The document provides information about fractions and decimals. It defines different types of fractions such as proper fractions, improper fractions, mixed numbers, and equivalent fractions. It explains how to convert between improper fractions and mixed numbers. It also describes rules for adding, subtracting, multiplying, and dividing fractions. Additionally, the document defines different types of decimals and explains how to convert between fractions and decimals. It provides examples of adding, subtracting, multiplying, and dividing decimals.

1. Article 1
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Fractions
A 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:

2. 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.
Numerator
The numerator is the top number of a fraction.

3. 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.

4. So, .

5. 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

6. 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

7. 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.

8. 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.

9. To make equivalent fractions we must multiply or divide the numerator and
denominator by the same number
EXAMPLE # 1 EXAMPLE # 2
3 3x2 6 8 8/4 2
5
= 5x2
= 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 / = 10 / = 5
8 2
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.

10. Example # 2
Example # 1
2/3 + 4/5 = ? 51/4 - 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.

11. 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.

12. 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

13. 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

14. 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.

15. 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 9
Divide numerator & :. 4.537 = 4 537/1000
100 25 denominator by 4
=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 xx
40 40
40 36
0 40
36
:. 3/8 = 0.375 4

16. :, 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.
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

17. 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.)

18. 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.

19. 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.)