Arithmetic

Working with fractions

Equivalent fractions, lowest terms, and the four operations — with the reasons behind the rules.

About 9 minutes

A fraction \frac{a}{b} means $a$ parts out of $b$ equal parts of a whole, and also the result of dividing $a$ by $b$ . The rules for calculating with fractions all follow from one idea: multiplying the top and bottom by the same non-zero number does not change the value.

Equivalent fractions and lowest terms

\frac{1}{2}, \frac{2}{4} and \frac{50}{100} are the same number written differently. To simplify a fraction, divide the numerator and denominator by their greatest common divisor (GCD). For \frac{42}{56} the GCD is 14, so the fraction reduces to \frac{3}{4}.

\frac{a}{b} = \frac{a \div g}{b \div g},\quad g = \gcd(a, b)

Adding and subtracting

Parts can only be added when they are the same size, so both fractions must first be written with a common denominator. The least common denominator is the least common multiple (LCM) of the two denominators.

Never add the denominators: \frac{1}{2} + \frac{1}{3} is not \frac{2}{5} — that would be less than \frac{1}{2} alone.

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Multiplying and dividing

To multiply, multiply the numerators together and the denominators together. “Half of a third” is \frac12 \times \frac13 = \frac16.

To divide, multiply by the reciprocal. Dividing by \frac{1}{4} asks how many quarters fit, which is the same as multiplying by 4.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Mixed numbers

A mixed number such as 2\tfrac{1}{3} means 2 + \frac13. Convert it to an improper fraction before multiplying or dividing: 2\tfrac13 = \frac{2 \times 3 + 1}{3} = \frac73.

Worked examples

Calculate \frac{1}{2} + \frac{1}{3}.

The LCM of 2 and 3 is 6.
\frac12 = \frac36 and \frac13 = \frac26.
Add the numerators: \frac{3 + 2}{6} = \frac56.

Answer: \frac{5}{6}

Calculate 1\tfrac{1}{2} \div \frac{3}{4}.

Convert: 1\tfrac12 = \frac32.
Multiply by the reciprocal: \frac32 \times \frac43 = \frac{12}{6}.
Simplify: $2$ .

Answer: 2

Common mistakes

Adding numerators and denominators separately.
Forgetting to flip the second fraction when dividing — or flipping the first one.
Leaving an answer unsimplified, such as \frac{6}{8} instead of \frac34.
Multiplying mixed numbers part by part: 2\tfrac12 \times 2\tfrac12 is \frac52 \times \frac52 = 6\tfrac14, not 4\tfrac14.

Practice problems

1.\frac{3}{4} - \frac{1}{6}
2.\frac{2}{3} \times \frac{9}{10}
3.\frac{5}{8} \div \frac{15}{16}
4.Simplify \frac{84}{126}

Practise this topic with checked answers

Tools for this lesson

Fraction CalculatorAdd, subtract, multiply, divide and simplify fractions and mixed numbers exactly, with step-by-step working, GCD and LCM.Percentage CalculatorFind a percentage of a number, percentage change, increase, decrease, difference, reverse percentages, discounts and tax.Rational Expression SimplifierCombine and reduce rational expressions by cancelling the polynomial GCD, and list the values excluded from the domain.

Next lesson: Solving linear equations

Equivalent fractions and lowest terms

\frac{1}{2}

\frac{1}{2},

\frac{2}{4}

\frac{2}{4} and

\frac{50}{100}

\frac{50}{100} are the same number written differently. To simplify a fraction, divide the numerator and denominator by their greatest common divisor (GCD). For

\frac{42}{56}

\frac{42}{56} the GCD is 14, so the fraction reduces to

\frac{3}{4}

\frac{3}{4}.

\frac{a}{b} = \frac{a \div g}{b \div g},\quad g = \gcd(a, b)

\frac{a}{b} = \frac{a \div g}{b \div g},\quad g = \gcd(a, b)

Adding and subtracting

Never add the denominators:

\frac{1}{2} + \frac{1}{3}

\frac{1}{2} + \frac{1}{3} is not

\frac{2}{5}

\frac{2}{5} — that would be less than

\frac{1}{2}

\frac{1}{2} alone.

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Multiplying and dividing

To multiply, multiply the numerators together and the denominators together. “Half of a third” is

\frac12 \times \frac13 = \frac16

\frac12 \times \frac13 = \frac16.

To divide, multiply by the reciprocal. Dividing by

\frac{1}{4}

\frac{1}{4} asks how many quarters fit, which is the same as multiplying by 4.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Worked examples

Calculate $\frac{1}{2} + \frac{1}{3}$ \frac{1}{2} + \frac{1}{3}.

The LCM of 2 and 3 is 6.
$\frac12 = \frac36$ \frac12 = \frac36 and $\frac13 = \frac26$ \frac13 = \frac26.
Add the numerators: $\frac{3 + 2}{6} = \frac56$ \frac{3 + 2}{6} = \frac56.

Answer: $\frac{5}{6}$ \frac{5}{6}

Calculate $1\tfrac{1}{2} \div \frac{3}{4}$ 1\tfrac{1}{2} \div \frac{3}{4}.

Convert: $1\tfrac12 = \frac32$ 1\tfrac12 = \frac32.
Multiply by the reciprocal: $\frac32 \times \frac43 = \frac{12}{6}$ \frac32 \times \frac43 = \frac{12}{6}.
Simplify: $2$ .

Answer: 2

Common mistakes

Adding numerators and denominators separately.

Forgetting to flip the second fraction when dividing — or flipping the first one.

Leaving an answer unsimplified, such as

\frac{6}{8}

\frac{6}{8} instead of

\frac34

\frac34.

Multiplying mixed numbers part by part:

2\tfrac12 \times 2\tfrac12

2\tfrac12 \times 2\tfrac12 is

\frac52 \times \frac52 = 6\tfrac14

\frac52 \times \frac52 = 6\tfrac14, not

4\tfrac14

4\tfrac14.

Practice problems

\frac{3}{4} - \frac{1}{6}

\frac{3}{4} - \frac{1}{6}

\frac{2}{3} \times \frac{9}{10}

\frac{2}{3} \times \frac{9}{10}

\frac{5}{8} \div \frac{15}{16}

\frac{5}{8} \div \frac{15}{16}

4.Simplify

\frac{84}{126}

\frac{84}{126}

Practise this topic with checked answers