Add and Subtract Fractions with Different Denominators

Find the least common denominator

Can you add one quarter and one dime? Not directly — you’d first convert them to a common unit, cents. One quarter is $25$ cents and one dime is $10$ cents, so together they’re worth $35$ cents, or $\tfrac{35}{100}$ of a dollar.

Similarly, to add fractions with different denominators, we first convert them to equivalent fractions with a common denominator.

Using fraction tiles to find a common denominator for $\tfrac{1}{2}$ and $\tfrac{1}{3}$: fourths don’t exactly cover both a half-tile and a third-tile; fifths don’t either; but sixths do — exactly three $\tfrac{1}{6}$ tiles cover the $\tfrac{1}{2}$ tile, and exactly two cover the $\tfrac{1}{3}$ tile. Twelfths would also work, since even smaller tiles can always cover both. The denominator of the smallest piece that covers both fractions is called the least common denominator (LCD). So the LCD of $\tfrac{1}{2}$ and $\tfrac{1}{3}$ is $6$ — the least common multiple of the denominators $2$ and $3$.

To find the LCD of two fractions, we find the LCM of their denominators — using only the denominators, not the numerators.

Example. Find the LCD for the fractions $\tfrac{7}{12}$ and $\tfrac{5}{18}$.

Factor each denominator into primes: $12 = 2 \cdot 2 \cdot 3$ and $18 = 2 \cdot 3 \cdot 3$. Listing the primes in matching columns and bringing each column down:

The LCM of $12$ and $18$ is $36$, so the LCD of $\tfrac{7}{12}$ and $\tfrac{5}{18}$ is $36$.

Convert fractions to equivalent fractions with the LCD

Earlier we saw that the LCD of $\tfrac{1}{4}$ and $\tfrac{1}{6}$ is $12$ — three $\tfrac{1}{12}$ pieces exactly cover $\tfrac{1}{4}$, and two cover $\tfrac{1}{6}$, so $\tfrac{1}{4} = \tfrac{3}{12}$ and $\tfrac{1}{6} = \tfrac{2}{12}$. We can find this algebraically using the Equivalent Fractions Property, without models.

Example. Convert $\tfrac{1}{4}$ and $\tfrac{1}{6}$ to equivalent fractions with denominator $12$, their LCD.

The number that multiplies $4$ to get $12$ is $3$; the number that multiplies $6$ to get $12$ is $2$. Multiply the numerator and denominator of each fraction by that number:

We do not reduce the resulting fractions — doing so would just get us back to the original fractions and lose the common denominator.

Example. Convert $\tfrac{8}{15}$ and $\tfrac{11}{24}$ to equivalent fractions with denominator $120$, their LCD.

The number that multiplies $15$ to get $120$ is $8$; the number that multiplies $24$ to get $120$ is $5$. Multiplying:

Add and subtract fractions with different denominators

Once two fractions share a common denominator, we can add or subtract them by combining the numerators.

Example. Add: $\tfrac{1}{2} + \tfrac{1}{3}$.

Find the LCD of $2$ and $3$: it’s $6$. Convert each fraction: $\tfrac{1 \cdot 3}{2 \cdot 3} + \tfrac{1 \cdot 2}{3 \cdot 2} = \tfrac{3}{6} + \tfrac{2}{6}$. Add the numerators:

Since $5$ and $6$ have no common factors, this fraction cannot be reduced further.

Example. Subtract: $\tfrac{1}{2} - \left(-\tfrac{1}{4}\right)$.

The LCD of $2$ and $4$ is $4$ — one fraction already has the LCD, so we only need to convert the other: $\tfrac{1 \cdot 2}{2 \cdot 2} = \tfrac{2}{4}$. Subtract:

Example. Add: $\tfrac{7}{12} + \tfrac{5}{18}$.

The LCD of $12$ and $18$ is $36$. Converting each fraction and adding:

Since $31$ is prime, it shares no factors with $36$, so the answer is already simplified.

When we use the Equivalent Fractions Property, there’s a quick way to find the number needed for each fraction: the “missing” factors of a denominator (compared to the LCD’s full factor list) tell you exactly what to multiply by. For instance, if the LCD $36 = 2 \cdot 2 \cdot 3 \cdot 3$ and one denominator is $12 = 2 \cdot 2 \cdot 3$ (missing one factor of $3$), multiply that fraction’s numerator and denominator by $3$.

Example. Subtract: $\tfrac{7}{15} - \tfrac{19}{24}$.

The LCD of $15$ and $24$ is $120$. Rewriting as equivalent fractions with denominator $120$ and subtracting, then removing the common factor of $3$:

Example. Add: $-\tfrac{11}{30} + \tfrac{23}{42}$.

The LCD of $30$ and $42$ is $210$. Converting and adding, then removing the common factor of $2$:

When one fraction has a variable in its numerator, we follow the same steps.

Example. Add: $\tfrac{3}{5} + \tfrac{x}{8}$.

