## Brief Overview of Fraction

Fraction is part of a whole. It could be anything such as half of a cake or a quarter of the pizza. In mathematical language, fraction is a decimal number that can be written in a/b form, where a is called ‘numerator’ and b is called ‘denominator’. Hence, we can write half of cake as ‘1/2^{nd} cake’ where 1 is numerator and 2 is denominator. Similarly, we will write quarter of pizza as ‘1/4^{th} pizza’ where again 1 is numerator and 4 is denominator.

Fractions can be of two types mainly. Proper fraction and Improper Fractions. The former is a kind of fraction when the numerator is less than the denominator (a < b), whereas in the latter, the numerator is greater than denominator (a > b).

There is another type of fraction called mixed fraction which is a combination of whole number and proper fraction. For example, $3\frac{1}{4},4\frac{1}{5},9\frac{3}{8}$ etc. In these examples 3,4, and 9 are whole numbers whereas 1/4, 1/5, 3/8, are proper fractions. Mixed fraction can be also solved into improper fractions and similarly improper fractions can be solved into mixed fractions.

Suppose we have the mixed fraction $2\frac{1}{3}$. We can convert it into improper fraction in three steps. Firstly, multiply the denominator 3 with the whole part 2 i.e., 3 × 2 = 6. Next, we add the numerator 1 with the answer of first step i.e., 1 + 6 = 7. Finally, we write it in a/b form where a is the sum obtained in second step and b is the denominator ‘3’. The result is 7/3 an improper fraction.

We can write the improper fraction into mixed fraction by dividing the numerator with the denominator. The remainder is obtained, and the mixed fraction is written as quotient followed by proper fraction of remainder/devisor.

## Adding Fractions

### Definition

Adding fractions means to add one fraction with the other. It is very normal to add two numbers such as a + b or b + a. However, in this section, we will be adding two fractions. Let say a / b and c / d are two fractions, then to add the two fractions we can write as (a / b) + (b / c). Now how sum these fractions? We will learn it in the coming sections.

### Method / Steps

Addition of fractions can be carried out using the following basic steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions. For example, (a / b) + (c / d) where a / b is first fraction or addend, and c / d is second fraction or addend.

**Step 2:** Inspect the denominators of both numbers and decide that the fractions to be added are like fractions or unlike fractions.

**Step 3:** If the denominators of the fractions to be added are not the same, this is called unlike fractions addition, we will take least common multiple (LCM) of the denominators. For example, we have 1/2 and 1/5 as unlike fractions to be added. Now to take LCM of 2 and 5 we must find the smallest number that is divisible by both. Thus 2 × 5 = 10 which is divisible by both 2 and 5. We need to make 10 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 5, and the second fraction by 2 such that the two fractions become (1 × 5) / (2 × 5) = 5/10 and (1 × 2) / (5 × 2) = 2/10. Now we can add the numerators since it is like fraction and write it over the denominator 10 as (5 + 2) / 10 = 7/10.

**Step 4: **If the denominator of both or all the fractions are the same, this is called like fractions additions, then simply add the numerators of all the fractions and write the answer of it over the (same) denominator. For example, a_{1}/b + a_{2}/b + a_{3}/b +a_{4}/b + . . . + a_{n}/b. Then we can write (a_{1 }+ a_{2 }+ a_{3 }+ . . . + a_{n}) / b

**Step 5: **Simplify or solve the fraction if required.

### Significance of Adding Fractions

Just like the numbers, we can now add the fractions as well but let’s take a minute to think what does it mean to add one fraction with the other?

We can explain it by giving an example of pizza. For example, one person eats quarter of pizza whereas the other had half. Now, if we want to know what fraction of pizza is left we must add the amount of pizza eaten by the two persons. We add 1/4 and 1/2 which gives 3/4. The leftover pizza is 1 – 3/4 = 1/4. Hence, we can say that quarter of pizza is left. This was just one practical example of showing significance of adding fractions.

### Techniques and Tricks

Addition of fractions can be carried out using the five steps mentioned above. However, there is also another technique by which we can add fractions. This is particularly for unlike fractions (having different denominator) since there is no need of trick for adding like fractions.

- Multiply the denominator of fraction 2 with the numerator of fraction 1.
- Then multiply the denominator of fraction 1 with the numerator of fraction 2.
- Add the result of both multiplication and write it in the numerator of the sum.
- Multiply both the denominators i.e., den1 × den2 and write it as the denominator of the sum.

## Adding Proper Fractions with Proper Fractions

### Example 1

Add the fractions 3/4 and 4/5.

#### Solution:

To add 3/4 and 4/5, we will use the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (3 / 4) + (4 / 5) where 3 / 4 is first fraction or addend, and 4 / 5 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 3/4 and 4/5 as unlike fractions to be added. Now to take LCM of 4 and 5 we must find the smallest number that is divisible by both. Thus 4 × 5 = 20 which is divisible by both 4 and 5. We need to make 20 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 5, and the second fraction by 4 such that the two fractions become (3 × 5) / (4 × 5) = 15/20 and (4 × 4) / (5 × 4) = 16/20.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 20 as (15 + 16) / 20 = 31/20.

