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MAT081 :: Lecture Note :: Week 08
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##### Overview
• [Tuesday] 10-Point Assessment #4 is due.
• [Tuesday] Review for 7-Point Departmental Exam #2 (of 4).
• [Thursday] Take second (2 of 4) 7-point Departmental Exam.

### Percents

The word percent means "per one hundred."

Percent values are suffixed by a `%` character.

A percent is a fraction where the denominator is 100.

```      5%  equals     5/100
15%  equals    15/100
120%  equals   120/100
37.5%  equals  37.5/100
```

Percent values can be written as a decimal number by multiplying the value by `0.01`.

```     4%  equals    4(.01)  equals  0.04
11%  equals   11(.01)  equals  0.11
110%  equals  110(.01)  equals  1.10
```

Decimal values can be written as percents by multiplying by 100%.

```   0.02  equals  0.02(100)%  equals    2%
0.33  equals  0.33(100)%  equals   33%
2.10  equals  2.10(100)%  equals  210%
```

Percents can be written as fractions by multiplying by `1/100` (or one one-hundreth).

```     5%  equals    5(1/100)  equals    5/100
25%  equals   25(1/100)  equals   25/100
110%  equals  110(1/100)  equals  110/100
```

PurpleMath.com:: Converting Between Decimals, Fractions, and Percents [opens new window]

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Converting Fractions Into Percents and Vice Versa

The following algorithm converts a fraction into a percent.

```   a)  divide the numerator by the denominator

b)  multiply the resulting decimal number by 100(%)
```

Examples.

```    3/8 = 0.375;     0.375 * 100(%) = 37.5%
1/2 = 0.5;       0.5 * 100(%) = 50%
7/11 = 0.6364;    0.6364 * 100(%) = 63.64%
11/7 = 1.5714;    1.5714 * 100(%) = 157.14%
```

The following algorithm converts a percent into a fraction.

```   a)  write the percent as a fraction (n% = n/100)

b)  reduce the fraction, if possible
```

Examples.

```   25% = 25/100;        25/100 reduces to 1/4
62% = 62/100;        62/100 reduces to 31/50
202% = 202/100;      202/100 reduces to 2 1/50
1% = 1/100;          1/100 is reduced
0.2% = 0.2/100;      0.2/100 reduces to 0.1/50 or 1/500
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Percent Change

The percentage increase between two values is calculated as follows.

```   new value - original value
-------------------------- * 100
original value
```

The percentage decrease between two values is calculated as follows.

```   original value - new value
-------------------------- * 100
original value
```
##### Examples

The Maricopa Community Colleges increased 2005-06 in-state tuition rates by `\$5` per credit-hour. The 2004-05 tuition rate was `\$55`.

```            old rate:  \$55   [2004-05 rate]
increase:  \$ 5   ["increase" implies addition]
new rate:  \$60   [sum of \$55 and \$5]

percentage increase:

60 - 55            5
-------     =     ---   =    0.0909
55              55

0.0909 * 100 = 9.09%
```

The Maricopa Community Colleges increased 2006-07 in-state tuition rates from `\$60 per credit-hour` to `\$65 per credit-hour`.

```   percentage increase:

65 - 60            5
-------     =     ---   =    0.0833
60              60

0.0833 * 100 = 8.33%
```

The Valley Metro Transit is considering decreasing bus fares by `\$0.25`. Current bus rates are `\$1.25` per two hours of bus riding.

```        current rate:  \$1.25
proposed decrease:  \$0.25  [the word "decrease" implies subtraction]
new rate:  \$1.00  [difference of \$1.25 and \$1.00]

percentage decrease:

1.25 - 1.00           0.25
-----------     =     ----   =    0.2000
1.25              1.25

0.2000 * 100 = 20%
```

PurpleMath.com:: "Percent of" Word Problems: General Increase and Decrease [opens new window]

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### "Percent of" Problems

Percent problems come in many forms. Three popular forms are often worded as follows.

```   What is 'a' percent of 'b'?

'a' is what percent of 'b'?

'a' is 'b' percent of what?
```

Here are some rules to map these type of percent word problems into equations.

```   "what" becomes a variable
"is" becomes an equals operator
"of" becomes a multiply operator
```

1) What is `5%` of `10`?

```   n = 5% * 10            ...variable is named 'n'
n = 5(.01) * 10        ...%5 converted to decimal
n = 0.05 * 10          ...word "of" became times
n = 0.5                ...decimal arithmetic

0.5 is 5% of 10        ...final answer
```

2) `7` is what percent of `49`?

```   7 = n * 49

7     n * 49
-- =  -------
49       49

1
- = n
7

.143 = n

.143 * 100 = 14.3 = n  (final answer:  14.3%)

7 is 14.3% of 49
```

3) `4` is `12%` of what?

```    4 = 12% * n

4 = 12(.01) * n

4    .12 * n
--- = -------
.12     .12

33.3 = n

4 is 12% of 33.3
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}