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MAT081 :: Lecture Note :: Week 06
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### Introduction to Decimal Numbers

A decimal number is written with the whole number followed by a dot (decimal point) followed by the fractional part. A decimal number falls between two integers that differ by one in value.

```   2 < 2.43 < 3
8 < 8.0001 < 9
99 < 99.9999 < 10
```

A decimal fraction is a fraction where the denominator is 10 raised to a positive integer exponent.

```   104 equals 10,000
103 equals  1,000
102 equals    100
101 equals     10
100 equals      1
10-1 equals     1/10      equals   0.1
10-2 equals     1/100     equals   0.01
10-3 equals     1/1000    equals   0.001
10-4 equals     1/10,000  equals   0.0001
...
10-9 equals     0 1/1,000,000,000  equals   0.000000001
```

To the left of the decimal point are the ones, tens, hundreds, thousands, and so on. On the fractional side of decimal number are the tenths, hundreths, thousandths, and so on. There are no oneths.

Decimal numbers that lie between zero and one (and zero and a negative one) are often prefixed with a zero.

```
.1 = 0.1       .375 = 0.375      -.55 = -0.55
```

Trailing zeroes after the decimal point are not necessary; however, in science, engineering, statistics and other fields, trailing zeros are retained to show a level of confidence in the accuracy of the number.

When writing a decimal number in English, use the word and to represent the decimal point.

```      7.59  is  seven and fifty-nine hundreths
0.459  is  four hundred fifty-nine thousandths
5000.29  is  five thousand and twenty-nine hundreths
233.056  is  two hundred thirty-three and fifty-six thousandths
```
##### A Nano-Moment
 Note: nano is a prefix meaning one-billionth (or `10-9` or `1/1,000,000,000` or `0.000000001`). ``` 1 nano... ------------- = 0.000000001 1,000,000,000 ``` In everyday-world, GDT replaces nano with "very, very, very small." For example, a nanofoo is a very, very, very small foo. It doesn't matter what foo is; whatever it is, it is very, very, very small. Let's get smaller (i.e. closer to zero)... A nanosecond is a very, very, very short second (i.e. a billionth of a second). ``` From Fall 2004: "Optical 'rulers' are lasers that emit pulses of light lasting just 10 femtoseconds (10 quadrillionths of a second, or 10 millionths of a billionth of a second)." ``` GDT::Blog:: Nanotech SmallBlog GDT::BAB:: A Mathematical Singularity GDT::Computing::Bit:: Nanosecond--Grace Hopper to ASU GDT::BAB:: Popular Phrase--Nano Giga

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

If necessary, re-write the problem vertically and lineup the decimal points. It is okay to pad decimal numbers with zeros.

```   example:
4.451 + 13.3

4.451
+ 13.3
--------
17.751

or

4.451
+ 13.300
--------
17.751

example:
5.00073 + 255.101

5.00073
+ 255.10100
-----------
260.10173
```

Subtraction works the same way; i.e., lineup the decimal points prior to doing the subtraction.

```   5.55 - 4.02

5.55
- 4.02
------
1.53
```

[top]

### Multiplying Decimals

Execute the multiply as if dealing with whole numbers (i.e. it is not necessary to lineup the decimal points).

Insert the decimal point in the product by starting at the right and moving a number of places equal to the sum of the decimal places in both numbers multiplied.

```   5.50 * 2.1

5.50
*    2.1
--------
550
+  1100
--------
11550

5.50 has 2 digits to the right of the decimal point.
2.1 has 1 digit to the right of the decimal point.

2+1 is 3; therefore, the decimal point goes left of
the 3rd digit from the right

11.550  or 11.55
```

[top]

### Dividing Decimals

If the divisor has a decimal point, then make it a whole number by moving the decimal point to the right.

Move the decimal point in the dividend to the right by the number of moves made in the divisor.

Execute a whole number division ignoring any decimal point in the dividend.

Insert a decimal point in the quotient directly above the decimal point in the dividend.

```
_________
3.5 ) 15.75

45
_________
35 )  157.5
-140
---
175
-175
---
0

Insert the decimal point into the quotient.

4.5
```

Recall that answers to division problems can be checked by multiplying the quotient (result) by the divisor. The product should equal the dividend.

```     4.5
* 3.5
-----
225
+135
-----
1575

Both 4.5 and 3.5 have 1 digit to the right of the decimal.
Insert decimal point into the product 2 digits left of the
right-most digit.

15.75
```

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

### Rounding Numbers (Common Method)

Find the rounding-to digit. If the digit immediately to the right of the rounding-to digit is five or larger, increase the rounding-to digit by one; otherwise, leave the rounding-to digit unchanged. In both cases, all digits to the right of the rounding-to digit become zero.

```              --------- nearest ---------
number     ten    hundred   thousand
----------------------------------------
13825      13830   13800     14000
77093      77090   77100     77000
72555      72560   72600     73000

--------- nearest ---------
number     tenth  hundreth  thousandth
--------------------------------------
1.28382     1.3     1.28       1.284
102.1291  102.1   102.13     102.129
0.05454     0.1     0.05       0.055
```

PurpleMath.com:: Rounding Numbers

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

### Reducing Factions

Fractions are often expressed in their lowest form (i.e. simplified or reduced).

The following technique can be used to reduce fractions.

```   1) Do a prime factorization of the numerator.
2) Do a prime factorization of the denominator.
3) Cancel (eliminate) "like" factors.
```

Example.

```               12
reduce:   ------
44

2 * 2 * 3
prime factorizations: ------------
2 * 2 * 11

2 * 2 * 3
cancel like factors:  ------------
2 * 2 * 11

3
11
```

Another reduction techique is to use the greatest common factor of the numerator and denominator.

```   x                 x ÷ gcf(x,y)
-   reduces to    ------------
y                 y ÷ gcf(x,y)
```

Example.

```               12
reduce:   ------
44

gcf(12, 44) equals 4

12 ÷ 4       3
------  =  ----
44 ÷ 4      11
```

Another fraction reducing strategy is as follows.

```   1) Determine if the numerator or the denominator is the
simplist prime factorization and do a prime factorization.
2) Cancel the prime factors that are also factors of the
fraction part that was not prime factorized.
```

Example.

```              25
reduce:  ------
205

do a prime factorization on the numerator

25     5 * 5
------ = -----
205     205

5 evenly divides 205 41 times
5 does not evenly divide 41

25       5
------ = -----
205      41
```

An observation about cancelling like terms...

```	a(c)  a   c   a
--- = - * - = -    c divided-by c is 1
b(c)  b   c   b
```

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

### 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}