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Section 2.4
Formulas and
Percents
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Objective 1
Solve a formula for a variable.
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Solving a Formula for a Variable
• We know that solving an equation is the
process of finding the number or numbers that
make the equation a true statement.
Formulas contain two or more letters,
representing two or more variables. The
formula for the perimeter P of a rectangle is
P  2l  2w where l is the length and w is the
width of the rectangle. We say that the formula
is solved for P, since P is alone on one side
and the other side does not contain a P.
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Solving a Formula for a Variable (cont)
• Solving a formula for a variable means
using the addition and multiplication
properties of equality to rewrite the
formula so that the variable is isolated on
one side of the equation.
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Solving a Formula for a Variable (cont)
To solve a formula for one of its variables, treat
that variable as if it were the only variable in the
equation. Think of the other variables as if they
were just numbers. Use the addition property of
equality to isolate all terms with the specified
variable on one side. Then use the
multiplication property of equality to get the
specified variable alone. The next example
shows how to do this.
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Area of a Rectangle
The area, A, of a rectangle with length l and
width w is given by the formula
A  lw .
l
w
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Perimeter of a Rectangle
The perimeter, P, of a rectangle with length
l and width w is given by the formula
P  2l  2w .
l
w
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Solving a Formula for a Variable
Solve the perimeter equation for w.
2w  2l
2w  2l  2l
2w
2w
2
P
 P  2l
 P  2l
P  2l

2
P  2l
w
2
Subtract 2l from both sides.
Simplify.
Divide both sides by 2.
Simplify.
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Example
Solve the formula y  mx  b for x.
y  mx  b
y  b  mx  b  b Subtract b from both sides.
y  b  mx
Simplify.
y b
x
Divide both sides by m
m
to find x.
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Objective 1: Example
1a. Solve the formula A  lw for l .
A  lw
A lw

w w
A
l
w
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Objective 1: Example
1b. Solve the formula 2l  2w  P for l .
2l  2w
2l  2w  2w
2l
2l
2
P
 P  2w
 P  2w
P  2w

2
P  2w
l
2
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Objective 1: Example
1c. Solve the formula T  D  pm for m.
T  D  pm
T  D  pm
T  D pm

p
p
T D
m
p
T D
m
p
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Objective 1: Example
x
1d. Solve the formula  4 y  5 for x.
3
x
 4y  5
3
x

3   4y   3  5
3

x
3   3  4y  3  5
3
x  12y  15
x  12y  12y  15  12y
x  15  12y
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Objective 2
Use the percent formula.
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Percents
• Percents are the result of expressing
numbers as a part of 100. The word
percent means per hundred or 1/100.
• If 45 of every 100 students take
Introductory Algebra, then 45% of the
students take Introductory Algebra. As a
fraction, it is written
45
100
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Writing Decimals as Percents
Using the definition of percent, you should
be able to write decimals as percents and
also be able to write percents as decimals.
Here is the rule for writing a decimal as a
percent.
1. Move the decimal point two places to the
right.
2. Attach a percent sign.
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Example
Express 0.47 as a percent.
0.47  47% (since percent means 1/100,
both sides here mean “47/100.”)
Express 1.25 as a percent.
1.25  125%
When we insert a percent sign, we move the
decimal point two places to the right.
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Writing Percents as Decimals
Use the following steps to write a percent
as a decimal.
1. Move the decimal point two places to the
left.
2. Remove the percent sign.
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Example
Express 63% as decimal.
63%  0.63
Express 150% as decimal.
150%  1.50
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Percent Formula
A
is
P percent
of
B
 P
A
·
B
In the formula,
A  PB
B  Base Number
P  Percent written as a decimal
A  The number compared to B
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Example
8 is what percent of 12?
8
is

8
8  P  12
P percent of
P
12
· 12
8 12P

12
12
0.66  P
P  67% Rounded to the nearest percent.
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Example
What is 12% of 8?
What
is
12 percent
A

0.12
8
of
·
8
A  0.12  8 
A  0.96
Thus, 12% of 8 is 0.96.
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Example
5 is 25% of what number?
5
5
is

25 percent
0.25
what number?
of
·
B
5  0.25B
5
0.25B

0.25
0.25
20  B
B  20
Thus, 5 is 25% is 20.
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Objective 2: Example
2a. What is 9% of 50?
Use the formula A  PB
A is P percent of B.
what? is 9% of 50
 0.09  50
A  4.5
4.5 is 9% of 50.
A
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Objective 2: Example
2b. 9 is 60% of what?
Use the formula A  PB
A is P percent of B.
9 is 60% of what?
9  0.60 
9
0.60B

0.60 0.60
15  B
B
9 is 60% of 15.
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Objective 2: Example
2c.
18 is what percent of 50?
Use the formula A  PB: A is P percent of B.
18 is what percent of 50
18 
P
18  P  50
18 P  50

50
50
0.36  P
 50
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Objective 2: Example (cont)
To change 0.36 to a percent, move the
decimal point two places to the right and
add a percent sign.
0.36  36%
18 is 36% of 50.
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Objective 3
Solve applied problems involving
percent change.
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Percent Increase and Decrease
Percents are used for comparing changes,
such as increases or decreases in sales,
population, prices, and production. If a
quantity changes, its percent increase or
percent decrease can be determined by
asking the following question:
The change is what percent of the
original amount?
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Percent Increase and Decrease
The question is answered using the percent
formula as follows:
Percent Increase
A

P
B
The increase is what
percent of the original
amount.
Percent Decrease
A

P
B
The decrease is what
percent of the original
amount.
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Objective 3: Example
3a. A television regularly sells for $940.
The sale price is $611. Find the percent
decrease in the television’s price?
Use the formula A  PB:
A is P percent of B.
The price
what
the original
decrease is percent of price?
329

P

940
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Objective 3: Example (cont)
329  P  940
329 940P

940 940
0.35  P
To change 0.35 to a percent, move the
decimal point two places to the right and add
a percent sign. 0.35  35%
There was a 35% decrease.
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Objective 3: Example (cont)
3b. Suppose you paid $1200 in taxes.
During year 1, taxes decrease by 20%.
During year 2, taxes increase by 20%.
What do you pay in taxes for year 2?
How do your taxes for year 2 compare
with what you originally paid, namely
$1200? If the taxes are not the same,
find the percent increase or decrease.
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Objective 3: Example (cont)
First, find the amount that the taxes
decreased from the original year to
year 1:
0.20  $1200  $240
Next, subtract this amount of decrease
from the original tax amount to obtain
the amount paid in year 1.
Amount paid in year 1:
$1200  $240  $960
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Objective 3: Example (cont)
Now, find the amount that the taxes
increased from year 1 to year 2:
0.20  $960  $192
Next, add this amount of increase to
the amount paid in year 1 to obtain the
amount paid in year 2.
Amount paid in year 2:
$960  $192  $1152
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Objective 3: Example (cont)
The taxes for year 2 are less than
those originally paid.
Since the taxes are not the same
($1200 original year, $1152 year 2), find
the percent decrease.
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Objective 3: Example (cont)
Find the tax decrease:
$1200  $1152  $48
The tax
what
the original
decrease is percent of tax?
 P

48  P  1200
48
1200P

1200 1200
0.04  P
48
1200
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Objective 3: Example (cont)
To change 0.04 to a percent, move the
decimal point two places to the right and add
a percent sign.
0.04  4%
The overall tax decrease is 4%.
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