The LCD of $5$ and $8$ is $40$. Rewriting as equivalent fractions: $\tfrac{3 \cdot 8}{5 \cdot 8} + \tfrac{x \cdot 5}{8 \cdot 5} = \tfrac{24}{40} + \tfrac{5x}{40}$. Adding the numerators:

We cannot combine $24$ and $5x$ since they aren’t like terms, so this is fully simplified.

Identify and use fraction operations

By now you’ve practiced all four fraction operations. Remember: you need a common denominator to add or subtract fractions, but not to multiply or divide them.

Example. Simplify: (a) $-\tfrac{1}{4} + \tfrac{1}{6}$; (b) $-\tfrac{1}{4} \div \tfrac{1}{6}$.

(a) The operation is addition. The LCD of $4$ and $6$ is $12$. Convert: $-\tfrac{3}{12} + \tfrac{2}{12} = -\tfrac{1}{12}$.

(b) The operation is division — no common denominator is needed. Multiply by the reciprocal of the second fraction: $-\tfrac{1}{4} \cdot \tfrac{6}{1} = -\tfrac{6}{4} = -\tfrac{3}{2}$.

Example. Simplify: (a) $\tfrac{5x}{6} - \tfrac{3}{10}$; (b) $\tfrac{5x}{6} \cdot \tfrac{3}{10}$.

(a) The operation is subtraction, and the fractions don’t share a denominator. The LCD of $6$ and $10$ is $30$. Rewriting: $\tfrac{25x}{30} - \tfrac{9}{30} = \tfrac{25x-9}{30}$.

(b) The operation is multiplication — no common denominator needed. Multiplying and removing common factors of $2$ and $3$: $\tfrac{5x \cdot 3}{6 \cdot 10} = \tfrac{x}{4}$.

Use the order of operations to simplify complex fractions

A complex fraction is a fraction in which the numerator or denominator contains a fraction — we saw this in the previous section, where we rewrote such fractions as division problems. Now consider complex fractions where the numerator or denominator itself needs to be simplified first. Following the order of operations, we simplify the numerator and denominator separately, then divide.

Example. Simplify: $\tfrac{\left(\tfrac{1}{2}\right)^2}{4+3^2}$.

Simplify the numerator: $\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$. Simplify the denominator: $4 + 3^2 = 4 + 9 = 13$. Divide the numerator by the denominator: $\tfrac{1}{4} \div 13 = \tfrac{1}{4} \cdot \tfrac{1}{13}$:

Example. Simplify: $\tfrac{\tfrac{1}{2}+\tfrac{2}{3}}{\tfrac{3}{4}-\tfrac{1}{6}}$.

Rewrite the numerator with LCD $6$ and the denominator with LCD $12$: $\tfrac{3}{6}+\tfrac{4}{6}$ over $\tfrac{9}{12}-\tfrac{2}{12}$. Add in the numerator and subtract in the denominator: $\tfrac{7}{6} \div \tfrac{7}{12}$. Rewrite as multiplication by the reciprocal and simplify:

Evaluate variable expressions with fractions

We’ve evaluated expressions before; now we can do the same with fractions. To evaluate an expression, substitute the value of the variable and simplify.

Example. Evaluate $x + \tfrac{1}{3}$ when (a) $x = -\tfrac{1}{3}$; (b) $x = -\tfrac{3}{4}$.

(a) Substitute $-\tfrac{1}{3}$ for $x$: $-\tfrac{1}{3} + \tfrac{1}{3} = 0$.

(b) Substitute $-\tfrac{3}{4}$ for $x$. The LCD of $4$ and $3$ is $12$: $-\tfrac{9}{12} + \tfrac{4}{12} = -\tfrac{5}{12}$.

Example. Evaluate $2x^2y$ when $x = \tfrac{1}{4}$ and $y = -\tfrac{2}{3}$.

In $2x^2y$, the exponent applies only to $x$. Substituting and simplifying the exponent first: $2\left(\tfrac{1}{4}\right)^2\left(-\tfrac{2}{3}\right) = 2 \cdot \tfrac{1}{16} \cdot \left(-\tfrac{2}{3}\right)$. Multiplying and removing common factors:

Example. Evaluate $\tfrac{p+q}{r}$ when $p = -4$, $q = -2$, and $r = 8$.

Substitute the values into the expression: $\tfrac{-4+(-2)}{8}$. Add in the numerator first: $\tfrac{-6}{8}$. Simplify:

Key terms

least common denominator (LCD) — the least common multiple of the denominators of two or more fractions. complex fraction — a fraction whose numerator or denominator (or both) itself contains a fraction, requiring the order of operations to simplify: numerator first, denominator next, then divide.

This section is adapted from Prealgebra 2e, Section 4.5: Add and Subtract Fractions with Different Denominators by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: described the coin and fraction-tile models for the LCD in prose instead of reproducing the diagrams, and presented the two-column worked examples as prose walkthroughs with typeset math; omitted the Be Prepared quiz, Manipulative Mathematics callout, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.