Article 1

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FractionsA fraction is a way of expressing division since the fraction bar indicates division.

The number below the bar is called the denominator.  It tells the number of equal parts into which the whole has been divided.

The number above the bar is called the numerator.  It tells how many of the equal parts are being considered.

Example 1:

The whole has been divided into 5 parts and 2 are being considered.

A proper fraction is a fraction whose numerator is less than its denominator.  If the numerator is greater than the denominator, then it is an improper fraction.

Example 2:

A number which consists of a whole number plus a fraction is a mixed number.  Mixed numbers can be written as an improper fraction and an improper fraction can be written as a mixed number.

Example 3:

Write as an improper fraction.

Example 4:

Write as an improper fraction.

A fraction is in lowest terms when the numerator and denominator have no common factor other than 1.  To write a fraction in lowest terms, divide the numerator and denominator by the greatest common factor.

Example 5:

Write as an improper fraction.

45 and 75 have a common factor of 15.

See also fraction operations.

NumeratorThe numerator is the top number of a fraction.

Since fractions are just shorthand for division, the numerator is the number that gets divided by the denominator. In a division problem, it is called the dividend.

Converting Fractions to Decimals To convert a fraction to a decimal, just divide the numerator by the denominator.

Example 1:

Write as a decimal.

Since 15 is larger than 3, in order to divide, we must add a decimal point and some zeroes after the 3.  We may not know how many zeroes to add but it doesn’t matter.  If we add to many we can erase the extras; if we don’t add enough, we can add more.

Sometimes, you may get a repeating decimal.

Example 2:

Write as a decimal.

So, .

Article 2

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FRACTIONS AND DECIMALS

 

TYPES OF FRACTIONS

 

A fraction is a part of a whole. There are many types of fractions.

 

Simple Fraction - a fraction in which the numerator and denominator are both integers. Also known as a common fraction.

Examples:      2     7    6     5

                      3 ,   3  , 7 ,   1

 

Proper Fraction - a fraction in which the numerator is less than the denominator.

                   Examples:  1     2     1

                4 ,  7 ,  8

 

Improper Fraction - a fraction in which the numerator is equal to or greater than the denominator. Improper fractions are usually changed to whole or mixed numbers.

  

        Examples:   5      7    11

                            3  ,   7 ,   8

 

Mixed Number - a number that is a combination of an integer and a proper fraction. Thus, it is "mixed."

 

     Examples:          2          7          1

                            2  3 ,   5  8  ,   2   2

 

Unit Fraction -a fraction in which the numerator is one.

 

         Examples:    1      1

                           5   , 1

An Integer Represented as a Fraction -a fraction in which the denominator is one.

 

        Examples:    2      3

                          1   ,   1

Complex Fraction - a fraction in which the numerator or the denominator, or both numerator and denominator, are fractions.

 

  3                7

  5                9                5

Examples:             7               4                1

                            8         ,           ,          3          ,

Reciprocal- the fraction that results from dividing one by that number.                   

           Example: 4 is the reciprocal of 1

                            1                                   4

 

 

Zero Fraction - a fraction in which the numerator is zero. A zero fraction equals zero.    0     Example: "3 = 0

 

Undefined Fraction - a fraction with a denominator of zero. (7/0 means 7 divided by 0, which is impossibility because nothing can be divided by 0. Therefore, the fraction remains undefined.)

 

Indeterminate Form - an expression having no quantitative meaning.

                 0

Example: 0

 

EQUIVALENT AND BASIC FRACTIONS

 

Fractions are used to express parts of a whole in regards to lengths, volumes, weights, and other measures. We can say that we have:

1/2 of a glass of water, 7/8 of a pizza or, 3/10 of the provinces in Canada are Prairie Provinces.

 

       

 

When two or more fractions have the same value, they are called equivalent fractions and the chart below shows this.

 

We can see from the chart above that:  1/2 = 2/4 = 3/6= 4/8 = 6/12  or

                                                                       1/3 = 2/6 = 4/12 etc.

