Fractions

A fraction represents a part of a whole. It is written as a/b, where a is the numerator (parts we have) and b is the denominator (total equal parts). The denominator cannot be zero.

Types of Fractions
Proper fraction: numerator < denominator. Example: 3/7.
Improper fraction: numerator ≥ denominator. Example: 9/4.
Mixed number: a whole number + a proper fraction. Example: 2 (1/4).
Like fractions: same denominator. Example: 3/8 and 5/8.
Unlike fractions: different denominators. Example: 1/3 and 1/5.
Equivalent fractions: different-looking fractions that represent the same value. Example: 1/2 = 2/4 = 4/8.

Simplest Form (Lowest Terms)
A fraction is in its simplest form when the HCF of numerator and denominator is 1.
To simplify: divide both numerator and denominator by their HCF.

Comparing Fractions
Like fractions: larger numerator = larger fraction.
Unlike fractions: find a common denominator (LCM) first, then compare numerators.

Addition and Subtraction of Fractions
Like fractions: add/subtract numerators, keep denominator.
Unlike fractions: convert to like fractions using LCM, then add/subtract.

Example 1

Write 5 equivalent fractions for 2/3.
Multiply numerator and denominator by 2, 3, 4, 5, 6:
4/6, 6/9, 8/12, 10/15, 12/18.

Example 2

Simplify 36/48.
HCF of 36 and 48 = 12. 36/48 = 3/4.

Example 3

Compare 3/4 and 5/6.
LCM of 4 and 6 = 12. 3/4 = 9/12 and 5/6 = 10/12. Since 9 < 10, we have 3/4 < 5/6.

Example 4

Add 2/5 + 3/7.
LCM of 5 and 7 = 35. 2/5 = 14/35, 3/7 = 15/35. Sum = 29/35.

Example 5

Convert 17/5 to a mixed number.
17 ÷ 5 = 3 remainder 2. So 17/5 = 3 (2/5).

Example 6

Subtract 2 (1/3) - 1 (3/4).
Convert: 7/3 - 7/4. LCM of 3 and 4 = 12. 28/12 - 21/12 = 7/12.

Example 7

Renu has 3/8 of a pizza and her brother has 2/8. How much do they have together?
3/8 + 2/8 = 5/8 of a pizza.

Key Facts
Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent fraction.
Every whole number can be written as a fraction: 5 = 5/1.
LCM of denominators is the key to adding/subtracting unlike fractions.

Common mistakes

Never add denominators: 1/3 + 1/4 is NOT 2/7.
When simplifying, divide by the HCF, not just any common factor.
Mixed numbers must be converted to improper fractions before multiplying or dividing.

Summary

Fractions represent parts of a whole. Equivalent fractions show the same value. Simplification uses HCF. Comparison and addition/subtraction of unlike fractions require a common denominator.

Practice Problems