Percent

expressing a number as a part of 100.

One of the simplest ways to solve percent problems is with the proportion:

%
100

You should always know 2 of the three things (part, whole, and %). You can cross-multiply to find what you don ’t know.
Example: 13 is what percent of 75?

%
100
13
 x
75 100
Example: What is 42% of 7000?
_
13

100
÷
75 = 17.3%

%
100 x

42
7000 100
7000

42
÷
100 = 2940
Example: 53 is 40% of what number? part whole

%
100
53

40 x 100
53

100
÷
40 = 132.5
Example:
Sales tax is 7%. How much is the tax on a $157 TV set?

%
100 x

7
157 100
157

7
÷
100 = $10.99

Example:
Garrigan is having a magazine sales drive. The goal for sales is $10,250. As of Monday they had sold $9,278.
What percent of the goal is this? part

% whole 100
9278
10250
9278

100

÷
%
100
10250
… = about 90.5%.
Example
You need to get 80% to earn a “B” in a course. You earned 377 points, which was just barely enough to get a
“B”. How many total points were available in the course?
%

100
377

80 x 100
377

100
÷
80 = 471.25
Example
(rounds to 471 points)
A pair of jeans that originally sold for $79 are on sale for 45% off. a. How much do you save? b. How much do you pay?

%
100 x

45
79 100
79

45
÷
100 = 35.55
(So you save $35.55)
79 – 35.55 = 43.45
(So you pay $43.45)
Example
Someone’s bar tab came to $11.75. They told the bartender to keep the change from $15. a. b.
How much money did they tip?
What percent of the bill did they tip?
15 – 11.75 = $3.25 part

% whole 100
3 .
25
11 .
75

100
3.25

100
÷
11.75
This was about a 28% tip.

Example
Change the fraction
7
8
to a percent.

%
100
7
 x
8 100
7

100
÷
8 = 87.5%
Example
Last July gas cost $1.74. Now it costs $1.89. What is the percent of increase?

First find how much it increased.
1.89 – 1.74 = .15
 For increase or decrease problems, the “whole” is always the original amount. part whole

%
100
.
15
1 .
74
 x
100
… about 8.6%
Also remember …

You can change between percents and decimals by moving the decimal point.

Move the decimal 2 places right to change a decimal to a percent

Move the decimal 2 places left to change a percent to a decimal
Example: Change .04275 to a percent.
.04275  4.275%
Change 70% to a decimal
70%  .70 or .7
INTEREST TERMS
Principal

Original amount of money you borrow or invest
Interest
 Money paid for the right to use someone else’s money over time
Simple Interest

You borrow money and pay it back after a given time period at a set percentage rate.

This is often used in agreements between friends or relatives.

Bonds also often pay simple interest.

Banks, credit cards, and loan companies almost never deal in simple interest.

Idea of simple interest
Years
0
1
2
3
4
5
6
Interest at 4%
$4
$4
$4
$4
$4
$4
Total Value
$100
$104
$108
$112
$116
$120
$124
Notice that you earn exactly the same amount of interest every year. The total value goes up at a steady rate.
Simple Interest Formulas
 I

P rt
 A

P

1
 rt

P … principal rate of interest
(original investment) r t
A
…
…
… time
(always a decimal)
(in years) accumulated amount
(total value after time, including interest
– also called “future value”)
You borrow $700 from your brother-inlaw and agree to pay him back with 6% interest. If you don’t pay him back until 3 years later, how much do you owe him?
A

P ( 1
 rt )
A

700 ( 1

.
06

3 )
A

$ 826
You owe him $826.00
Mr. Rich purchased a bond for $7,500. When he dies 24 years later, his estate redeems the bond, which earned 4.25% simple interest.
How much interest does he earn?
I

7500

.
0425

24
$ 7650
What was the total value of the bond?
A

7500

1

.
0425

24

A

$ 15 , 150

Lisa borrowed $300 from her sister. She paid back the loan three months later, including 15% interest. How much did she pay back?
NOTE: The time must be in years … 3 months is ¼ year.
A

300

1

.
15

.
25
 b.
I
$ 311 .
25
When little Billy is born, his grandmother buys him a bond at a cost of $3164. When Billy heads off to college
18 years later, he cashes in the bond, which is now worth $5000. What rate of interest did the bond earn?
FIRST  How much interest was earned?
5000

3164

1836

I
NOW  Find the rate of interest.
I

P rt
1836

3164
 r

18
1836

56952 r
.
0322
 r
The bond earned about 3.22%.
Many big cities have “Pay-Day Loan” services that advance people money and then have the people sign over their paychecks to cover the loan.
Nick’s regular paycheck is $642.19. He goes to the loan service 5 days before payday. They take out a small service charge and advance him $612.19. Then, 5 days later, he signs over his check. a.
I

642 .
How many dollars in interest did Nick pay?
19

612 .
19

$ 30 .
00
In years, what was the term of the loan? c. d.
5 year
365
What amount was the principal?
P

612 .
19
What was the interest rate?

I

P rt
30

612 .
19
 r

5
365
30

8 .
3861543 r
3 .
5773251
… This is 357.7%.
 r
Compound Interest

Interest accumulates both on the principal and on interest that is already accumulated.

In the real world, most investments and loans involve compound interest.
Idea of compound interest
Years Interest at 4% Total Value
0
1
2
3
4
5
6
$4.00
$4.16
$4.33
$4.50
$4.68
$4.87
$100
$104.00
$108.16
$112.49
$116.99
$121.67
$126.54
Notice that you earn more interest each year than you did the year before. The total value increases more and more rapidly as you go on.
Compound Interest Formula
A

P


1 r m 


 
P … principal t r
A …
…
… accumulated amount
(sometimes called
“future value”) rate of interest time
(sometimes called
“present value”)
(always a decimal)
(years) m … # of times compounded per year quarterly monthly

 semiannually  weekly annuall


4
12
2
52
1 daily  ???
Traditionally daily compou nding has used the “bankers’ rule” of 360 days/year. However, recently most banks have switched to 365.

Sally inherits $12,345.67 when her grandfather dies. She invests the money in an account that earns 8% interest, compounded monthly. How much will her account be worth 9 years from now?
.
08


12

9

12345 .
67


1
12 

You can type this straight in on a calculator if you’re careful with parentheses.
12345.67(1+.08/12)^(12*9)
=$25302.82
Dean invests $300 at 5% interest, compounded daily. How much money will he have in 7 months?


12
7
12 


300 1
.
05
12
300(1+.05/12)^(12*7/12)
= $308.86
Find the present value of $4000 invested at 3% interest for 10 years, compounded annually.
A

P


1 r m 


 
We need to find “P”.
4000
4000

P
 



2976 .
38
P


1
P
.
03
1 



1

10

.
343916379
Effective Rate of Return

The rate of simple interest that would need to be paid to earn the same amount in a year that compound interest pays.

Allows you to compare interest with different compounding periods.
Effective Rate Formula r eff
 

1 r m 

 m

1
Note that the parentheses are the same as the compound interest formula.
 Idea is that you’re figuring out how much compound interest $1 earns in 1 year

An account earns 7% interest, compounded weekly. Find the effective rate of return. r eff
 
1
.
07

52
52

1
A
B
(1+.07/52)^52-1
=.0724576961
… about 7¼%
Which account is the better deal?
A  8.25%, compounded
B daily
 8.4%, compounded semiannually
 (1+.0825/365)^365
-1 = .0859885495
 (1+.084/2)^2-1
= .085764
A is the better deal.