Subtracting Fractions with Unlike Denominators Formula

Subtracting fractions with unlike denominators are subtracting fractions with different denominators by first rewriting them with a common denominator.

The Formula

abβˆ’cd=adβˆ’bcbd(orΒ useΒ LCD)\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \quad \text{(or use LCD)}

When to use: To find 34βˆ’13\frac{3}{4} - \frac{1}{3}, convert to twelfths: 912βˆ’412=512\frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Same idea as addition, just subtract.

Quick Example

34βˆ’13=912βˆ’412=512\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}

Notation

abβˆ’cd\frac{a}{b} - \frac{c}{d} β€” rewrite with LCD, then subtract: adβˆ’bcbd\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 34βˆ’13\frac{3}{4} - \frac{1}{3}, convert to twelfths: 912βˆ’412=512\frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Same idea as addition, just subtract.

Formal View

abβˆ’cd=adβˆ’bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} where b,dβ‰ 0b, d \neq 0

Worked Examples

Example 1

easy
Subtract 34βˆ’16\frac{3}{4} - \frac{1}{6}.

Answer

712\frac{7}{12}

First step

1
Find the LCD of 44 and 66: LCD=12\text{LCD} = 12.

Full solution

  1. 2
    Convert: 34=912\frac{3}{4} = \frac{9}{12} and 16=212\frac{1}{6} = \frac{2}{12}.
  2. 3
    Subtract: 912βˆ’212=712\frac{9}{12} - \frac{2}{12} = \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 78\frac{7}{8} of a race, then stopped. If the race is 11 km long, what fraction of the race is left? If another runner has already finished 25\frac{2}{5} of the remaining distance, how much of the total race has that runner covered?

Example 3

hard
Subtract 78βˆ’23βˆ’16\frac{7}{8} - \frac{2}{3} - \frac{1}{6}.

Common Mistakes

  • Subtracting numerators and denominators separately - match denominators first, then subtract only the numerators.
  • Rewriting the numerator without scaling it like the denominator - 34=912\frac{3}{4}=\frac{9}{12} means multiply top and bottom by 3.
  • Subtracting the larger from the smaller out of order - keep the minuend first; 13βˆ’34\frac{1}{3}-\frac{3}{4} is negative.

Why This Formula Matters

Subtraction needs a shared unit just as addition does β€” you cannot take fourths away from thirds until both are twelfths. Missing this gives wrong differences and breaks later work like subtracting mixed numbers and rational expressions. Recognizing it by "Do the fractions have different denominators that must be matched before subtracting?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from subtracting fractions with like denominators and adding fractions with unlike denominators and dividing fractions in a mixed problem set.

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 34βˆ’13\frac{3}{4} - \frac{1}{3}, convert to twelfths: 912βˆ’412=512\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?

abβˆ’cd\frac{a}{b} - \frac{c}{d} β€” rewrite with LCD, then subtract: adβˆ’bcbd\frac{ad - bc}{bd}

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

Subtraction needs a shared unit just as addition does β€” you cannot take fourths away from thirds until both are twelfths. Missing this gives wrong differences and breaks later work like subtracting mixed numbers and rational expressions. Recognizing it by "Do the fractions have different denominators that must be matched before subtracting?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from subtracting fractions with like denominators and adding fractions with unlike denominators and dividing fractions in a mixed problem set.

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

The procedure for subtracting fractions with unlike denominators is the easy part; the trap is subtracting numerators and denominators separately. Asking "Do the fractions have different denominators that must be matched before subtracting?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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