Chapter 3

 

Section 3.1 Least Common Multiples (LCM)

The Least Common Multiple of at least two numbers is the smallest number that they will all multiply into.

Method 1:  Finding the LCM by using a list of multiples

Consider the LCM(10, 15)  15 has multiples of 15, 30, 45, 60 and 10 has multiples of 10, 20, 30, 40, 50, 60.  Since 30 and 60 are in both lists, and 30 < 60, then 30 is the correct `answer.

Method 2:  Finding the LCM by finding prime factorizations.

Consider the LCM( 16, 24 )  16 = 24 while 24 = 23 ∙ 3.  The only the number 24 differs from the number 16 by 2, therefore 2 ∙ 24 = 48.  Likewise the number 16 differs from the number 24 by 3 which means 3 ∙ 16 = 48 hence

LCM( 16, 24 ) = 48

Method 3:  Finding the LCM by the use of division by prime numbers.  Consider LCM(24, 32, 40).  Dividing it through I yield

          2       |  24  32  40

          2       |  12  16  20

          2       |    6    8  10

                        3    4    5

Even though 4 is not a prime number, because there exists no number that divides all three except 1, then

LCM( 24, 32, 40 ) = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 4 ∙ 5 = 480 which is confirmed by my calculator.  If one wishes to find a LCM of 3 or more numbers on a calculator, the above example could be found by LCM( LCM( 24, 32 ), 40 ) since most calculators do not examine more than two numbers at once.

 

Section 3.2 Addition and Applications

If the denominators are the same on all fractions, the numerators add right up.  An example of this would be:

3/12 + 5/12 + 1/12 =  ( 3 + 5 + 1 )/12 = 9/12 = 3/4

If the denominators are different, find the LCM of the denominators, multiply the numerators by the same number needed to make the denominator = LCM of all the denominators, then just add all of the numerators together and simplify.  Consider 1/3 + 5/8 + 5/12 = 8/24 + 15/24 + 10/24 =

( 8 + 15  + 10 )/24 = 33/24 = 19/24 = 13/8

 

Section 3.3 Subtraction, Order and Applications

Just like addition above the denominators should be = LCM of the denominators; then subtraction of the numerators is legal, just like addition above.  Consider 1/2 + 3/12;

LCM( 2, 12 ) = 12.  Then 1/2 = 6/12 implies

 6/12 - 3/12 = ( 6 - 3  )/12 = 3/12 = 1/4

To determine the order of the fractions, first the denominators must be the same.  Then look at the order of the numerators (see section 1.4).  An example would be:

1/2 □ 3/7.  To make this equation true, first make the denominators = by finding LCM( 2, 7 ).  LCM( 2, 7 ) = 14 then we have 7/14 □ 6/14 .  By comparing the numerators,

7 > 14 then 7/14 > 6/14

 

Section 3.4  Mixed Numerals

53/10 = 5 + 3/10 = 50/10 + 3/10 = 53/10 but 53/10 ≠ 5 ∙ 3/10 the terms must be added, not multiplied.  Mixed numerals are considered proper fractions where as if any of the

numerators > the denominators, it is considered an improper fraction.  Likewise, writing mixed numerals into a

fraction is changing a proper fraction into an unsimplifyed one.  Changing a fraction into mixed numerals or changing an improper fraction into a proper fraction.  An example of this is:

42/5 = 82/5 here        ___8__________

                             5  )  42

                                    40

                                      2

8 is the number of times 5 goes into 42 with a remainder of 2.  Then the 8 becomes the leading number, the remainder becomes the numerator, and the denominator is the divisor.

Read page 190 of Basic Mathematics 9th Edition by Marvin L. Bittinger  about proper and improper fractions.

 

Section 3.5 Addition and Subtraction of Mixed Numerals with

          Applications

First add or subtract the fractions, then add or subtract the whole number for example:

43/8

      + 71/4

43/8

      + 72/8

       115/8

Likewise in subtraction

93/8

      - 71/4

93/8

      - 72/8

        21/8

Notice that multiplying the fraction by 1 does not affect the whole number.  If, in addition, the fraction becomes improper, then simplify it and add the number to the whole number.  An example of this would be:

45/8

      + 71/2

45/8

      + 74/8

       11 + 11/8 = 121/8

If subtracting and the minuend fraction is smaller than the subtrahend fraction, subtract 1 from the whole number and add it to the fraction.  An example of this is:

93/8

      - 73/4

93/8

      - 76/8

88/8 + 3/8

      - 76/8

811/8

      - 76/8

        15/8

 

Section 3.6 Multiplication and Division Using Mixed Numerals

          and Applications

In multiplication, convert the mixed numeral to fraction notation, multiply and simplify (into a mixed number if applicable).  An example of this is:

43/4 ∙ 22/3 = ( 16/4 + 3/4 ) ∙ ( 6/3 + 2/3 ) = 19/4 ∙ 8/3  = ( 19 ∙ 8 )/( 4 ∙ 3 ) here the 8 = 2 ∙ 4 and the 4’s cancel out leaving

( 19 ∙ 2 )/3  = 38/3  and simplifying leaves 122/3.  Division is done much the same way.  1/96 ¸ 1/96 = 1/96 ∙ 96/1 = 96/96 = 1.  With mixed numerals, first put into fraction form, then divide, and finally simplify if need be.  Looking at:

23/5 ¸ 11/5 = ( 10/5 + 3/5 ) ¸ ( 5/5 + 1/5 ) = ( 10 + 3 )/5  ¸ ( 5 + 1 )/5 =

 13 /5  ¸ 6/5 = 13/5  ∙ 5/6 = 13/6.

