1Cademy - Adding $$\frac{3}{5} + \frac{x}{8}$$

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Least Common Denominator

Example

Adding $\frac{3}{5} + \frac{x}{8}$

Add two fractions with unlike denominators where one numerator is a variable:

$\frac{3}{5} + \frac{x}{8}$

Step 1 — Find the LCD: Factor each denominator: $5 = 5$ and $8 = 2 \cdot 2 \cdot 2$ . Since the two factorizations share no primes, the LCD is $2 \cdot 2 \cdot 2 \cdot 5 = 40$ .

Step 2 — Rewrite as equivalent fractions with the LCD: The denominator $5$ is missing three factors of $2$ (i.e., $8$ ), and the denominator $8$ is missing the factor $5$ :

$\frac{3 \cdot 8}{5 \cdot 8} + \frac{x \cdot 5}{8 \cdot 5} = \frac{24}{40} + \frac{5x}{40}$

Step 3 — Add the numerators: Combine over the common denominator: $\frac{24 + 5x}{40}$ .

Step 4 — Check for simplification: The constant $24$ and the variable term $5x$ are not like terms, so the numerator cannot be simplified further. The answer is $\frac{24 + 5x}{40}$ .

This example shows that when a sum involves both a number and a variable expression in the numerator, the result stays as a single fraction with an unsimplified numerator — unlike terms cannot be combined.

Updated 2026-04-21

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OpenStax Elementary Algebra

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