How to Add Fractions

When you hear sentences like "Only 1/4th of the cylinder is filled with gas" or "1/3rd of the cake was eaten up by Jane", what do you comprehend from them? It implies that if the cylinder were divided into 4 parts, only 1 part of the cylinder is full and 3 parts are empty. Similarly, if cake were divided into 3 parts, 1 part was eaten by Jane leaving 2/3rd of the cake behind. Any entity, as per mathematics is divided into a specific number of parts of which, a few parts are used to indicate a section of the whole entity. Such measuring techniques are known as fractions. Let me state an example here - There are 10 fruits in a basket, of which 4 are oranges and 5 are apples. So what fraction of fruits in the basket are apples? 5/10, right? But remember, whenever you deal with fractions, make sure you simplify them. 5/10 can be further simplified to 1/2, as the denominator and the numerator have a common number by which both are divisible.

Adding Fractions

Every fraction has a numerator and a denominator. Numerator signifies the part of the whole entity whereas denominator indicates the whole total. So when you are given two fractions to add, there are three possibilities. They can be fractions with different numerators and same denominators, same numerators and different denominators or different numerators and different denominators. Not to miss on a very important point, remember that only if the numerator is equal to or less than the denominator, it is called a proper fraction, else the fraction is known as an improper fraction. So let's scour through all the three possibilities!

With Same Denominators
When you have two fractions with same denominators, check whether they are proper fractions first. If they are, add the numerators and the result will be a fraction too. If the fractions to be added are improper fractions, then you may have to simplify the fraction.

Example
2/9 + 3/9 = 5/9 (Proper fraction)
10/9 + 20/9 = 30/9 ~ 10/3 (Improper fraction) ~ 3⅓ (Mixed number)

With Different Denominators
When there are different denominators for the fractions, then first figure out a least common denominator (LCD), which is divisible by both the denominators.

Example 1: 4/9 + 4/8.
How do you find an LCD for these denominators? Very simple! Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81,... and multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72,.... Here the least common divisor for both is 72. Multiply the respective numerators with the multiplicands to get the appropriate fractions. For the first fraction 4/9, the multiplicand is 8 which needs to be multiplied by both numerators and denominators. For the second fraction 4/8, the multiplicand is 9. So 4/9 will be written as (4 x 8)/(9 x 8) and 4/8 will be written as (4 x 9)/(8 x 9).

4/9 + 4/8 = 32/72 + 36/72 = 68/72 ~ 17/13 (Improper fraction) ~ 1 4/13 (Mixed number)

Example 2: 1/3 + 1/6.
In these fractions, 3 and 6 have a common divisor 6. Also 6 happens to be the least common denominator.

1/3 + 1/6 = (1 x 2)/(3 x 2) + (1 x 1)/(6 x 1) = 2/6 + 1/6 = 3/6 ~ 1/2

With Different Denominators and Numerators
Now when you are given a pair of fractions with different numerators and different denominators, adding is comparatively complex than the above cases. But the rule of LCD remains the same. Let's learn with an example!

Example: 3/5 + 5/6.
The common LCD for these denominators is 30. In cases where denominators are such that either of them is not divisible by the other, then multiply the denominators and you get the LCD which will serve as the common denominator. This method is also known as the cross multiplication method. In this case it is 5 x 6 = 30.

3/5 + 5/6 = (3 x 6)/(5 x 6) + (5 x 5)/(6 x 5) = 18/30 + 25/30 = 43/30 (Improper fraction) ~ 1 13/30 (Mixed number)

So with the above stated examples and explanation, hope you have understood how to add fractions. The thumb rule one must remember when adding fractions with different denominators is that the denominators must be same. So students, your homework will no longer be boring as adding fractions will get over in a jiffy for you!