 

 

 

 

 

 

To make equivalent fractions we must multiply or divide the numerator and denominator by the same number

                                                                     

EXAMPLE # 1   EXAMPLE # 2

3

=3 x 2

=6   8

=8 / 4

=2

5 5 x 2 10   12 12 / 4 3

 

A basic fraction is formed when we can no longer divide both the numerator and denominator by any number other than the number 1.

 

EXAMPLE # 1   EXAMPLE # 2

36

=

36 / 2

=18 / 3

=6

 64

=

64 / 8

=

8 / 2

=

4

42 42 / 2 21 / 3 7   80 80 / 8

10 / 2

5

  

ADDITION AND SUBTRACTION OF FRACTIONS

 

Rules For Adding and/or Subtracting Fractions

 

1. Convert all mixed fractions to improper fractions.

2. Find a common denominator.

3. Keeping the denominator the same, either add or subtract the numerators.

4. Convert your answers to mixed fractions if necessary and reduce your fraction.

              

Example # 1  Example # 2

2 / 3 + 4 / 5 = ?   5 1 / 4 - 3  5 / 6 = ?10 / 15 + 12 / 15 = 22 / 15   21 / 4 - 23 / 6 = ?      = 1 7 / 15   63 /12 - 46 / 12 = 17 / 12                  = 1 5 / 12

 

                                               

            MULTIPLICATION AND DIVISION OF FRACTIONS

 

Rules For Multiplying Fractions

 

1. Convert all mixed fractions into improper fractions.

2. Reduce the fractions if possible by finding the GCF.

3. Multiply the numerators (top parts) together.

4. Multiply the denominators (bottom parts) together.

5. Convert your fractions to mixed fractions and reduce if necessary.

 

                            EXAMPLE # 1                                              EXAMPLE # 2

                    3/4 X 14/15 = ?                                              2 1/ 2 X 3 3/ 4 = ?

                 1/2  X   7/5 = 7/10                                         5/2 X 15/4 = 7 5/8 = 9 3/8

 

 

Rules For Dividing Fractions

1. Convert all mixed fractions to improper fractions.

2. Write the reciprocal or multiplicative inverse of the divisor. (Flip the second fraction.)

3. Proceed as you would in a multiplication question.

                        

 

                            EXAMPLE # 1                                                     EXAMPLE # 2

                            4/9  / 10/12 = ?                                                 3 1/3 / 2 2/5 = ?

                          4/9  /  12/10 = 8/15                                                  10/3 / 12/5 =

10/3 X 5/12 = 25/18 = 1 7/18

DECIMAL NOTATION

 Numbers have different values depending on where they are placed in a string of numbers.In the case of decimals, the first number to the right of the decimal is in the tenths spot, the second number is in the hundredths spot, the third number is in the thousandths spot and so on. The number 27.6581 written below shows the value of each digit.

 

2 7 . 6 5 8 1

tens ones . tenths hundredths thousandths ten-thousandths

 

When we write 27.6581 in expanded form we would get either:

(2 x 10) + (7 x 1) + (6 x 0.1) + (5 x 0.01) + (8 x 0.001) + (1 x 0.0001) or

(2 x 10) + (7 x 1) + (6 X l/10) + (5 x l/100 + (8 x l/1000) + (1 x l/10 000)

 

In word form 27.6581 would be: twenty-seven and six thousand five hundred eight-one ten thousandths.

     The number in standard form is 27.6581

PLACE VALUE CHART

 

 

 

TYPES OF DECIMALS

 

In the broadest sense, a decimal is any numeral in the base ten number system. Following are several types of decimals.

 

Decimal Fraction - a number that has no digits other than zeros to the left of the decimal point.

Examples: 0. 349 ,  .84  ,  0.3001

 

Mixed Decimal - an integer and a decimal fraction.

Examples: 8.341 ,  27.1 ,  341.7

 

Similar Decimals - decimals that have the same number of places to the right of the decimal point.

Examples:    3. 87 and .12      ,     14.015 and 3. 396

 

Decimal Equivalent of a Proper Fraction - the decimal fraction that equals the proper fraction.

Examples: .25 = 1 / 4   ,    .3 = 3 / 10

 

Finite (or Terminating) Decimal - a decimal that has a finite number of digits.

