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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."
2
7
1
2 3, 5 8 , 2 2
Examples:
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.
Examples:
3
5
7
8
7
9
4
,
,
5
1
3
,
Reciprocal- the fraction that results from dividing one by that number.
Example: 4 is the reciprocal of 1
1
4
1
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.
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To make equivalent fractions we must multiply or divide the numerator and denominator by
the same number
EXAMPLE # 1
3
5
3x2
5x2
=
EXAMPLE # 2
6
10
=
8
12
=
8/4
12 / 4
=
2
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
36
42
=
36 / 2
42 / 2
=
18 / 3
21 / 3
=
6
7
EXAMPLE # 2
64 / 8
8/2
= 80 / 8 = 10 / 2
64
80
4
= 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.
2/3
10 / 15
+
+
Example # 1
4/5
12 / 15
=
=
=
?
22 / 15
1 7 / 15
51/4
21 / 4
63 /12
-
Example # 2
3 5/6
=
?
23 / 6
=
?
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
3/4 X 14/15 = ?
1/2 X 7/5 = 7/10
EXAMPLE # 2
2 1/ 2 X 3 3/ 4 = ?
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.)
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3. Proceed as you would in a multiplication question.
EXAMPLE # 1
4/9 / 10/12 = ?
4/9 / 12/10 = 8/15
EXAMPLE # 2
3 1/3 / 2 2/5 = ?
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
tens
ones
.
.
6
tenths
5
8
1
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
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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
proper fraction.
Examples: .25 = 1 / 4 ,
- the decimal fraction that equals the
.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.
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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
Since the last number to the right of the decimal is in
is in the thousandths place, the denominator is
the hundredths spot, the denominator is 100.
1000.
:. 0. 36
= 36
Reduce if possible
100
= 36
100
9
Divide numerator &
denominator by 4
25
:. 4.537
=
4 537/1000
=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
8
) 3.000
2 4xx
60
56x
40
40
0
:. 3/8
= 0.375
3 4/9 = 31/ 9
Mixed to Improper
3.444 ...
9)31.00
27 xx
40
36
40
36
4
:, 3 4/9
= 3. 6
COMPUTATION INVOLVING DECIMALS
Adding and Subtracting Decimal Numbers
Whenever 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.
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Multiplying Decimal Numbers
When 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: # 1
EXAMPLE # 2
2.6
8.3.) 21.5.8
Division using the powers of 10
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.)
EXAMPLE # 1
Find the sum of 82.635, 325.68
and 53.47
82.635
+ 325.68
+ 53.47
461.785
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 # 2
Find the difference between 7836.25 and
4532.78
7836. 25
3303. 47
- 4532. 78
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