Home Previous Next

MAT081 :: Lecture Note :: Week 06
Assessments | Handouts | Math Resources {MathBabbler: Email | Facebook | Twitter}
GDT::Bits:: Time  |  Weather  |  Populations  |  Special Dates

Overview
Assignments

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)."
External Hyperlink(s)

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


Adding and Subtracting Decimals

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

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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
   final answer: ----
                  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

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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]

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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.

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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]

YouTube.com::Math Education Professor Dor Abrahamson

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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]

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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]

{TopOfPage} {MathWords} {PurpleMath} {KhanAcademy}
{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

External Hyperlinks

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


Percent Exercises

_______% of the five cells are red.
                   
_______% of the five cells are green.
                   
_______% of the ten cells are not white.
                                       
Shade 45% of the five cells.
                   
Shade 0.5% of the five cells.
                   

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


Home Previous Next