Math Core

You can only add things that are measured in the same unit. Thirds and fourths are different-sized pieces, so before you add $\dfrac{1}{3} + \dfrac{1}{4}$ you need to rewrite both in a common unit. This lesson reviews that process and extends it to negative fractions, using the same integer rules you already know.

Same denominator

When the denominators match, the pieces are the same size. Add or subtract the numerators and keep the denominator.

\frac{5}{9} + \frac{2}{9} = \frac{7}{9} \qquad\qquad \frac{3}{8} - \frac{7}{8} = \frac{3 - 7}{8} = -\frac{4}{8} = -\frac{1}{2}

The second example shows that subtracting can give a negative result. That's fine: just follow the integer rules for the numerators.

Different denominators

Adding or subtracting fractions

Find a common denominator, ideally the least common denominator (LCD), which is the least common multiple of the denominators.
Rewrite each fraction as an equivalent fraction with that denominator.
Add or subtract the numerators. Keep the denominator.
Simplify.

Worked example: Unlike denominators

Find $\dfrac{5}{6} + \dfrac{3}{8}$ .

Multiples of 8: 8, 16, 24. Since 24 is also a multiple of 6, the LCD is 24.

\frac{5}{6} + \frac{3}{8} = \frac{20}{24} + \frac{9}{24} = \frac{29}{24} = 1\frac{5}{24}

Tip

Any common multiple works as a denominator, including the product of the two denominators. The LCD just keeps the numbers smaller. If you use $6 \times 8 = 48$ above, you get $\dfrac{58}{48}$ , which simplifies to the same $\dfrac{29}{24}$ .

Adding and subtracting negative fractions

Negative fractions follow the same rules as integers:

Subtracting a number is the same as adding its opposite: $a - b = a + (-b)$ .
Subtracting a negative is adding a positive: $a - (-b) = a + b$ .
To add numbers with different signs, subtract their absolute values and keep the sign of the one farther from zero.

Put the negative sign on the numerator, find a common denominator, and then it's just integer arithmetic on the numerators.

1/4 - 3/4 = -1/2: start at 1/4 and move 3/4 to the left

Worked example: Mixed signs

Find each sum or difference.

$-\dfrac{2}{3} + \dfrac{1}{4}$
$\dfrac{1}{6} - \left(-\dfrac{3}{4}\right)$
$-\dfrac{3}{10} - \dfrac{2}{5}$

Solutions.

LCD 12: $\dfrac{-8}{12} + \dfrac{3}{12} = \dfrac{-8 + 3}{12} = -\dfrac{5}{12}$ .
Subtracting a negative means adding: $\dfrac{1}{6} + \dfrac{3}{4}$ . LCD 12: $\dfrac{2}{12} + \dfrac{9}{12} = \dfrac{11}{12}$ .
LCD 10: $\dfrac{-3}{10} - \dfrac{4}{10} = \dfrac{-3 - 4}{10} = -\dfrac{7}{10}$ .

Common mistake

Never add the denominators. $\dfrac{1}{3} + \dfrac{1}{4}$ is not $\dfrac{2}{7}$ . A quick check shows why: $\dfrac{2}{7}$ is less than $\dfrac{1}{3}$ , but adding a positive number should make the total bigger. The correct sum is $\dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}$ .

Mixed numbers

The most reliable method, especially when negatives or borrowing are involved, is to convert mixed numbers to improper fractions first.

Worked example: Subtracting mixed numbers

Find $2\dfrac{1}{4} - 3\dfrac{2}{3}$ .

Convert: $2\dfrac{1}{4} = \dfrac{9}{4}$ and $3\dfrac{2}{3} = \dfrac{11}{3}$ . The LCD is 12.

\frac{9}{4} - \frac{11}{3} = \frac{27}{12} - \frac{44}{12} = \frac{27 - 44}{12} = -\frac{17}{12} = -1\frac{5}{12}

Does the sign make sense? You are subtracting a bigger number from a smaller one, so the answer should be negative. It is.

Worked example: A word problem

A diver is at $-4\dfrac{1}{2}$ meters (below the surface). She rises $2\dfrac{3}{4}$ meters. What is her new position?

Rising means adding: $-4\dfrac{1}{2} + 2\dfrac{3}{4} = -\dfrac{18}{4} + \dfrac{11}{4} = -\dfrac{7}{4} = -1\dfrac{3}{4}$ .

She is at $-1\dfrac{3}{4}$ meters, still below the surface.