Fractions — Add, Subtract, Multiply, Divide

To add and subtract fractions, make the denominators match first. To multiply fractions, multiply across. To divide fractions, multiply by the reciprocal of the second fraction.

That is the whole idea, but one condition matters: the second fraction in a division problem cannot be $0$. If it were $0$, the reciprocal would not exist and the division would be undefined.

What A Fraction Means

A fraction $\frac{a}{b}$ means $a$ parts of size $\frac{1}{b}$, with $b \ne 0$. The numerator counts how many parts you have, and the denominator tells you the size of each part.

That is why $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$. Halves and thirds are different-sized pieces, so you must rewrite them in the same unit before adding.

Fraction Rules At A Glance

For $b \ne 0$, $d \ne 0$, and $c \ne 0$ in the division rule:

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ $$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$ $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$ $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$

For addition and subtraction, these formulas work because $bd$ is a common denominator. In actual homework, you often use the least common denominator instead because it keeps the numbers smaller.

One Worked Example For All Four Operations

Use the same pair each time:

$$\frac{2}{3} \quad \text{and} \quad \frac{1}{4}$$

Add Fractions

The least common denominator of $3$ and $4$ is $12$, so rewrite both fractions:

$$\frac{2}{3} = \frac{8}{12}$$ $$\frac{1}{4} = \frac{3}{12}$$

Now the pieces match:

$$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$$

Subtract Fractions

Use the same common denominator:

$$\frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}$$

Multiply Fractions

There is no need for a common denominator here:

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

Divide Fractions

Keep the first fraction, flip the second one, and multiply:

$$\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \cdot \frac{4}{1} = \frac{8}{3}$$

This answer is greater than $1$, which makes sense. Dividing by $\frac{1}{4}$ asks how many quarter-sized pieces fit into $\frac{2}{3}$.

Why Common Denominators Matter

Addition and subtraction combine amounts of the same size. If the pieces are different sizes, the numerators alone do not tell the whole story.

Multiplication and division are different. Multiplication scales one amount by another, and division compares how many times one fraction fits into another, so a common denominator is not the key step there.

Common Fraction Mistakes

1. Adding both numerators and denominators. In general, $\frac{a}{b} + \frac{c}{d} \ne \frac{a+c}{b+d}$.
2. Finding a common denominator when multiplying or dividing. That extra step is not needed.
3. Flipping the first fraction during division. Only the second fraction is inverted.
4. Forgetting to simplify, such as leaving $\frac{2}{12}$ instead of $\frac{1}{6}$.
5. Dividing by a zero fraction. $\frac{a}{b} \div \frac{0}{d}$ is undefined.

When Students Use Fraction Operations

You use fraction operations in measurement, recipes, rates, probability, algebra, and any problem where quantities are parts of a whole.

The choice of operation depends on the question:

• Add or subtract when you are combining or comparing amounts.
• Multiply when you need a fraction of a fraction.
• Divide when you want to know how many groups fit or what one fraction is relative to another.

The One Thing to Remember

The denominator rules depend on the operation. Adding and subtracting require a common denominator first; multiplying and dividing do not. Keeping that distinction straight is what separates a correct fraction calculation from the most common error.