Adding and Subtracting Fractions Less Than 1

To add fractions, you have to have "common" (shared) denominators. As the proverb says, you can only add apples to apples, not apples to oranges. In the context of adding fractions, you can't combine, say, 1/4 and 2/5. Because they are fractions of different types (one being fourths and the other being fifths), they are "apples and oranges". To add them, you first have to convert to an all-apples denominator; in this case, you'd convert to twentieths, getting 5/20 and 8/20 as your all-apples fractions.

Simplify: 1/4 + 2/5

Before I can add these fractions, I have to find their common denominator.

The lowest (smallest) common denominator is just the Least Common Multiple (LCM) of the two denominators, 4 and 5. The prime factorizations and LCM of the denominators 4 and 5 are:

4 = 2 x 2

5 = 5

LCM = 2 x 2 x5 = 20In other words, I have to convert the fourths and fifths into twentieths. I'll do this by multiplying by a useful form of 1 whole. In the case of the first fraction, 1/4, the 4 needs to become a 20, so I need to multiply the 4 by 5. To keep the fraction equal to its original value (equivalent fractions), I'll have to multiply the top by 5, too. In other words, I'll multiply the fraction by 5/5, which is just a useful form of the number 1:

1/4 x 5/5 = 5/20

Because I multiplied by (a useful form of) 1, I haven't changed the actual value of the fraction. All I've changed is how the value is stated.

In the case of the second fraction, 2/5, the 5 needs to become a 20, so I have to multiply the 5 by 4. To keep the fraction equal to the same value, I also have to multiply the top by 4, too. In other words, I'll multiply by 4/4, which is just a useful form of 1:

2/5 x 4/4 = 8/20The fourths and fifths are now both twentieths; I'm finally in an all-apples situation. Only now can I actually add the fractions. To add these "apples", I add the numerators:

1/4 + 2/5 = 5/20 + 8/20 = 5 + 8/20 = 13/20

Simplify: 1/4 + 2/5

Before I can add these fractions, I have to find their common denominator.

The lowest (smallest) common denominator is just the Least Common Multiple (LCM) of the two denominators, 4 and 5. The prime factorizations and LCM of the denominators 4 and 5 are:

4 = 2 x 2

5 = 5

LCM = 2 x 2 x5 = 20In other words, I have to convert the fourths and fifths into twentieths. I'll do this by multiplying by a useful form of 1 whole. In the case of the first fraction, 1/4, the 4 needs to become a 20, so I need to multiply the 4 by 5. To keep the fraction equal to its original value (equivalent fractions), I'll have to multiply the top by 5, too. In other words, I'll multiply the fraction by 5/5, which is just a useful form of the number 1:

1/4 x 5/5 = 5/20

Because I multiplied by (a useful form of) 1, I haven't changed the actual value of the fraction. All I've changed is how the value is stated.

In the case of the second fraction, 2/5, the 5 needs to become a 20, so I have to multiply the 5 by 4. To keep the fraction equal to the same value, I also have to multiply the top by 4, too. In other words, I'll multiply by 4/4, which is just a useful form of 1:

2/5 x 4/4 = 8/20The fourths and fifths are now both twentieths; I'm finally in an all-apples situation. Only now can I actually add the fractions. To add these "apples", I add the numerators:

1/4 + 2/5 = 5/20 + 8/20 = 5 + 8/20 = 13/20