x = 2x + 12.
6
4. Solve x = √36 • 16.
ANSWER
ANSWER
24
5. A triangle has angle measures 18º and 82º. Find the
measure of the third angle.
ANSWER 80º

2. 6.1

3. Example 1
6.1
Simplify the ratio.
a. 64 m : 6 m
b.
5 ft
20 in.
SOLUTION
a.
Write 64 m : 6 m as 64 m . Then divide out the units
6m
and simplify.
64 m
6m
= 32
3
= 32 : 3
b. To simplify a ratio with unlike units, multiply by a
conversion factor.
5 ft = 5 ft . 12 in. =
20 in.
20 in.
1 ft
60 = 3
1
20

4. Guided Practice
6.1
Simplify the ratio.
1. 24 yards to 3 yards
ANSWER 8 : 1
2. 150 cm : 6 m
ANSWER 1 : 4

5. Example 2
6.1
Painting You are planning to
paint a mural on a rectangular
wall. You know that the perimeter
of the wall is 484 feet and that the
ratio of its length to its width is
9 : 2. Find the area of the wall.
SOLUTION
STEP 1
Write expressions for the length and width.
Because the ratio of length to width is 9 :
2, you can represent the length by 9x and the
width by 2x.

6. Example 2
6.1
STEP 2
Solve an equation to find x.
2l + 2w
2(9x) + 2(2x)
22x
x
STEP 3
=
=
=
=
P
484
484
22
Formula for perimeter of rectangle
Substitute for l, w, and P.
Multiply and combine like terms.
Divide each side by 22.
Evaluate the expressions for the length and
width. Substitute the value of x into each
expression.
Length = 9x = 9(22) = 198
Width = 2x = 2(22) = 44
The wall is 198 feet long and 44 feet wide, so its area is
198 ft . 44 ft = 8712 ft 2
.

7. Example 3
6.1
ALGEBRA The measures of the angles in
CDE are
in the extended ratio of 1 : 2 : 3. Find the measures of
the angles.
SOLUTION
Begin by sketching the triangle. Then use
the extended ratio of 1 : 2 : 3 to label the
measures as x°, 2x°, and 3x°.
x° + 2x° + 3x° = 180°
6x = 180
x =
30
Triangle Sum Theorem
Combine like terms.
Divide each side by 6.
The angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

8. Guided Practice
6.1
3. The perimeter of a rectangular room is 48 feet and
the ratio of its length to its width is 7 : 5. Find the
length and width of the room.
ANSWER 14 ft, 10 ft
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°

9. 6.1

10. Example 4
6.1
Solve the proportion.
ALGEBRA
a.
5 = x
10
16
SOLUTION
a.
5
10
x
16
Write original proportion.
5 . 16 =
10 . x
Cross Products Property
80
10 x
Multiply.
x
Divide each side by 10.
=
=
8 =

11. Example 4
6.1
ALGEBRA
b.
Solve the proportion.
2
1
= 3y
y+1
SOLUTION
b.
1
=
y+1
2
3y
Write original proportion.
1 . 3y
=
2 (y + 1)
Cross Products Property
3y
=
2y + 2
Distributive Property
y
=
2
Subtract 2y from each side.

12. Guided Practice
6.1
Solve the proportion.
5. 2 = 5
x
8
16
ANSWER
5
1
4
6.
= 3x
x–3
ANSWER 12
7. y – 3 = y
7
14
ANSWER
6

13. Example 5
6.1
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.
SOLUTION
Write and solve a proportion involving two ratios that
compare the number of trees with the area of the land.

14. Example 5
6.1
270
n
=
2
150
270 . 150 = 2 . n
20,250 = n
number of trees
Write proportion.
area in acres
Cross Products Property
Simplify.
There are about 20,250 trees in the 150 acre area.

15. 6.1

16. Example 6
6.1
Find the geometric mean of 24 and 48.
SOLUTION
x =
ab
Definition of geometric mean
=
24 . 48
Substitute 24 for a and 48 for b.
=
24 . 24 . 2
Factor.
= 24
2
Simplify.
The geometric mean of 24 and 48 is 24
2 ≈ 33.9.

17. Guided Practice
6.1
Find the geometric mean of the two numbers.
9.
12 and 27
ANSWER
10.
18 and 54
ANSWER
11.
18
18
3
16 and 18
ANSWER
12
2

18. Exit Slip
6.1
1. Simplify the ratio 1200 cm : 1.8 m.
ANSWER
20 : 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

19. Exit Slip
6.1
3.
Solve
ANSWER
4.
1 = 3
s+8
36
4
Find the geometric mean of 42 and 12.
ANSWER
6 14

20. Exit Slip
6.1
5.
The extended ratio of the angles of a triangle is
5: 12: 13. Find the angle measures of the triangle.
ANSWER
6.
30 , 72°, 78°
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.

21. 6.1
Homework
Pg 376-379
#12, 16, 28, 50, 58