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Subtracting Fractions with a Common Denominator

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Subtracting Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Fraction Subtraction

If \(a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c\) are numbers where \(c\ne 0,\) then

\(\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}\)

To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example

Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)

Solution

 \(\frac{23}{24}-\frac{14}{24}\)
Subtract the numerators and place the difference over the common denominator.\(\frac{23-14}{24}\)
Simplify the numerator.\(\frac{9}{24}\)
Simplify the fraction by removing common factors.\(\frac{3}{8}\)

Example

Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)

Solution

 \(\frac{y}{6}-\frac{1}{6}\)
Subtract the numerators and place the difference over the common denominator.\(\frac{y-1}{6}\)

The fraction is simplified because we cannot combine the terms in the numerator.

Example

Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)

Solution

Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}.\)

 \(-\frac{10}{x}-\frac{4}{x}\)
Subtract the numerators.\(\frac{-10-4}{x}\)
Simplify.\(\frac{-14}{x}\)
Rewrite with the negative sign in front of the fraction.\(-\frac{14}{x}\)

Now let’s do an example that involves both addition and subtraction.

Example

Simplify: \(\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}.\)

Solution

 \(\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}\)
Combine the numerators over the common denominator.\(\frac{3+\left(-5\right)-1}{8}\)
Simplify the numerator, working left to right.\(\frac{-2-1}{8}\)
Subtract the terms in the numerator.\(\frac{-3}{8}\)
Rewrite with the negative sign in front of the fraction.\(-\frac{3}{8}\)

Optional Video: Subtracting Fractions With Like Denominators

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