FRACTIONS

Definitions of Fraction

 A part of a whole
 A portion of a totality
 A subset of a bigger set
 A ratio or comparison between two quantities
 A quotient between two numbers

Parts of a Fraction

1. Numerator (N) – it shows the number of parts taken from the whole.
2. Denominator (D) – it indicates into how many parts the whole is divided.
3. Fraction Line/Bar – it is the line between the numerator and denominator.

Kinds of Fraction

1. Proper Fraction – it is a fraction wherein the numerator is smaller than the denominator (N < D).
Ex: 1/2, 2/5, 3/4, 1/3, 5/8
2. Improper Fraction – it is a fraction wherein the numerator is bigger than the denominator (N > D).
Ex: 5/3, 9/4, 12/5, 8/7, 5/4
3. Mixed Fraction – it is a fraction which consists a whole number and a proper fraction.
Ex: 2 1/2, 3 1/4, 6 2/3, 4 1/4, 1 2/5

Expressing Improper Fraction to Mixed Fraction (vice-versa)

 An improper fraction can be changed to a mixed number by actually dividing the numerator by the
denominator. Hence, = Q + wherein N= numerator, D= denominator, Q= quotient & R= remainder.

Ex: =2
 To change a mixed number to an improper fraction, multiply the denominator by the whole number and add the
numerator to the product. The denominator of the fraction is retained.
( )
Ex: = =

Simplifying Fractions

 It means reducing fractions to its lowest term.

 A fraction is simplified if it is in lowest term. When a fraction is in its reduced form or lowest term, the greatest
common factor (GCF) of the numerator and denominator is one or simply there is no more common prime
factor. A fraction in simplified form is referred to as a simple fraction.
 To reduce fractions to lowest term, divide the numerator and the denominator by its GCF or cancel the common
factors.
 Ex: = = or = =

More Examples:

1. 10/15 = 2/3
2. 18/24 = 3/4
3. 20/8 = 5/2
4. 21/18 = 7/6

Equal/Equivalent Fractions

 Fractions are said to be equivalent if their cross-products are equal. Likewise, it can be applied in inequalities (<
or >).
1. = if and only if ad = bc

2. < if and only if ad < bc

3. > if and only if ad > bc

Ex: = because (5)(12) = (6)(10)

60 = 60

Similar and Dissimilar Fractions

1. Similar Fractions – fractions are similar if they have the same denominator.
Ex: 1/5, 2/5, 3/5 & 4/5 are similar fractions.
2. Dissimilar Fractions – fractions are dissimilar if they do not have the same denominators or if the denominators
are not equal.
Ex: 3/4, 1/6, 3/7 & 2/3 are dissimilar fractions.

Changing Dissimilar Fractions to Similar Fractions

 We can change dissimilar fractions to similar fractions by getting the least common denominator (LCD/LCM) of
the denominators of fractions. Divide the LCD by the denominator of the given fraction and then multiply the
quotient by its numerator. The LCD is retained as the denominator of the similar fractions.

Ex: Change 1/2, 1/4, 2/3, and 5/12 to similar fractions.

Sol’n: the LCD of 2, 4, 3 and 12 is 12.

= =

= =

= =
 = =

Therefore, 6/12, 3/12, 8/12, and 5/12 are similar fractions.

Ordering, Comparing and Arranging Fractions

1. Based on the Numerator – if the fractions have the same denominator or they are similar fractions, the fraction
with a bigger numerator is bigger than the fraction with a smaller numerator.
Ex: 5/17 < 8/17 < 11/17 < 15/17 < 25/17
11/12 > 10/12 > 5/12 > 3/12 > 1/12

2. Based on the Denominator – when the numerators of two or more fractions are equal, a fraction with a smaller
denominator is bigger than the fraction with a bigger denominator.
Ex: 19/25 < 19/18 < 19/15 < 19/10 < 19/3
14/4 > 14/7 > 14/9 > 14/11 > 14/13

The Fraction of a Number

 To find the fractional part of a whole number, multiply the numerator of a fraction by the whole number and
use the denominator of the fraction to find its quotient. Reduce to its lowest term.
Examples:
1. What is 2/3 of 45?
Ans: x 45 = = 30
2. What is 5/6 of 63?
Ans: x 63 = = 52
3. What is 3/10 of 20?
Ans: x 20 = =6

Operations on Fractions

Addition and Subtraction of Fractions

 To add or subtract similar fractions, copy the common denominator of the fractions, then add or subtract their
numerators.
 To add or subtract dissimilar fractions, change the given fractions to similar fractions, then add or subtract their
numerators.

Examples:

1. 1/5 + 3/5 + 6/5 = 10/5 = 2

2. 2/7 + 3/7 + 1/7 = 6/7
( ) ( )
3. 2/3 + 1/4 = = 11/12
( ) ( )
4. 3 + = 7/2 + 2/5 = = 39/10 or 3
 5. 7/9 – 2/9 = 5/9
6. 16/24 – 12/24 = 4/24 = 1/6
( ) ( )
7. 4 – 2 = 14/3 – 5/2 = = 13/6 or 2

Multiplication of Fractions

 To multiply fractions, get the product of the numerators over the product of the denominators. Factor and
cancel common factors if necessary.
Examples:
1. 1/2 x 3/4 = 3/8
2. 2/4 x 5/8 x 1/6 = 10/192 = 5/96
3. 6/5 x 5/6 = 30/30 = 1
( )( )( )( )( )( )( )( )( )
4. 3/4 x 1/2 x 4/5 x 5/7 x 2/3 x 12 = = 12/7 or 1
( )( )( )( )( )( )
5. 5 x 2 = 17/3 x 9/4 = 153/12 or 12

Division of Fractions

 To divide fractions, multiply the dividend (first fraction) by the reciprocal of the divisor (second fraction).
Examples:
1. 5/8 ÷ 3/4 = 5/8 x 4/3 = 20/24 = 10/12 = 5/6
2. 3 ÷ 1 = 19/5 ÷ 7/4 = 19/5 x 4/7 = 76/35 or 2
3. ? ÷ 2/3 = 4/5
4/5 x 2/3 = 8/15
4. 4/7 ÷ ? = 3/5
4/7 ÷ 3/5 = 4/7 x 5/3 = 20/21