**Step 5: **The fraction 31/20 which is an improper fraction is already in its simplified form. Hence, we can write.

(3 / 4) + (4 / 5) = 31 / 20

### Example 2

Add the fractions 21/42 and 53/68.

#### Solution:

We will add 21/42 and 53/68 using the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (21 / 42) + (53 / 68) where 21 / 42 is first fraction or addend, and 53 / 68 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 21/42 and 53/68 as unlike fractions to be added. Now to take LCM of 42 and 68 we must find the smallest number that is divisible by both. Thus 2 × 3 × 2 × 7 × 17 = 1428 which is divisible by both 42 and 68. We need to make 1428 in the denominator of both fractions to make them like fractions. So, we will multiply and divide the first fraction by 34, and the second fraction by 21 such that the two fractions become (21 × 34) / (42 × 34) = 714/1428 and (53 × 21) / (68 × 21) = 1113/1428.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 1428 as (714 + 1113) / 1428 = 1827/1428.

**Step 5: **The fraction 1827/1428 is an improper fraction which can be simplified further. Divide both the numerator and denominator by 21 such that 21 × **87** = 1827 and 21 × **68** = 1428. The simplified sum is 87/68.

(21 / 42) + (53 / 68) = 87 / 68

## Adding Proper Fractions with Improper Fractions

### Example 1

Add the fractions 10/20 and 25/5.

#### Solution:

We will add the proper fraction 10/20 and improper fraction 25/5 using the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (10 / 20) + (25 / 5) where 10 / 20 is first fraction or addend, and 25 / 5 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 10/20 and 25/5 as unlike fractions to be added. Now to take LCM of 20 and 5 we must find the smallest number that is divisible by both. Thus 5 × 4 = 20 which is divisible by both 20 and 5. We need to make 20 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 1, and the second fraction by 4 such that the two fractions become (10 × 1) / (20 × 1) = 10/20 and (25 × 4) / (5 × 4) = 100/20.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 20 as (10 + 100) / 20 = 110/20.

**Step 5: **The fraction 110/20 is an improper fraction which can be simplified further. Divide both the numerator and denominator by 10 such that 10 × **11** = 110 and 10 × **2** = 20. The simplified sum is 11/2.

(10 / 20) + (25 / 5) = 11 / 2

### Example 2

Add the fractions 19/20 and 29/10.

#### Solution:

To add the proper fraction 19/20 and improper fraction 29/10, we use the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (19 / 20) + (29 / 10) where 19 / 20 is first fraction or addend, and 29 / 10 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 19/20 and 29/10 as unlike fractions to be added. Now to take LCM of 20 and 10 we must find the smallest number that is divisible by both. Thus 10 × 2 = 20 which is divisible by both 20 and 10. We need to make 20 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 1, and the second fraction by 2 such that the two fractions become (19 × 1) / (20 × 1) = 19/20 and (29 × 2) / (10 × 2) = 58/20.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 20 as (19 + 58) / 20 = 77/20.

**Step 5: **The fraction 77/20 is an improper fraction which is already in the simplified form. Hence, we can write.

(19 / 20) + (29 / 10) = 77 / 20

## Adding Improper Fractions with Improper Fractions

### Example 1

Add the fractions 12/6 and 18/3.

#### Solution:

We will add the two improper fractions 12/6 and 18/3 using the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (12 / 6) + (18 / 3) where 12 / 6 is first fraction or addend, and 18 / 3 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 12/6 and 18/3 as unlike fractions to be added. Now to take LCM of 6 and 3 we must find the smallest number that is divisible by both. Thus 3 × 2 = 6 which is divisible by both 6 and 3. We need to make 6 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 1, and the second fraction by 2 such that the two fractions become (12 × 1) / (6 × 1) = 12/6 and (18 × 2) / (3 × 2) = 36/6.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 6 as (12 + 36) / 6 = 48/6.

**Step 5: **The fraction 48/6 is an improper fraction which can be simplified further. In this case, 48 is dividend and 6 is the divisor and if we recall the 6 – times table i.e., 6 × **8 **= 48. Hence, 8 is the simplified sum.

(12 / 6) + (18 / 3) = 8

### Example 2

Add the improper fractions that are 21/3 and 110/11.

#### Solution:

To add the two improper fractions 21/3 and 110/11 we use the following steps.

