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MAT081 :: Lecture Note :: Week 07
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### Ratios

A ratio is the "relationship in quantity, amount, or size between two or more things." [source::m-w.com::Merriam-Webster Online]

The general form of a ratio is as follows is `n:m`, where `n` is the quantity of one thing and `m` is the quantity of another thing. For example, if a recipe calls for 3 parts lemon and 5 parts lime, then the ratio is `3:5`.

Sometimes ratios are written using the word to instead of a colon. For example, 3 parts lemon to 5 parts lime.

Fractions can be written as ratios.

```    1/7  is   1:7  or   1 to 7
3/8  is   3:8  or   3 to 8
15/7  is  15:7  or  15 to 7
```

Which of the following, if any, are valid ratios?

```   8:8   1.5:1    4:2.25    0:55    4:0    2:-1
```

A rate is a special kind of ratio, indicating a "relationship between two measurements with different units."

PurpleMath.com:: Ratios [opens new window]

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

### Introduction to Rates

A rate is a ratio that "relate" different "units." Merriam-Webster Online defines rate as "a fixed ratio between two things."

Example rates.

```   65 miles per hour (mpg)   [ratio -- 65:1]
You can travel 65 miles in one hour.

\$0.59 per pound           [ratio -- 0.59:1]
It will cost you \$0.59 to buy one pound of foo.

60 beats per minute       [ratio -- 60:1]
A good heart rate for a man at rest is 60 beats in one minute.
{"beat" is one complete pulsation of the heart}
```
 It is possible for us to manually measure our heart rate. We were given a ratio of `60 beats per minute`, but a minute is a long time; therefore, we can approximate our heart rate by counting the heart beats for `10` seconds and multiply the final count by `6`. [There are `60` seconds per minute.] TopEndSports.com:: Heart rate measurement

The term per implies a ratio (i.e. fraction i.e. division).

```   Receive 10 coupons per every 3 purchases.
_
ratio... 10:3   fraction... 10/3   division... 3.3
```

There are many "forms" of rates. Birth rates, death rates, tax rates, and numerous other financial rates (e.g. mortgage rates, commission rates, interest rates, inflation rates just to name a few). There are also discount rates, tipping rates, data transfer rates, phone rates, turnover rates, failure rates and so on.

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

### Proportions

A proportion is a "statement of equality between two ratios in which the first of the four terms divided by the second equals the third divided by the fourth." [source::m-w.com::Merriam-Webster Online]

Given two ratios, `n:m` and `a:b`, `n` is `a` as `m` is to `b`.

Two ratios are equal if the cross-products are equal.

```     /----- * ----\
|            |
2:4  equals   10:20
|                 |
\------- * -------/

2 * 20 = 40
4 * 10 = 40
```

From the book "Zero: The Biography of a Dangerous Idea"...

```   "To the Pythagoreans, ratios and proportions controlled
musical beauty, physical beauty, and mathematical beauty.
Understanding nature was as simple as understanding the
mathematics of proportions."
```

PurpleMath.com:: Proportions: Introduction [opens new window]

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

### 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}