Addition and Subtraction of Fractions

Adding Like Fractions

Simple numerator sum.

Adding like fractions is super simple because all the pieces are already the same size. You only add the top numbers.
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Key Points

  1. Add the numerators (top numbers) together.
  2. Keep the denominator (bottom number) exactly the same. Do not add the bottom numbers!
  3. For example: 15+25=35\frac{1}{5} + \frac{2}{5} = \frac{3}{5}.

Olympiad Fact

Always verbalize the fraction (e.g., 'two-fifths') to keep the unit of measurement clear in your mind. This helps prevent adding the denominators incorrectly.

Adding Unlike Fractions

Converting before combining.

You cannot add unlike fractions together because their pieces are different sizes. You have to make their piece sizes match first.
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Key Points

  1. First, find the Least Common Multiple (LCM) of the two bottom numbers to use as your new matching denominator.
  2. Figure out what number you need to multiply the old bottom number by to reach this new LCM denominator.
  3. Multiply both the top and bottom numbers of the fraction by that exact same amount.
  4. Now that both fractions share the same matching bottom number, simply add their top numbers together!
  5. Keep the matching bottom number exactly the same in your final answer.

Olympiad Fact

While the universal cross-multiplication formula (ab+cd=ad+bcbd)(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}) is effective, finding the lowest common denominator (LCM) is the preferred method in competitions to keep results simpler.

Subtracting Like Fractions

Simple numerator difference.

Subtracting like fractions works just like adding them. Because the pieces are the same size, you only subtract the top numbers.
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Key Points

  1. Subtract the second top number from the first top number.
  2. The bottom number stays exactly the same. Do not subtract the bottom numbers!
  3. For example, 57−27=37\frac{5}{7} - \frac{2}{7} = \frac{3}{7}.

Olympiad Fact

Always remember to simplify your result. A common trap in competition math is providing the correct difference without reducing it to its simplest form.

Subtracting Unlike Fractions

Common ground for subtraction.

Just like addition, you cannot subtract fractions with different bottom numbers. You must make them match before doing any math.
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Key Points

  1. First, find the Least Common Multiple (LCM) of the two bottom numbers to use as your new matching denominator.
  2. Figure out what number you need to multiply the old bottom number by to reach this new LCM denominator.
  3. Multiply both the top and bottom numbers of the fraction by that exact same amount.
  4. Once the bottom numbers match, subtract the second top number from the first top number.
  5. Keep the matching bottom number exactly the same in your final answer.

Olympiad Fact

Shortcut formula: ab−cd=(a×d)−(b×c)b×d\frac{a}{b} - \frac{c}{d} = \frac{(a \times d) - (b \times c)}{b \times d}. Use the cross-products (a×d)(a \times d) and (b×c)(b \times c) as your new numerators, and (b×d)(b \times d) as your shared common denominator.

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