Looking at both, the basic rules of multiplication and division apply.

 

Section 3.7 Order of Operations, Estimation

The same rules in Section 1.9 (page 85) of Basic Mathematics 9th Edition by Marvin L. Bittinger are used in this case. 

 

 

Chapter 4

 

Section 4.1 Decimal Notation, Order, and Rounding

Decimal comes from the Latin word decimal meaning a tenth part.  In 10.53, the dot is called a decimal point.  .53 = 53/100 in the terms of U. S. currency, $.53 = $ 53/100  = 53 cents.  See page 233 for the place value chart.  10.53 written out is

ten dollars and fifty three hundredths.  The fraction notation of 0.0045 is 0045/10,000.  Since .0045 = 45, it is written as 45/10,000.  In comparing decimal terms look for the first number that is largest, not the greatest number of digits.  Consider 3.12452 and 3.1246, which is larger?  3.12452 has the greatest number of digits, but 3.1246 is the largest of the two terms because 6 ten thousandths is greater than 52 ten thousandths.  Compare each digit until the largest one is found; above, 6 > 5 which implies 3.1246 > 3.12452.

 

Section 4.2 Addition and Subtraction

In both addition and subtraction, start to the right and work left keeping the decimal point in the same position.  Let’s start with an addition example with not the same amount of digits.

3.16

      + 3.5443

          6.7043

Like wise in subtraction, the example is:

          10.2

       -   9.991

           0.009

 

Section 4.3 Multiplication

This is done like regular multiplication but the decimal point is placed where the number of digits are added together from the right of the decimal point on each of the factors.  An example of this would be:

          10.012

      X  9.1

         10012

    + 901080

       911092

because the first factor has 3 digits to the right of the decimal point and the second factor has 1 digit to the right of the decimal point, the decimal point is placed 4 digits from the right.  The final answer is:

       91.1092.

For the second example consider 20.100 ∙ 1.0010.  In this example, the first factor can be simplified to 20.1 as

100/1000 = 1/10.  The second factor can be reduced in the same way as 0010/10,000 = 001/1000 = 1/1000.  Notice that the two 0’s to the left of the 1 can not be reduced out as the statement would not be true.  With the simplified factors, the equation looks like:

          20.1

     X    1.001

          20.1201

Here the answer is simplified.

Next, we discussed large numbers (see chart on page 255).

If converting cents to dollars, move the decimal point to the left two spaces; so 40¢ = $.40.  Likewise in converting dollars to cents, move the decimal point two spaces to the right.  For instance, consider $1.40 = 140¢.

 

Section 4.4 Division

Put both the dividend and the divisor into fraction form and divide.  Dividing 32.2 ¸ 1.01 then I obtain:

          322/10 ¸ 101/100 = 322/10 ∙ 100/101 = 32200/1010 = 3220/101

If dividing by .001, then move the decimal point to the right the number 3 spaces. Consider 14.1 ¸ .001, doing the above instructions I yield:

                    14100        

               1 )  14100

                    14100

                            0

Doing the same equation by changing both to fractions I obtain:

          141/10 ¸ 1/1000 = 141/10 ∙ 1000/1 = 141000/10 = 14100

The two are the same.

Likewise if the divisor is 100, move the decimal point to the left 2 spaces.

 

Section 4.5 Converting from Fractional Notation to Decimal

Notation

If the fraction is or can be reduced to a whole number then you are finished.  An example of this is:

6/2 = 3.  0’s can be added after the decimal point like:

3.00 = 3.

If the denominator can be easily made into 10, 100, 1000, etc. multiply the numerator and denominator by the number it would take to do this like:

4/5 = 4/5 ∙ 2/2 = 8/10 = 0.8 = .8

If the fraction can not be easily made into the above, divide it out.  An example of this would be:

1/4 =           .25

          4  )  1.0

                    8

                   .20

                    20

                      0

If the decimal repeats continually with out end we put a line above it and it is called reputan or a repeating decimal.  An example of this would be 1/3.  Dividing 1/3 out I get

                .333…

          3 ) 1.0

                 .9

                 .10

                    9

                 .010

                      9

                   and so on.

 

Because it never resolves it is written as:

 _

.3

 

Section 4.6 Estimating

This is done like in Section 1.4 of Basic Mathematics 9th Edition by Marvin L. Bittinger except with the decimal notation.  One rounds to the nearest specific digit like 1.54319 to the nearest hundredth is 1.54.

 

Section 4.7 Applications and Problem Solving

To review the 5 point process, it is:

1       :  Familiarize

2       :  Translate

3       :  Solve

4       :  Check

5       :  State

‘is’ is always defined as =

Example:  A checkbook costs $5.00 each to make.  They sell for $15.00 each.  The profit is $10.00 ( $15.00 - $5.00) each.  What would the profit be after selling 50?

1: We want to find the profit so let p = the profit.

2:  The profit   is   $10.00   multiplied    by  50

     (4243 2  123     123     123

            p           =    10.00            ∙               50

3:  p = 10.00 ∙ 50 implies

     p = 500.00

4:  ( 15.00 – 5.00) is the profit for each checkbook = 10

     we are selling 50 checkbooks so 10 ∙ 50 = 500 so the

     answer checks out.

5:  We will get $500.00 profit.

 

 

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