Download Chapter 9.5--Equations Containing Fractions Chapter 9.5

Chapter 9.5--Equations Containing Fractions
Chapter 9.5--Equations Containing Fractions
Part 1: Solve Equations Containing Fractions
If we have an equation containing fractions we usually multiply both sides of the
equation with the common denominator of the fractions. Then we can solve the equation
without fractions. Of course, there is a hazard to this technique! You may get what looks
like a bona fide solution that really is not a solution at all. We call these extraneous
solutions and they must be discarded.
4x
x−4
Question 9: Solve
Multiply both sides by
4x
x−4
5 
5x
x−4
x−4
x − 4  5x − 4 
5x
x−4
x − 4
4x  5x − 20  5x
Look Ma! No Fractions!!!
4x  20
x 5
Always check your answers.
20
1
Question 12: Solve
Multiply by
 5?  ? 55
? Yes
1
x
x4
 3−
4
x4
x4
x  3x  4 − 4
x  3x  12 − 4
−2x  8
x  −4
Check your answer
−4
−44
Notice we are trying to divide by zero! Ouch!
So -4 is not a real solution.
There is no solution.
Question 16 Now you solve this one:
5
3n−8

n
n2
Cross-multiply
5n  10  3n 2 − 8n
Moving everything to the left:
3n 2 − 13n  10  0
3n − 10n − 1  0
n  10
or 1
3
W Clarke
1
12/3/2004
Chapter 9.5--Equations Containing Fractions
Part 2: Solve Proportions
A ratio is a fraction relating two quantities. A proportion is an equation that sets two
ratios equal to each other.
Example: Speed is a ratio. It is the fraction of distance over time. When we say miles per
hour, we mean the number of miles divided by the number of hours.
To solve proportions, all we have to do is multiply both sides by the product of the two
denominators. In actual practice this means we multiply each fraction by the denominator
of the other fraction. We call this cross-multiplication.
16
 64x
Question 27: Solve the proportion:
9
Multiply by 9x . This is equivalent to cross-multiplying.
16x  9  64
Now divide by 16
x  964
 9  4  36
16
6
2
Question 36: Now you solve this proportion: x−1  2x1
Cross multiply
22x  1  6x − 1
4x  2  6x − 6
−2x  −8
x 4
2
Question 43: A simple syrup is made by dissolving 2 c of sugar in 3 c of boiling water.
At this rate, how many cups of sugar are required for 2 c of boiling water.
sugar
Use the ratio
wat er
2
2/3

22
s
s
2
Multiply both sides by 2
2
3
Multiply top and bottom by 3
12
2
s
So we need to use 6 c of sugar.
W Clarke
2
12/3/2004
Chapter 9.5--Equations Containing Fractions
Question 48: Leonardo da Vinci measured various distances on the human body in order
to make accurate drawings. He determined that in general, the ratio of the kneeling height
3
of a person to his or her standing height is 4 . Using this ratio determine the standing
height of a person who has a kneeling height of 48 in.
kneeling
Use
st anding
3
4

48
s
Cross multiply
3s  48  4
s  484
 64
3
The person is 64 inches tall.
Part 3: Solving similar triangles.
Triangles are similar if corresponding pairs of angles are equal and corresponding sides
are in proportion.
In the diagram below, Triangles ABC and DEF are similar.
F
A
B
n
m
j
i
k
C
E
p
D
We get the following proportions:
j
j
i
k
i
k
m  n,
m  p,
n  p
Notice that in writing these proportions, I always put sides from the first triangle on top
and those from the second triangle on the bottom. Of course if you flipped all of the
fractions you would still have valid proportions.
W Clarke
3
12/3/2004
Chapter 9.5--Equations Containing Fractions
Question 65: Find the perimeter of triangle ABC.
D
C
10 m
7.5 m
A
4 in
F
B
AC
7.5

4
5
4 7.5
5
BC
10

30
5
4
5
E
5m
Now multiply both sides by 7.5
AC 

 6 in
BC  8 in
Question 71 Now you do this one: Given MP and NQ intersect at O, NO measures 25 ft,
MO measures 20 ft, and PO measures 8 ft. Find the length of QO.
Q
20 ft
M
O
8 ft
P
25 ft
N
We assume the triangles are similar.
QO
25

8
20
Multiply by 25
QO 
W Clarke
825
20
4
 10
12/3/2004
Chapter 9.5--Equations Containing Fractions
Question 83: Three people put their money together to buy lottery tickets. The first
person put in $25, the second person put in $30, and the third person put in $35. One of
their tickets was a winning ticket. If they won $4.5 million, what was the first person's
share of the winning?
First person’s input
Use the ratio
T ot al

First person’s share
T ot al W innings
25
253035

f
4.5
Multiply by 4.5 (million)
254.5
90
 1. 25  f
So the first person's portion is $1.25 million.
W Clarke
5
12/3/2004