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How to Work Out Fractions Without Calculator

Reviewed by Calculator Editorial Team

Working with fractions without a calculator can be challenging, but with the right methods and practice, you can master these operations. This guide covers all the essential fraction operations: addition, subtraction, multiplication, division, simplification, and conversion between fractions and decimals.

Adding Fractions

To add two or more fractions, follow these steps:

  1. Find a common denominator for all fractions.
  2. Convert each fraction to have the common denominator.
  3. Add the numerators together.
  4. Simplify the resulting fraction if possible.

Formula: a/b + c/d = (ad + bc)/bd

Remember that the denominators must be the same before you can add the numerators.

Example

Add 1/4 and 3/8:

  1. The denominators are 4 and 8. The least common denominator is 8.
  2. Convert 1/4 to 2/8.
  3. Add 2/8 + 3/8 = 5/8.

Subtracting Fractions

The process for subtracting fractions is similar to adding them:

  1. Find a common denominator.
  2. Convert each fraction to have the common denominator.
  3. Subtract the numerators.
  4. Simplify the resulting fraction if possible.

Formula: a/b - c/d = (ad - bc)/bd

Example

Subtract 3/8 from 5/6:

  1. The denominators are 8 and 6. The least common denominator is 24.
  2. Convert 3/8 to 9/24 and 5/6 to 20/24.
  3. Subtract 20/24 - 9/24 = 11/24.

Multiplying Fractions

Multiplying fractions is straightforward:

  1. Multiply the numerators together.
  2. Multiply the denominators together.
  3. Simplify the resulting fraction if possible.

Formula: (a/b) × (c/d) = (a × c)/(b × d)

Example

Multiply 2/3 by 4/5:

  1. Multiply numerators: 2 × 4 = 8.
  2. Multiply denominators: 3 × 5 = 15.
  3. The result is 8/15, which is already in simplest form.

Dividing Fractions

Dividing fractions involves multiplying by the reciprocal:

  1. Find the reciprocal of the second fraction (swap numerator and denominator).
  2. Multiply the first fraction by this reciprocal.
  3. Simplify the resulting fraction if possible.

Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d)/(b × c)

Example

Divide 3/4 by 2/5:

  1. The reciprocal of 2/5 is 5/2.
  2. Multiply 3/4 × 5/2 = 15/8.

Simplifying Fractions

To simplify a fraction, follow these steps:

  1. Find the greatest common divisor (GCD) of the numerator and denominator.
  2. Divide both the numerator and denominator by the GCD.

If the numerator and denominator have no common factors other than 1, the fraction is already in simplest form.

Example

Simplify 8/12:

  1. The GCD of 8 and 12 is 4.
  2. Divide numerator and denominator by 4: 8 ÷ 4 = 2, 12 ÷ 4 = 3.
  3. The simplified fraction is 2/3.

Converting Fractions

You can convert fractions to decimals and vice versa:

Fraction to Decimal

  1. Divide the numerator by the denominator.

Decimal to Fraction

  1. Write the decimal as a fraction with denominator 1.
  2. Multiply numerator and denominator by 10, 100, etc., until the numerator is a whole number.
  3. Simplify the resulting fraction if possible.

Some decimals, like 1/3, convert to repeating decimals (0.333...).

FAQ

What is the easiest way to add fractions?

The easiest way is to find a common denominator first, then add the numerators. The least common denominator is the smallest number that both denominators divide into evenly.

How do I know when a fraction is in simplest form?

A fraction is in simplest form when the numerator and denominator have no common factors other than 1. You can check this by finding the greatest common divisor (GCD) of the numerator and denominator.

What's the difference between improper and proper fractions?

A proper fraction has a numerator smaller than its denominator (e.g., 3/4), while an improper fraction has a numerator larger than or equal to its denominator (e.g., 5/2). Improper fractions can be converted to mixed numbers.