# Fractions (mathematics) | Definition, Explanations, Exercises and Answers # Fractions (mathematics)

Fractions, in mathematics, is naively a number of parts considered after dividing a whole number into equal parts. For example, the fraction 56/8 denotes the quotient of 56 by 8. It is equal to 7 because 7×8 = 56. In this fraction, 56 is called the numerator and 8 the denominator.

Numbers that can be represented by fractions of whole numbers are called rational numbers. The set of rationals is denoted Q.

There is a more general and more abstract definition of fractions. If (A, +, .) is an integral unitary commutative ring, we can create the field of fractions of A. Its elements are written (by analogy to fractions of integers) a/b and have the same operational properties (sum, product, simplification, …) as the fractions of Q.

## How to calculate fractions easily

##### I – Sum and difference of fractions

When two fractions have the same denominator, we add or subtract the numerator and keep the denominator.

Example: 15/12 + 14/12 = 29/12 and 15/12 – 15/12 = 1/12

When two fractions don’t have the same denominator, we first make them have the same denominator and then add or subtract the numerator.

Examples:

15/4 + 11/16 = 15×3/4×3 + 11×2/6×2 = 45/12 + 22/12 = 67/12

15/4 – 11/6 = 15×3/4×3 – 11×2/6×2 = 45/12 – 22/12 = 23/12

##### II – Multiplication and division

To multiply two fractions, multiply the denominator and multiply the numerator.

Example: 5/8×7/12 = 5×7/8×12 = 35/96

Specifically, to multiply a fraction by a natural number, we simply multiply the numerator by this number and the denominator remains.

Example: 5/8×3 = 5×3/8 = 15/8

To divide by two fractions, multiply the first fraction by the reciprocal of the second fraction.

Example: 5/8 : 7/2 = 5/8×12/7 = 1×12/8×7 = 5×3/2×7 = 15/14

### Franctions Exercises

##### 1. Examples

2/3 +3/4 = 17/12

6/5+7/5 = 13/5

15/4 – 5/2 = 5/4

3/4 – 1/4 = 1/2

1/8 × 4/2 = 1/4

7/4 × 2 = 7/2

1/2 × 3/4 = 3/8

3/2 × 2 = 3

1/5 : 6/5 = 1/6

3/4 : 2/3 = 9/8

2/3 + 5/6 = 2×2/3×2 + 5/6 =4/6 + 5/6 = 9/6 = 3/2

##### 2. Exercises on the rules for calculating fractions: we share a heritage. I take a quarter of it and my brother two-thirds of the rest. How much of the inheritance did my brother receive?
Answer:After my 1/4 share, 3/4 of the inheritance remains. But my brother takes 2/3 of the rest, that is 2/3 of the remaining 3/4: we multiply them.