copyright © 2013, 2009, 2006 pearson education, inc. 1 1 section 2.4 formulas and percents...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 2.4

Formulas andPercents

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1


Objective #1 Solve a formula for a variable.


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.



• 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.

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.


The area, A, of a rectangle with length l and width w is given by the formula

A = lw.

l

w

l

Area 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

Perimeter of a Rectangle


Solve the perimeter equation for w.2w + 2l = P2w + 2l – 2l = P – 2l Subtract 2l from both

sides.

2w = P – 2l Simplify.

Divide both sides by 2.

Simplify.

2 2

2 22

2

w P l

P lw




EXAMPLEEXAMPLE

SOLUTIONSOLUTION

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.

xm

by

Divide both sides by m to find x.


1a. Solve the formula A lw for l.

A lw

A lww wA

lw

Objective #1: Examples


1a. Solve the formula A lw for l.

A lw

A lww wA

lw




1b. Solve the formula 2 2l w P for l.

2 2

2 2 2 2

2 2

2 22 2

22

l w P

l w w P w

l P w

l P w

P wl


1c. Solve the formula T D pm for m.

T D pm

T D pm

T D pmp p

T Dm

p

T Dm

p




1d. Solve the formula 4 53x

y for x.

4 53

3 4 3 53

3 3 4 3 53

12 15

12 12 15 12

15 12

xy

xy

xy

x y

x y y y

x y


Objective #2 Use the percent formula.


• 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

100

45

Percents


1. Move the decimal point two places to the right.

2. Attach a percent sign.

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.


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.

Writing Decimals as Percents

EXAMPLEEXAMPLE


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.

Writing Percents as Decimals


Express 63% as a decimal.

63% = 0.63

Express 150% as a decimal.

150% = 1.50

Writing Percents as Decimals

EXAMPLEEXAMPLE


A = P · BIn the formula,

A = PBB = Base NumberP = Percent written as a decimalA = The number compared to B

A is P percent of B

Percent Formula


8 is what percent of 12?

8 = P · 12

8 = P · 12

8 is P percent of 12

8 12

12 120.66

67% Rounded to the nearest percent.

P

P

P

Using the Percent Formula

EXAMPLEEXAMPLE


8 is what percent of 12?

8 = P · 12

8 = P · 12

8 is P percent of 12

8 12

12 120.66

67% Rounded to the nearest percent.

P

P

P


EXAMPLEEXAMPLE


What is 12% of 8?

A = 0.12 · 8

A = 0.12(8)

A = 0.96

Thus, 12% of 8 is 0.96.

What is 12 percent of 8


EXAMPLEEXAMPLE


5 is 25% of what number?

5 = 0.25 · B

5 = 0.25B

5 is 25 percent of what number?

5 0.25

0.25 0.2520

20, Thus 5 is 25% of 20.

B

B

B


EXAMPLEEXAMPLE



2a. What is 9% of 50?

Use the formula : is percent of .A PB A P B

What is 9% of 50?

0.09 50A 4.5A

4.5 is 9% of 50.



2a. What is 9% of 50?


What is 9% of 50?

0.09 50A 4.5A

4.5 is 9% of 50.



2b. 9 is 60% of what?


9 is 60% of what?

9 0.60 B 9 0.60

0.60 0.6015

B

B

9 is 60% of 15.



2c. 18 is what percent of 50?


18 is ofwhat percent 50?

18 50P 18 50

18 5050 50

0.36

P

P

P

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.



2c. 18 is what percent of 50?


18 is ofwhat percent 50?

18 50P 18 50

18 5050 50

0.36

P

P

P


18 is 36% of 50.


Objective #3 Solve applied problems involving percent change.


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?

The question is answered using the percent formula as follows:

Percent Increase Percent Decrease The is what of the original The is what of the original increase percent amount decrease percent amount

= A P B = A P B



3a. A television regularly sells for $940. The sale price is $611. Find the percent decrease in the television’s price.

Use the formula : is percent of .A PB A P B Find the price decrease: $940 $611 $329

The price what the originaldecrease is percent of price?

329 940P

329 940

329 940940 9400.35

P

P

P


There was a 35% decrease.



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.

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

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

The taxes for year 2 are less than those originally paid.



Since the taxes are not the same ($1200 year 1, $1152 year 2), find the percent decrease. Find the tax decrease: $1200 $1152 $48

The tax what the originaldecrease is percent of tax?

48 1200P

48 1200

48 12001200 12000.04

P

P

P

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%.

CONTINUECONTINUE

copyright © 2013, 2009, 2006 pearson education, inc. 1 1 section 2.4 formulas and percents...

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