Operations on Fractions: Rules, Formulas and Examples

Operations on Fractions include addition, subtraction, multiplication and division of fractional numbers. The method depends on whether denominators are equal or different. This page explains how to find the least common denominator, simplify results, multiply numerators and denominators, and divide by multiplying by the reciprocal.

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What Are Operations on Fractions?

Operations on fractions are arithmetic calculations performed on fractions using addition, subtraction, multiplication or division. The rules depend mainly on the relationship between the numerators and denominators.

For addition and subtraction, fractions are first expressed with a common denominator. For multiplication, multiply the numerators and denominators directly. For division, multiply the first fraction by the reciprocal of the second fraction. A final answer should be reduced to its lowest terms. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Operations on Fractions Formula & Tricks

Important Formulas

Addition of Fractions
a/b + c/d = (ad + bc)/bd

Multiply each numerator by the other denominator, add the products, and multiply the denominators.

Subtraction of Fractions
a/b − c/d = (ad − bc)/bd

Multiply each numerator by the other denominator, subtract the second product from the first, and multiply the denominators.

Multiplication of Fractions
a/b × c/d = ac/bd

Multiply the numerators together and the denominators together. Cancel common factors before or after multiplication.

Division of Fractions
a/b ÷ c/d = a/b × d/c = ad/bc, where c/d ≠ 0

Keep the first fraction, change division to multiplication, and take the reciprocal of the second fraction.

Mixed Number Conversion
p q/r = (pr + q)/r

To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator.

Quick Tricks

Use the least common denominator

For addition or subtraction, use the least common multiple of the denominators instead of multiplying all denominators. This keeps the intermediate numbers smaller.

Example: For 1/6 + 3/8, the LCM of 6 and 8 is 24. Thus, 1/6 + 3/8 = 4/24 + 9/24 = 13/24.
Cross-cancel before multiplication

In multiplication or division, cancel common factors between a numerator and a denominator before multiplying. This reduces calculation and prevents large numbers.

Example: 3/14 × 7/9 = (3 × 7)/(14 × 9). Cancelling 3 with 9 and 7 with 14 gives 1/6.
Convert division to multiplication immediately

For fraction division, retain the first fraction, reverse the second fraction, and multiply. Do not reverse both fractions.

Example: 5/8 ÷ 10/3 = 5/8 × 3/10 = 15/80 = 3/16.

Operations on Fractions Concepts

Adding Fractions with Equal and Unequal Denominators

Fractions with equal denominators are added by adding their numerators and retaining the denominator; fractions with unequal denominators require a common denominator.

If the denominators are equal, a/b + c/b = (a + c)/b. If the denominators differ, find their LCM, convert each fraction, add the numerators, and retain the common denominator. Reduce the result if possible.

Example: 2/7 + 3/7 = 5/7, while 1/4 + 2/3 = 3/12 + 8/12 = 11/12.

Subtracting Fractions

Fraction subtraction follows the same denominator rule as addition: use the common denominator first, then subtract the numerators.

For equal denominators, a/b − c/b = (a − c)/b. For unequal denominators, convert both fractions to equivalent fractions with the LCM as denominator. If the numerator of the result is negative, retain the negative sign.

Example: 5/6 − 1/4 = 10/12 − 3/12 = 7/12.

Multiplying Fractions

To multiply fractions, multiply the numerators and denominators separately; a common denominator is not required.

The rule is a/b × c/d = ac/bd. Common factors may be cancelled diagonally before multiplication. When multiplying a fraction by a whole number n, write n as n/1.

Example: 4/15 × 9/8 = 36/120 = 3/10. Equivalently, cancel 4 with 8 and 9 with 15 before multiplying.

Dividing Fractions

To divide by a fraction, multiply by its reciprocal, which is obtained by interchanging its numerator and denominator.

The rule is a/b ÷ c/d = a/b × d/c, provided c/d is not zero. The reciprocal of a non-zero fraction a/b is b/a. Division by zero is undefined.

Example: 7/10 ÷ 14/25 = 7/10 × 25/14 = 175/140 = 5/4.

Operations with Mixed Numbers

Convert mixed numbers to improper fractions before performing multiplication or division, and usually before addition or subtraction.

For a mixed number p q/r, the improper fraction is (pr + q)/r. After completing the operation, convert an improper fraction back to a mixed number when required.

Example: 2 1/3 + 1 1/6 = 7/3 + 7/6 = 14/6 + 7/6 = 21/6 = 7/2 = 3 1/2.

Operations on Fractions Video Lessons

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Fraction Operations: Add, Subtract, Multiply, Divide

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Practice Operations on Fractions Questions

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1Find the simplified value of the following expression: (34 × 3-2) ÷ (33 × 3-5) × (36)-1→ 2((√3 + √2)²)/(√3 - √2) = ?→ 3Find the value of { ( 1/3 )⁻² - ( 1/2 )⁻³ } ÷ ( 1/4 )⁻² + { ( (-3)/2 )⁴ × ( 5/3 )⁴ } .→ 4Simplify: (2√3)/√12→ 5Simplify: ((6⁴ × 4⁵ × 6⁵))/((6² × 4² × 6³))→ 6If Y^x = 121 , what will (Y - 1)^((x - 2)) = ?→ 7Simplify ((32)6)5 and express it as a power of 9. What is the exponent of 9?→ 8If 6x - 6x-1 = 180, then the value of (2x-1)/(2x+3) is:→ 9Simplify: √2 + 1/√2 + 1/(2+√2) - 1/(2-√2)→ 10Solve √(30-√(30-√(30-....)))→

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Quick Revision Notes

Operations on Fractions: Quick Revision

Use these rules to calculate and simplify fractional expressions correctly.

  • For addition and subtraction, first make the denominators equal.
  • Use the LCM of the denominators as the least common denominator.
  • For multiplication, multiply numerators and denominators; cancel common factors before multiplying when possible.
  • For division, multiply by the reciprocal of the divisor.
  • A fraction can be simplified by dividing its numerator and denominator by their greatest common divisor.
  • Convert mixed numbers to improper fractions before carrying out arithmetic operations.
  • Division by zero is undefined, so the divisor fraction cannot have a zero numerator.

Operations on Fractions FAQs

How do you add fractions with different denominators?

Find the LCM of the denominators, convert both fractions to that denominator, add the numerators, and simplify. For example, 2/5 + 1/3 = 6/15 + 5/15 = 11/15.

Can fractions be multiplied without making their denominators equal?

Yes. Multiply the numerators and denominators directly. For example, 2/3 × 5/7 = 10/21.

What is the reciprocal of 8/11?

The reciprocal of 8/11 is 11/8 because the numerator and denominator are interchanged.

How is fraction division different from fraction subtraction?

For subtraction, use a common denominator and subtract numerators. For division, multiply the first fraction by the reciprocal of the second fraction.

What is 3/4 − 5/6?

The LCM of 4 and 6 is 12. Therefore, 3/4 − 5/6 = 9/12 − 10/12 = −1/12.

How do you simplify the result of a fraction operation?

Divide the numerator and denominator by their greatest common divisor. For example, 18/24 becomes 3/4 after division by 6.

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