Consider two fractions such as one-fourth and two-thirds. Remember that a fraction is made up of a numerator that tells how many parts there are and is the top number; and a denominator that tells what size part each is and is the bottom number. So in one-fourth, one is the numerator and four is the denominator. If you have a sandwich that is cut into four equal parts, each part is one-fourth of the sandwich. In the fraction two-thirds, two is the numerator and three is the denominator. If you have a pie that is cut into three equal parts, each part is one-third.
Suppose that you have two fractions which have denominators that do not share a common factor. Perhaps you have one-fourth of one pie and two-thirds of another pie. Your task may be to discover how much pie you have in all, which requires you to add one-fourth to two-thirds. You must find the least common denominator first. Multiply the denominators: 3 x 4 = 12. Since both 3 and 4 are factors of 12 and neither is a factor of the other, 12 is the least common denominator. Now you can adjust both fractions. Multiply the numerator of one-fourth by 3 to obtain a new fraction: 1 x 3 = 3. So one fourth becomes its equivalent, three-twelfths. Multiply the numerator of two-thirds by 4 to obtain a new fraction: 2 x 4 = 8. So two-thirds becomes its equivalent, eight-twelfths. Now, since the least common denominator is shared by both fractions, you can add the numerators to discover that you have eleven-twelfths of a pie.
Suppose, on the other hand, that you have two fractions in which the denominator of one is a factor of the other such as two-fifths and four-tenths. The denominator of two-fifths, 5, is a factor of the denominator of four-tenths, 10. The least common denominator will be 10, so to be able to work with this pair of fractions you only need to change the first one. Since 5 x 2= 10, and 10 will be the common denominator, all you need to do is multiply the numerator of the same fraction, 2 by 2: 2 x 2 = 4. The new pair of fractions that share the least common denominator becomes four-tenths and five-tenths. If you were asked to add these fractions, the sum would be nine-tenths.
Suppose, finally, that you have two fractions with denominators that share a factor such as seven-twelfths and three-eighteenths. A common denominator could be found by multiplying the two denominators as in the first example: 12 x 18 = 216. However, 120 is not the least common denominator. Notice that both denominators, 12 and 18 both share a common factor of 6. 12 = 6 x 2 and 18 = 6 x 3. To find the least common denominator, multiply the common factor, 6, by the two unshared factors, 2 and 3. The product of these three numbers, 3 x 2 x 6 = 36. This is the least common denominator because both 12 and 18 are factors of 36. Now you can adjust the numerators. Seven, of seven-twelfths, will be multiplied by 3 because you have to multiply 12 by 3 to equal the new denominator. The new fraction becomes twenty-one-thirty-sixths. Three, of three-eighteenths, must be multiplied by 2 because you have to multiply 18 by 2 to equal 36. The new fraction becomes six- thirty-sixths. Now, with two fractions that share the least common denominator of 36, you can add and subtract these fractions as you need to.