Fractions

Fractions

Fractions are numeric expressions in the form of p/q, where 'p' and 'q' are whole numbers such that 'q' cannot be zero. 'p' is referred to as the numerator while 'q'  is referred to as the denominator.

Various types of fractions

1. Proper Fraction: Proper fractions represent a portion of a whole. They have numerators smaller than their denominators. Even the whole number 0 can be expressed as a proper fraction.
   Examples: 1/2, 1/4, 3/4, 3/7, 5/7, 9/11, 11/13, 17/21, 7/100, 100/101, etc.
2. Improper Fraction: Improper fractions represent a combination of a whole number and a part of the whole. They have numerators greater than or equal to their denominators. All natural numbers can be expressed as improper fractions.
   Examples: 3/2, 5/4, 9/4, 10/7, 11/7, 12/11, 13/17, 23/21, 101/100, etc.
3. Mixed Fraction: When an improper fraction is represented as an integer followed by a proper fraction, it is called a mixed fraction. 
   Examples: 5 ? is written as 17/3, 7 ?  is written as 39/5, etc.
4. Like Fractions: These are fractions with the same denominator but different numerators.
   Examples: 1/3 and 2/3, 3/5 and 4/5, 11/7 and 13/7, 26/15 and 28/15, etc.
5. Unlike Fractions: Unlike fractions have different denominators.
   Examples: 1/3 and 1/5, 3/5 and 4/7, 11/7 and 13/11, 17/11 and 5/13, etc.
6. Vulgar fraction: A vulgar fraction is defined as a fraction with a denominator that is a whole number other than 10, 100, 1000, and so on.

Examples: 3/5, 5/4, 3/4, 11/7, 11/17, 32/11, 23/27, 100/29, 110/ 43, etc. 

Note: If both the numerator and denominator of a fraction are multiplied by the same nonzero number, the value of the fraction remains unchanged.

Simplification of Fractions using BODMAS Rule

When simplifying fractions, the BODMAS rule is a helpful guideline that ensures accurate and systematic simplification. BODMAS stands for Brackets, Orders (exponents and roots), Division and Multiplication (from left to right) and Addition and Subtraction (from left to right).

Example: Simplify the following expression:
                                     [(3 ½)2 − 2 ?] ÷ 7/6 + 3/46 of 3 ?

a) −8 ¼
b) 8 ¼
c) −8 ¾
d) 8 ¾

Answer: d) 8 ¾

[(3 ½)2 − 2 ?] ÷ 7/6 + 3/46 of 3 ?

=  [(7/2)2 − 7/3] ÷ 7/6 + 3/46 of 23/6 (Changed to Improper Fractions) 
=  [49/4 − 7/3] ÷ 7/6 + 3/46 × 23/6 (Changed ‘of’ into multiplication symbol ‘×’ )
=  [(147 − 28)/12] ÷ 7/6 + 3/46 × 23/6 (Subtraction of two fractions)
=  [119/12] ÷ 7/6 + 1/4 (Division of two fractions)
=  [119/12 × 6/7] + 1/4 (Multiplication of two fractions)
=  17/2 + 1/4 (Addition of two fractions)
= (34 + 1)/4
= 35/4
= 8 ¾ (Changed to Mixed Fractions)

Decimals

A decimal number has a whole number followed by a decimal point. Digits following the decimal point have a value less than 1. 

An example of a decimal number is 418.273. Here, 4 is in the hundreds place, 1 is in the tens place, 8 is in the units place, 2 is in the tenth place, 7 is in the hundredth place and 3 is in the thousandth place.

The decimal number 418.273 is read as four hundred eighteen point two seven three. All the digits after the decimal point are read one digit at a time.

Example: A piece of cloth is 49.95 metres long. How many pieces can be cut from it if each length is 1.35 metres?

a) 17 pieces
b) 27 pieces
c) 37 pieces
d) 47 pieces

Answer: c) 37 pieces

Explanation: Length of a piece of cloth = 49.95 m
Length of one piece of cloth cut from it = 1.35 m
Number of piece = 49.95 ÷ 1.35 = 37 pieces