Exact Calculations with Fractions

To convert a mixed number, multiply the whole number by the denominator, add the original numerator, and place this total over the original denominator. Answers should always be given in their simplest form, meaning the numerator and denominator share no common factors other than 1. An improper fraction is not considered unsimplified just because it is top-heavy.

Addition and Subtraction

To add or subtract fractions, they must have a common denominator. This is found by identifying the lowest common multiple (LCM) of the denominators.

Once the denominators match, you only add or subtract the numerators. The denominator remains entirely unchanged.

Worked Example: Addition

Calculate $\frac{3}{4} + \frac{1}{6}$

• Step 1: Find a common denominator. The LCM of 4 and 6 is 12.

• Step 2: Create equivalent fractions.

$$\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$ $$\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$$

• Step 3: Add the numerators.

$$\frac{9 + 2}{12} = \frac{11}{12}$$

Worked Example: Subtraction of Mixed Numbers

Calculate $4\,\frac{1}{5} - 2\,\frac{2}{3}$

• Step 1: Convert to improper fractions. $4\,\frac{1}{5} = \frac{21}{5}$ and $2\,\frac{2}{3} = \frac{8}{3}$.

• Step 2: Find the common denominator. The LCM of 5 and 3 is 15.

$$\frac{21 \times 3}{5 \times 3} = \frac{63}{15}$$ $$\frac{8 \times 5}{3 \times 5} = \frac{40}{15}$$

• Step 3: Subtract the numerators.

$$\frac{63 - 40}{15} = \frac{23}{15}$$

• Step 4: Convert back to a mixed number. $23 \div 15 = 1$ remainder $8$, which is $1\,\frac{8}{15}$.

Multiplication and Division

Multiplying fractions does not require a common denominator. You simply multiply the numerators together and the denominators together.

To divide by a fraction, you must multiply by its reciprocal. The reciprocal is found by flipping the fraction upside down. A useful mnemonic for division is KFC: Keep the first fraction, Flip the second fraction, Change the sign to multiply.

Worked Example: Multiplication

Calculate $2\,\frac{1}{3} \times \frac{3}{5}$

• Step 1: Convert the mixed number to an improper fraction.

$$\frac{7}{3} \times \frac{3}{5}$$

• Step 2: Multiply numerators and denominators.

$$\frac{7 \times 3}{3 \times 5} = \frac{21}{15}$$

• Step 3: Simplify the fraction. Divide top and bottom by their highest common factor (3).

$$\frac{7}{5} = 1\,\frac{2}{5}$$

Worked Example: Division

Calculate $\frac{3}{4} \div \frac{5}{8}$

• Step 1: Apply KFC. Keep $\frac{3}{4}$, flip $\frac{5}{8}$ to get its reciprocal, and change to multiplication.

$$\frac{3}{4} \times \frac{8}{5}$$

• Step 2: Multiply.

$$\frac{24}{20}$$

• Step 3: Simplify fully. Divide top and bottom by 4.

$$\frac{6}{5} = 1\,\frac{1}{5}$$