Examples:   . 3  ,    . 2765    ,      . 38412

 

Infinite (or Nonterminating) Decimal - a decimal that has an unending number of digits to the right of the decimal point.

Examples: , √3, √33, √37, 34.12794 . . .

 

Repeating (Or Periodic) Decimal - Non terminating decimals in which the same digit or group of digits repeats. A bar is used to show that a digit or group of digits repeats. The repeating set is called the period or repent end. All rational numbers can be written as finite or repeating decimals.

Examples: .3, .37

 

Nonrepeating (or Nonperiodic) Decimal- decimals that are non-terminating and non repeating. Such decimals are irrational numbers.

Examples: 'IT, . √3

CONVERTING DECIMALS to FRACTIONS

When converting decimals to fractions, the very last number to the right of the decimal tells us what our denominator will be when we write the fraction. The denominator will be one of the following: 10, 100, 1000, 10 000, 100 000, etc., depending upon the place value of the last number to the right of the decimal. The examples below show how this conversion is done.

 

        EXAMPLE #1     EXAMPLE #2

  Convert 0.36 to a fraction.   Convert 4.537 to a fraction.

Since the last number to the right of the decimal is in the hundredths spot, the denominator is 100.

Since the last number to the right of the decimal is in the thousandths place, the denominator is 1000.

:. 0. 36 =  36   Reduce if possible       

      100          

  = 36               9   Divide numerator &    denominator by 4

:. 4.537 = 4  537/1000

    100       25        

  = 9              25          

 

CONVERTING FRACTIONS to DECIMALS

To convert a fraction into a decimal we divided the denominator (bottom part of the fraction) into the numerator (top part of the fraction). If the fraction is a mixed fraction, we must first convert it into an improper fraction before we divide. The examples below show how this is done.

 

EXAMPLE #1 EXAMPLE #2

Convert 3 / 8 into a decimal. Convert  3  4 / 9 into a decimal.

     0.375      3   4/9 = 31/ 9    Mixed to Improper

8 ) 3.000

            2 4xx   3.444 ...  60 9)31.00

    56x   27 xx40   4 040   360   40

    36:. 3/8 = 0.375    4

  :, 3  4/9 = 3. 6

 

COMPUTATION INVOLVING DECIMALS

Adding and Subtracting Decimal NumbersWhenever a question requires you to add or subtract numbers with decimals, you must remember to line up your decimals before you find the sum or difference, as shown in the examples below.

EXAMPLE # 1

Find the sum of 82.635, 325.68 and 53.47

 

     82.635

+ 325.68

+   53.47

        461.785

EXAMPLE # 2

 

Find the difference between 7836.25 and 4532.78

      7836. 25

      3303. 47

  - 4532. 78

 

              

 

    

 

 

 

 

 

 

 

Multiplying Decimal NumbersWhen multiplying numbers with decimals, it is not necessary to line up the decimals as we did when we were adding and subtracting. The examples below show how we multiply numbers with decimals.

EXAMPLE # 1

Find the product of 4.54 and 2.5

                    4.54

                   x 2.5

                  2 270

                  9 080

                11.350

(The factors have a total of 3 numbers to the right of the decimal point so we move the decimal 3 places to the left in the product.)

EXAMPLE # 2

Multiplication using the powers of 10

2.54 x 10 = 25.4

2.54 x 100 = 254

2.54 x 1000 = 2540 2.54 x 10 000 = 25 400

(The decimal moves to the right the same number of places as there are zeros.)

 

 

 

 

 

 

 

 

 

 

Dividing Decimal Numbers

             The example below shows the procedure and rules to follow when dividing numbers with decimals.

EXAMPLE # 2

 

Division using the powers of 10

 

25.4 + 10 = 2.54

25.4 + 100 = 0.254

25.4 + 1000 = 0.0254

25.4 + 10 000 = 0.00254

(The decimal moves to the left the same number of places as there are zeros in the divisor.)

 

 

EXAMPLE: # 1

                        2.6

8.3.) 21.5.8

              16   6 x

4        98

4       98

        0

 ( We must move the decimal to the right of the

divisor and we must move the decimal the same number of places to the right in the dividend. The decimal is now placed directly above in the quotient.)