Download Algebra Unit IV - Notes Section 3.6

Problems of the Day
1.)  3k
 5(6  k )  4(1  2k )
No Solution
2)
0.4 y  1.2  0.3 y  0.6
3)
2(v  8)  7  5(v  2)  3v  19
1
1 1
1
4)
a  a
3
2 2
3
y = 18
All real Numbers
a=–1
Ratios and Proportions
A ratio is a comparison of two numbers by division. The
ratio of x to y can be expressed in the following ways:
x to y, x : y, or
x
y
For example: Your school’s basketball team has won 7
games and lost 3 games. What is the ratio of wins to losses?
Because we are comparing wins to losses the first number in
our ratio should be the number of wins and the second
number is the number of losses. The ratio is 7 to 3.
An equation stating that two ratios are equal is called a
proportion.
So for example, we can state that
proportion.
2 8

3 12
is a
We could also write that proportion as 2 : 3 = 8 : 12.
Example 1 – Write an example of another proportion.
Investigation….
Look at the following proportions.
Discuss with your group if you notice
anything about all of these examples.
1 5

2 10
12 4

9 3
7 14

9 18
18 9

4 2
2 8

3 12
Try these:
Solve for x in the following
proportions using common sense, not
yet algebra:
2 x
1) 3  15
3 x6
3) 
4
12
x = 10
x=3
4 16

2)
3x 24
x=2
a c
Any proportion can be written in the form
 or a:b = c:d,
b d
where a and d are considered the “extremes” and b and c are
considered the “means”.
*** In a proportion, the product of the extremes is equal to the
product of the means. (ad = bc)
2 8
Numerical example: 
3 12
, so in this example 2∙12 = 3∙8
24 = 24
Determining if Ratios are Proportional
When asked to determine if ratios are
proportional, you are really being asked to
determine if the product of the means
equal the product of the extremes.
Example:
3 9
 ?
Does: 4 12
4 3
 ?
3 2
Check: 3(12) = 4(9)?
36 = 36?
Check: 4(2) = 3(3)?
8 = 9? X
Example 3 -
Example 2 -
Solve 5  35
3 x
4 6x

Solve
5 15
4(15) = 5(6x)
3(35) = 5(x)
105 = 5x
5
60 = 30x
30 30
5
21 = x
x = 21
2=x
x=2
Example 4 -
2
5

Solve the proportion
x  3 30
2(30) = 5(x + 3)
60 = 5x + 15
Be Careful!
Use
parenthesis if
the expression
has more than
one term
when cross
multiplying!!
-15
-15
45 = 5x
5
5
9=x
x=9
CHECK
2
5

x  3 30
2
5

9  3 30
2
5

12 30
2  30  5 12
60  60
Example 5 -
3(6) = 4(x – 2)
18 = 4x – 8
+8
+8
26 = 4x
4
4
6½ = x
x = 6.5
3 x2

4
6
CHECK
3 x2

4
6
3 6 .5  2

4
6
3 4 .5

4
6
3  6  4  4 .5
18  18
Example 6 -
8(3 + y) = 4(-y)
24 + 8y = -4y
+4y
+4y
24 + 12y = 0
-24
-24
12y = -24
y=-2
3 y  y

4
8
CHECK
3 y  y

4
8
3  (2)  (2)

4
8
1 2

4 8
1 8  2  4
88
Example 7 -
Trent goes on a 30-mile bike ride every Saturday. He
rides the distance in 4 hours. At this rate, how far can he
ride in 6 hours?
miles
hours
30 m

4
6
30 · 6 = 4(m)
180 = 4m
45 = m
miles
hours