**Step 1: **Write the two fractions as the two addends and put the plus ( + ) symbol in between the two fractions such that (21 / 3) + (110 / 11) where 21 / 3 is first fraction or addend, and 110 / 11 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 21/3 and 110/11 as unlike fractions to be added. Now to take LCM of 3 and 11 we must find the smallest number that is divisible by both. Thus 3 × 11 = 33 which is divisible by both 3 and 11. We need to make 33 in the denominator of both fractions to make it like fraction. So, we will multiply and divide the first fraction by 11, and the second fraction by 3 such that the two fractions become (21 × 11) / (3 × 11) = 231/33 and (110 × 3) / (11 × 3) = 330/33.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 33 as (231 + 330) / 33 = 561/33.

**Step 5: **The fraction 561/33 is an improper fraction which can be simplified further. In this case, 561 is dividend and 33 is the divisor and if we use the 33 – times table i.e., 33 × **17 **= 561. Hence, 17 is the simplified sum.

(21 / 3) + (110 / 11) = 17

## Adding Mixed Fractions

### Example 1

Add the mixed fraction 2(4/7) and the proper fraction 42/60.

#### Solution:

To add the two fractions 2(4/7) and 42/60 we use the following steps. Since there is a mixed fraction as well. We will first get rid of the mixed fraction in an additional step i.e., Step 0.

**Step 0: **There is one mixed fraction which is 2(4/7). We will convert it into improper fraction by multiplying the denominator 7 with whole 2 and adding 4 to it i.e., (7 × 2) + 4. The result is 18 which becomes the new numerator. Hence, mixed fraction can be written as 18/7

**Step 1: **Next we will do the same procedure as we are doing in the rest of the examples. We will write the two addends and put the plus ( + ) symbol in between the two fractions such that (18 / 7) + (42 / 60) where 18 / 7 is first fraction or addend, and 42 / 60 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 18/7 and 42/60 as unlike fractions to be added. Now to take LCM of 7 and 60 we must find the smallest number that is divisible by both. Thus 7 × 10 × 6 = 420 which is divisible by both 7 and 60. We need to make 420 in the denominator of both fractions to make them like fractions. So, we will multiply and divide the first fraction by 60, and the second fraction by 7 such that the two fractions become (18 × 60) / (7 × 60) = 1080/420 and (42 × 7) / (60 × 7) = 294/420.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 420 as (1080 + 294) / 420 = 1374/420.

**Step 5: **The fraction 1374/420 is an improper fraction which can be simplified further. We divide both the numerator and denominator by 6. This is because both belongs to the 6 – times table i.e., 6 × **229 **= 1374 and 6 × **70 **= 420. Thus the simplified sum is 229/70

2(4 / 7) + (42 / 60) = 229/70

### Example 2

Add the two mixed fractions that are 6(8/2) and 9(10/5)

#### Solution:

To add the two mixed fractions 6(8/2) and 9(10/5) we use the following steps. Since we are dealing with mixed fractions. We will first get rid of these mixed fractions in an additional step i.e., Step 0.

**Step 0: **The first mixed fraction which is 6(8/2). It can be converted it into improper fraction by multiplying the denominator 2 with whole 6 and adding 8 to it i.e., (2 × 6) + 8. The result is 20 which becomes the new numerator. Hence, mixed fraction can be written as 20/2. Similarly, the other mixed fraction can be written as 55/5.

**Step 1: **Next we will do the same procedure as we are doing in the rest of the examples. This is just like adding two improper fractions. We will write the two addends and put the plus ( + ) symbol in between them such that (20 / 2) + (55 / 5) where 20 / 2 is first fraction or addend, and 55 / 5 is second fraction or addend.

**Step 2:** By inspecting the denominators, we can see that the fractions to be added are unlike fractions.

**Step 3:** Since this is an unlike fractions addition, we will take least common multiple (LCM) of the denominators. We have 20/2 and 55/5 as unlike fractions to be added. Now to take LCM of 2 and 5 we must find the smallest number that is divisible by both. Thus 2 × 5 = 10 which is divisible by both 2 and 5. We need to make 10 in the denominator of both fractions to make them like fractions. So, we will multiply and divide the first fraction by 5, and the second fraction by 2 such that the two fractions become (20 × 5) / (2 × 5) = 100/10 and (55 × 2) / (5 × 2) = 110/10.

**Step 4: **We have now two like fractions and we can add the numerators and write it over the denominator 10 as (100 + 110) / 10 = 210/10.

**Step 5: **The fraction 210/10 is an improper fraction which can be simplified further. Here, 110 is the dividend and 10 is the divisor. We can use the 10 – times table i.e., 10 × **21 **= 210. Hence, the reduced sum of the two given mixed fractions is 21.

6(8/2) + 9(10/5) = 21

# References

[1] | “Adding Fractions,” MathsIsFun.com, 2021. [Online]. Available: https://www.mathsisfun.com/fractions_addition.html. [Accessed 30 9 2021]. |

[2] | “Adding Fractions with Unlike Denominators,” Varsity Tutors, 2007. [Online]. Available: https://www.varsitytutors.com/hotmath/hotmath_help/topics/adding-fractions-unlike-denominators. [Accessed 1 10 2021]. |

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