Addition and subtraction of fractions

Adding Fractions

Adding Fractions With The Same Denominator

In order to add fractions with the same denominator, you only need to add the numerators while keeping the original denominator for the sum.

${\frac {1}{5}}+{\frac {3}{5}}={\frac {1+3}{5}}={\frac {4}{5}}$

Adding fractions with the same denominator is the rule but it begs the question why? Why can’t (or shouldn't) I add both numerators and denominators?

${\frac {1}{4}}+{\frac {1}{4}}={\frac {1+1}{4+4}}={\frac {2}{8}}$

To make sense of this try taking a 12 inch ruler and drawing a 3 inch horizontal line (1/4 of a foot) and then on the end add another 3 inch line (1/4 of a foot). What is the total length of the line? It should be 6 inches (1/2 a foot) and not 2/8 of a foot (3 inches). In essence it seems we can only add like items and like items are terms that have the same denominator and we add them up by adding up numerators.

Adding Fractions With Different Denominators

When adding fractions that do not have the same denominator, you must make the denominators of all the terms the same. We do this by finding the least common multiple of the two denominators.

Least common multiple of 4 and 5 is 20; therefore, make the denominators 20:

${\frac {1}{4}}+{\frac {2}{5}}={\frac {1\times 5}{4\times 5}}+{\frac {2\times 4}{5\times 4}}={\frac {5}{20}}+{\frac {8}{20}}$

Now that the common denominators are the same, perform the usual addition:

${\frac {5+8}{20}}={\frac {13}{20}}$

Subtracting Fractions

Subtracting Fractions With The Same Denominator

To subtract fractions sharing a denominator, take their numerators and subtract them in order of appearance. If the numerator's difference is zero, the whole difference will be zero, regardless of the denominator.

${\frac {6}{13}}-{\frac {2}{13}}={\frac {6-2}{13}}={\frac {4}{13}}$

Subtracting Fractions With Different Denominators

To subtract one fraction from another, you must again find the least common multiple of the two denominators.

Least common multiple of 4 and 6 is 12; therefore, make the denominator 12:

${\frac {3}{4}}-{\frac {1}{6}}={\frac {3\times 3}{4\times 3}}-{\frac {1\times 2}{6\times 2}}={\frac {9}{12}}-{\frac {2}{12}}$

Now that the denominator is same, perform the usual subtraction.

${\frac {9-2}{12}}={\frac {7}{12}}$

Licensing

Content obtained and/or adapted from:

• Fractions, Wikibooks under a CC BY-SA license