Subtracting Fractions with Unlike Denominators Formula

The Formula

\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \quad \text{(or use LCD)}

When to use: To find \frac{3}{4} - \frac{1}{3}, convert to twelfths: \frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Same idea as addition, just subtract.

Quick Example

\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}

Notation

\frac{a}{b} - \frac{c}{d} — rewrite with LCD, then subtract: \frac{ad - bc}{bd}

What This Formula Means

Subtracting fractions with different denominators by first rewriting them with a common denominator, then subtracting numerators.

To find \frac{3}{4} - \frac{1}{3}, convert to twelfths: \frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Same idea as addition, just subtract.

Formal View

\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} where b, d \neq 0

Worked Examples

Example 1

easy
Subtract \frac{3}{4} - \frac{1}{6}.

Solution

  1. 1
    Find the LCD of 4 and 6: \text{LCD} = 12.
  2. 2
    Convert: \frac{3}{4} = \frac{9}{12} and \frac{1}{6} = \frac{2}{12}.
  3. 3
    Subtract: \frac{9}{12} - \frac{2}{12} = \frac{7}{12}.

Answer

\frac{7}{12}
Just as with addition of unlike fractions, you must first find equivalent fractions with a common denominator. Once the pieces are the same size, subtracting numerators gives the result.

Example 2

medium
A runner completed \frac{7}{8} of a race, then stopped. If the race is 1 km long, what fraction of the race is left? If another runner has already finished \frac{2}{5} of the remaining distance, how much of the total race has that runner covered?

Common Mistakes

  • Subtracting numerators and denominators independently: \frac{3}{4} - \frac{1}{3} = \frac{2}{1}
  • Finding the LCD but only converting one fraction
  • Flipping the subtraction order of the numerators

Why This Formula Matters

Completes the set of fraction addition/subtraction skills needed for algebra and real-world problem solving.

Frequently Asked Questions

What is the Subtracting Fractions with Unlike Denominators formula?

Subtracting fractions with different denominators by first rewriting them with a common denominator, then subtracting numerators.

How do you use the Subtracting Fractions with Unlike Denominators formula?

To find \frac{3}{4} - \frac{1}{3}, convert to twelfths: \frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Same idea as addition, just subtract.

What do the symbols mean in the Subtracting Fractions with Unlike Denominators formula?

\frac{a}{b} - \frac{c}{d} — rewrite with LCD, then subtract: \frac{ad - bc}{bd}

Why is the Subtracting Fractions with Unlike Denominators formula important in Math?

Completes the set of fraction addition/subtraction skills needed for algebra and real-world problem solving.

What do students get wrong about Subtracting Fractions with Unlike Denominators?

Subtracting numerators in the wrong order after converting, leading to negative results unexpectedly.

What should I learn before the Subtracting Fractions with Unlike Denominators formula?

Before studying the Subtracting Fractions with Unlike Denominators formula, you should understand: subtracting fractions like denominators, equivalent fractions, least common multiple.

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