Download Honors Geometry Section 8.2 A Ratios and Proportions

Honors Geometry
Section 8.2 A
Ratios and Proportions
A ratio is a comparison of two
numbers by division.
Ratios can be written in two ways,
as a fraction or using a colon.
You are required to simplify all
ratios and you may not have a
decimal or a fraction as one of the
terms in a ratio.
Examples: number of boys in class
today _____number
of girls in class
8
today _____Write
a ratio of:
10
8 4

a) boys to girls
10 5
b) girls to boys 10 : 8 or 5 : 4
c) boys to students 8 : 18 or 4 : 9
In addition to the conversions for length in the previous
unit, you must know the following weight and volume
conversions.
1 ton (T) = __________
2,000 pounds (lb)
4
1 gallon (gal) = __________
quarts (qt)
16
2
1 quart (qt) = __________
pints (pt)
1 pound (lb) = __________ ounces (oz)
2
1 pint (pt) = __________
cups (c)
If the two terms in the ratio do not
have the same units, you must
convert both terms to like units
before simplifying.
Examples: Simplify.
a) 2.5 ft
b)
4in
500 yd
.75mi
30in 15

4in
2
c)
1500 ft 25

3960 ft 66
d) 3mm
28oz
4lb
.18cm
28oz 7

64oz 16
3mm 30 5


1.8mm 18 3
 A and  B are supplementary. If ,
mA  70 what is the ratio of mA to
mB ?
mB  180  70  110
70
7

110 11
Two complementary angles are in
the ratio of 7:13. Find the measure
of each.
7 x  13 x  90
20 x  90
x  4.5
7(4.5)  31.5
13(4.5)  58.5
The measures of the angles of a
triangle are in the ratio 8:5:3. Find the
measures of the angles of the triangle.
8 x  5 x  3 x  180
16 x  180
x  11.25
8(11.25)  90
5(11.25)  56.25
3(11.25)  33.75
A ratio with more than two terms is called an extended
ratio.
A proportion is an equation
relating two ratios.
If two ratios
could write
a
b
and
a c

b d
or
c
d
are equal, we
a:b  c: d
.
The following property makes proportion problems very easy to
solve.
Cross-Multiplication Property
If
a c

b d
, then
ad  bc
Examples: Solve for x.
1. 4  5
2.
x 7
5 x  28
x  5.6
4.
x 7

4 5
5 x  28
x  5.6
3
2

x2 x
2( x  2)  3 x
2 x  4  3x
x4
5.
3.
5x  8 5x

7
6
6(5 x  8)  35 x
30 x  48  35 x
 48  5 x
x  9.6
4 x

5 7
5 x  28
x  5.6
As you can see from examples 1, 4
and 5 above, the terms of a
proportion can be rearranged, or
even changed, and the resulting
proportion will be equivalent to
the original proportion as long as
the cross-products remain
equivalent.
Example: Which of the following proportions are
a 3
equivalent to b  7 ? 7 a  3b
1.
a 7

3 b
2.
a b

3 7
no
3.
a  b 10

b
7
7(a  b)  10b
7 a  7b  10b
7 a  3b
yes
yes
4.
7 b

a 3
no