The keyword how to subtract fractions refers to the standard algebra procedure for computing the difference of two rational numbers. The essential idea is that subtraction is only valid when the fractions describe the same-sized parts, so denominators must match before numerators are subtracted.
Core rule If two fractions have the same denominator, subtract the numerators: \[ \frac{a}{d}-\frac{b}{d}=\frac{a-b}{d}. \] If denominators differ, convert both fractions to an equivalent form with a common denominator (often the least common denominator).
Case 1: Subtracting fractions with the same denominator
Consider \(\frac{11}{15}-\frac{4}{15}\). The denominator already matches, so only the numerators change.
\[ \frac{11}{15}-\frac{4}{15}=\frac{11-4}{15}=\frac{7}{15}. \]
Case 2: Subtracting fractions with different denominators
A linear process is used: find a common denominator, rewrite as equivalent fractions, subtract, then simplify.
Worked example: \(\frac{5}{12}-\frac{1}{18}\)
Step 1: Find the least common denominator
The least common denominator is the least common multiple of \(12\) and \(18\):
\[ 12=2^2\cdot 3, \qquad 18=2\cdot 3^2 \quad\Longrightarrow\quad \mathrm{LCD}=2^2\cdot 3^2=36. \]
Step 2: Rewrite both fractions with denominator 36
\[ \frac{5}{12}=\frac{5\cdot 3}{12\cdot 3}=\frac{15}{36}, \qquad \frac{1}{18}=\frac{1\cdot 2}{18\cdot 2}=\frac{2}{36}. \]
Step 3: Subtract numerators and simplify
\[ \frac{15}{36}-\frac{2}{36}=\frac{13}{36}. \]
Since \(13\) does not divide \(36\), \(\frac{13}{36}\) is already in simplest form.
Mixed numbers: subtracting by converting to improper fractions
For problems such as \(2\frac{1}{6}-\frac{3}{4}\), convert the mixed number to an improper fraction first.
\[ 2\frac{1}{6}=\frac{2\cdot 6+1}{6}=\frac{13}{6}. \]
Then find a common denominator for \(\frac{13}{6}-\frac{3}{4}\). The least common denominator of \(6\) and \(4\) is \(12\).
\[ \frac{13}{6}=\frac{26}{12}, \qquad \frac{3}{4}=\frac{9}{12}, \qquad \frac{26}{12}-\frac{9}{12}=\frac{17}{12}=1\frac{5}{12}. \]
Visualization: number-line view of fraction subtraction
Fraction subtraction can be interpreted on a number line: \(\frac{5}{12}-\frac{1}{18}\) means starting at \(\frac{5}{12}\) and moving left by \(\frac{1}{18}\). The endpoints align nicely when expressed with the least common denominator \(36\): \(\frac{15}{36}-\frac{2}{36}=\frac{13}{36}\).
Common errors and how to avoid them
Mistake Subtracting denominators: \(\frac{a}{b}-\frac{c}{d}=\frac{a-c}{b-d}\) (incorrect).
Fix Only numerators are subtracted after denominators match; the denominator stays the same.
Mistake Forgetting to simplify the final fraction.
Fix Divide numerator and denominator by their greatest common divisor when possible.
Summary
The procedure for how to subtract fractions is consistent: make denominators equal (using the least common denominator if needed), subtract the numerators over the common denominator, and simplify the result (converting to a mixed number when appropriate).