The lowest common denominator is the most important term to understand. You cannot add or subtract any fractions if the denominators are not the same. To find the LCD, write down the multiples of both denominators, looking for the lowest multiple they both have in common. For example, 20 is the lowest common multiple of four and five. To convert fractions so that they have the same denominator, multiply the numerator and denominator by the same factor. For example, 1/4 = (1 x 5) / (4 x 5) = 5/20. Because 1/4 and 5/20 have the same value, these are called equivalent fractions.
Reducing fractions applies the opposite property, dividing both numerator and denominator by the same factor until the fraction is written in its lowest form, 5/20 = (5 ÷ 5) / (20 ÷ 5) = 1/4.
This method only applies to mixed numbers that have the same denominators. If the denominators are not the same, the fraction components of the mixed number must first be converted to equivalent fractions with the same denominator. Then, simply separate the whole numbers from their fraction components. Subtract the numerators of the fractions and keep the denominator the same. Next, subtract the whole numbers and write the difference before the fraction. For example, you would regroup the problem 5 2/5 - 3 1/5 to read (5 - 3) + (2/5 - 1/5) = 2 1/5.
An improper fraction has a numerator larger than the denominator. To convert a mixed number to an improper fraction, multiply the denominator by the whole number, add the numerator and write the answer over the original denominator. For example, 3 2/5 = 17/5. Converting mixed numbers to improper fractions eliminates the grouping technique. The fractions still need to have the same denominator before you can subtract the numerators. Leave answers in improper fraction form or convert them back to mixed numbers by dividing the numerator by the denominator. The quotient is the whole number and the remainder goes over the original denominator -- for example, 11/5 = 2 1/5.
Sometimes, the subtrahend, or subtracted number, is larger than the minuend. With this technique, you borrow from the whole number and add that borrowed one to the fractional unit so that the minuend is larger. For example, to solve 4 1/5 -- 3 4/5 you need to borrow one from the whole number (4 -- 1 = 3) and add it to the fraction (5/5 + 1/5 = 6/5) so that the first fraction is larger than the second. (Although you borrowed 1, or 1/1, you still have to find a common denominator for (1/1 + 1/5). In this step, you have to perform a shortcut around the equivalent fraction step and use the fraction 5/5, which still equals one and has the necessary denominator.) Then simply subtract using the normal procedure.