Subtracting Fractions

Subtracting fractions with different denominators requires a common denominator to combine the numerators into a single fraction.

ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

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What each symbol means

What Subtracting Fractions takes
aa
bb
cc
dd
Subtracting Fractions →
SymbolMeaning
aaThe numerator (top number) of the first fraction you are subtracting from; it tells you how many parts you have.
bbThe denominator (bottom number) of the first fraction; it tells you how many equal parts the whole is divided into and cannot be zero.
ccThe numerator (top number) of the second fraction that you are subtracting away; it tells you how many parts you are removing.
ddThe denominator (bottom number) of the second fraction; it tells you how many equal parts that fraction's whole is divided into and cannot be zero.

When to use it

Reach for this when you need to find the difference between two fractions with unlike denominators.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Subtract two fractions with different denominators

Problem

Calculate 34−16\frac{3}{4} - \frac{1}{6}.
  1. 34−16\frac{3}{4} - \frac{1}{6}

    We need a common denominator before subtracting; the denominators are 4 and 6.

  2. LCM(4,6)=12\text{LCM}(4, 6) = 12

    The least common multiple of 4 and 6 is 12, so that will be our common denominator.

  3. 3⋅34⋅3−1⋅26⋅2\frac{3 \cdot 3}{4 \cdot 3} - \frac{1 \cdot 2}{6 \cdot 2}

    Multiply the first fraction by 33\frac{3}{3} and the second by 22\frac{2}{2} to get a denominator of 12.

  4. 912−212\frac{9}{12} - \frac{2}{12}

    Now both fractions have the same denominator, so we can subtract the numerators.

  5. 9−212=712\frac{9 - 2}{12} = \frac{7}{12}

    Subtract 9−2=79 - 2 = 7 to get the final answer.

Answer: 712\frac{7}{12}

Since the denominators were different, we found a common denominator using the LCM. Then we converted each fraction to an equivalent one with that denominator, and finally subtracted the numerators while keeping the denominator the same.

2. Subtract fractions with negative numbers

Problem

Calculate −23−15-\frac{2}{3} - \frac{1}{5}.
  1. −23−15-\frac{2}{3} - \frac{1}{5}

    We're subtracting a positive fraction from a negative one; the denominators are 3 and 5.

  2. LCM(3,5)=15\text{LCM}(3, 5) = 15

    The least common multiple of 3 and 5 is 15.

  3. −2⋅53⋅5−1⋅35⋅3-\frac{2 \cdot 5}{3 \cdot 5} - \frac{1 \cdot 3}{5 \cdot 3}

    Convert both fractions to have denominator 15.

  4. −1015−315-\frac{10}{15} - \frac{3}{15}

    Now we have a common denominator and can combine the fractions.

  5. −10−315=−1315=−1315\frac{-10 - 3}{15} = \frac{-13}{15} = -\frac{13}{15}

    Subtract the numerators: −10−3=−13-10 - 3 = -13. The result is negative because we're subtracting from a negative value.

Answer: −1315-\frac{13}{15}

When the first fraction is negative, the result will be more negative. We still use the same process: find a common denominator, convert both fractions, then subtract the numerators while paying attention to the signs.

3. Find the distance difference on a road trip

Problem

On a cross-country road trip, the first driving leg covers 23\frac{2}{3} of the total distance, and the second driving leg covers 14\frac{1}{4} of the total distance. By how much is the first leg longer than the second leg?
  1. 23−14\frac{2}{3} - \frac{1}{4}

    To find the difference, we subtract the second leg's fraction from the first leg's fraction.

  2. LCM(3,4)=12\text{LCM}(3, 4) = 12

    The least common multiple of 3 and 4 is 12, so we'll use 12 as the common denominator.

  3. 2⋅43⋅4−1⋅34⋅3\frac{2 \cdot 4}{3 \cdot 4} - \frac{1 \cdot 3}{4 \cdot 3}

    Multiply the first fraction by 44\frac{4}{4} and the second by 33\frac{3}{3} to convert to denominator 12.

  4. 812−312\frac{8}{12} - \frac{3}{12}

    Simplify to get fractions with the common denominator.

  5. 8−312=512\frac{8 - 3}{12} = \frac{5}{12}

    Subtract the numerators: 8−3=58 - 3 = 5.

Answer: 512\frac{5}{12}

Word problems with fractions often require you to subtract to find the difference between two quantities. Here, both legs are expressed as fractions of the total distance, so we could directly compare them using subtraction after finding a common denominator.

Common mistakes

Where Subtracting Fractions usually goes wrong
Answer came out wrong
Students write 34−16=2−2\frac{3}{4} - \frac{1}{6} = \frac{2}{-2} by subtracting both numerators and denominators.
Denominators must stay the same or be converted to a common denominator. Never subtract denominators: 34−16=912−212=712\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}.
Students write 34−16=3−14=24\frac{3}{4} - \frac{1}{6} = \frac{3-1}{4} = \frac{2}{4} by using only the first denominator without converting.
Find a common denominator first: 34−16=912−212=712\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}.
For 14−34\frac{1}{4} - \frac{3}{4}, students write 3−14=24\frac{3-1}{4} = \frac{2}{4} instead of 1−34=−24\frac{1-3}{4} = -\frac{2}{4}.
Keep the order exactly: 14−34=1−34=−24=−12\frac{1}{4} - \frac{3}{4} = \frac{1-3}{4} = \frac{-2}{4} = -\frac{1}{2}.
The mistakeWhy it is wrongThe fix
Students write 34−16=2−2\frac{3}{4} - \frac{1}{6} = \frac{2}{-2} by subtracting both numerators and denominators.Denominators describe the size of parts, not quantities to subtract; changing the denominator destroys the meaning of the fraction.Denominators must stay the same or be converted to a common denominator. Never subtract denominators: 34−16=912−212=712\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}.
Students write 34−16=3−14=24\frac{3}{4} - \frac{1}{6} = \frac{3-1}{4} = \frac{2}{4} by using only the first denominator without converting.Without a common denominator, the numerators represent parts of different sizes, so you cannot simply subtract them any more than you could subtract apples from oranges.Find a common denominator first: 34−16=912−212=712\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}.
For 14−34\frac{1}{4} - \frac{3}{4}, students write 3−14=24\frac{3-1}{4} = \frac{2}{4} instead of 1−34=−24\frac{1-3}{4} = -\frac{2}{4}.Subtraction is not commutative; the order matters. Subtracting a larger numerator from a smaller one always gives a negative result.Keep the order exactly: 14−34=1−34=−24=−12\frac{1}{4} - \frac{3}{4} = \frac{1-3}{4} = \frac{-2}{4} = -\frac{1}{2}.

Tips and when to use something else

  • If both fractions already have the same denominator, just subtract the numerators and keep the denominator—there's no need to find a common denominator.
  • Always use the Least Common Multiple (LCM) of the denominators instead of just multiplying them; this keeps your numbers smaller and easier to simplify.
  • Some texts use the formula ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} directly, but finding a common denominator using LCM usually requires less simplification at the end.
  • Don't forget to simplify your final answer by finding the Greatest Common Factor of the numerator and denominator.

Related formulas

Adding FractionsAdd fractions with different denominators by finding a common denominator, then combining the numerators to get your answer.Simplifying FractionsSimplify a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor.Equivalent FractionsMultiply or divide both parts of a fraction by the same non-zero number to create an equivalent fraction with the same value.Least Common MultipleThe least common multiple is the smallest positive number that all given numbers divide into evenly; use it when adding fractions with different denominators.

Frequently asked questions

Do I always need to find a common denominator?
Only if the denominators are different. If both fractions already have the same denominator, subtract the numerators directly. For different denominators, you must find a common denominator before subtracting.
What is the fastest way to subtract fractions?
If the denominators are already the same, just subtract the numerators. Otherwise, find the Least Common Multiple (LCM) of the denominators to minimize the size of the numbers you're working with. Some students use the formula ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} directly, which works but often produces large numbers that need to be simplified.
How do you subtract a fraction from a whole number?
Rewrite the whole number as a fraction with denominator 1. For example, 5−23=51−235 - \frac{2}{3} = \frac{5}{1} - \frac{2}{3}. Then proceed as usual: find a common denominator (which would be 3), convert to 153−23=133\frac{15}{3} - \frac{2}{3} = \frac{13}{3}.
Do negative fractions follow the same subtraction rules?
Yes. Subtracting a negative fraction is the same as adding a positive one: 14−(−14)=14+14=24=12\frac{1}{4} - (-\frac{1}{4}) = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}. Always find a common denominator first, then subtract the numerators, being careful with signs.

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Reviewed 2026-09-18