Table of Contents

Last modified on August 28th, 2024

Adding fractions involves three basic steps:

- Identifying whether the denominators are the same. If they are the same, we move on to the next step. If they are different, we find their common denominators.
- Adding the numerators
- Simplifying the sum if needed.

To add fractions with like denominators, we just add the numerators and then write the sum over the common denominator.

Let us add the fractions ${\dfrac{2}{7}+\dfrac{4}{7}}$

**Identifying the Denominators**

Here, the denominators are 7

**Adding the Numerators**

${\dfrac{2+4}{7}}$

= ${\dfrac{6}{7}}$, which cannot be simplified or reduced further.

Thus, the sum is ${\dfrac{6}{7}}$

Let us add the unlike fractions ${\dfrac{2}{15}+\dfrac{1}{6}}$

**Identifying the Denominators**

Here, the denominators are 15 and 6, that are different.

**Finding the LCM of the Denominators **

The LCM of 15 and 6 is 30

**Making the** **Denominators the Same**

Converting ${\dfrac{2}{15}}$ and ${\dfrac{1}{6}}$ to their equivalent fractions with 30 as the denominator,

${\dfrac{2\times 2}{15\times 2}}$ = ${\dfrac{4}{30}}$

${\dfrac{1\times 5}{6\times 5}}$ = ${\dfrac{5}{30}}$

Now, we have ${\dfrac{4}{30}+\dfrac{5}{30}}$

**Adding the Numerators**

On adding the numerators,

${\dfrac{4+5}{30}}$

= ${\dfrac{9}{30}}$

**Simplifying**

${\dfrac{9\div 3}{30\div 3}}$

= ${\dfrac{3}{10}}$

Thus, the sum is ${\dfrac{3}{10}}$

**Add: ${\dfrac{7}{9}}$ and ${\dfrac{11}{3}}$**

Solution:

Here, the denominators are 9 and 3

The LCM of 9 and 3 is 9

Now, ${\dfrac{7}{9}+\dfrac{11}{3}}$

= ${\dfrac{7}{9}+\dfrac{11\times 3}{3\times 3}}$

= ${\dfrac{7}{9}+\dfrac{33}{9}}$

= ${\dfrac{7+33}{9}}$

= ${\dfrac{40}{9}}$

= ${4\dfrac{4}{9}}$

Thus, the sum is ${4\dfrac{4}{9}}$

**Problem: **Adding a **NEGATIVE FRACTION**

**Add: ${\dfrac{5}{12}+\left( \dfrac{-1}{6}\right)}$**

Solution:

Here, the denominators are 12 and 6

The LCM of 12 and 6 are 12

Now, ${\dfrac{5}{12}+\left( \dfrac{-1}{6}\right)}$

= ${\dfrac{5}{12}+\left( \dfrac{-1\times 2}{6\times 2}\right)}$

= ${\dfrac{5}{12}+\left( \dfrac{-2}{12}\right)}$

= ${\dfrac{5-2}{12}}$

= ${\dfrac{3}{12}}$

= ${\dfrac{1}{4}}$

Thus, the sum is ${\dfrac{1}{4}}$

Co-prime denominators are the denominators with no common factors between them other than 1.

Given, ${\dfrac{2}{7}+\dfrac{1}{5}}$

**Identifying the Denominators**

First, we identify the denominators and determine whether they are co-prime.

Here, the denominators are 7 and 5, that are co-prime.

**Multiplying the First Fraction By the Denominator of the Second Fraction and Vice-Versa**

${\dfrac{2\times 5}{7\times 5}+\dfrac{1\times 7}{5\times 7}}$

= ${\dfrac{10}{35}+\dfrac{7}{35}}$

**Adding the Numerators**

${\dfrac{10+7}{35}}$

= ${\dfrac{17}{35}}$

Thus, the sum is ${\dfrac{17}{35}}$

Let us add: ${5+\dfrac{3}{8}}$

**Converting the Whole Number to Fraction**

${5}$ = ${\dfrac{5}{1}}$

Now, we have ${\dfrac{5}{1}+\dfrac{3}{8}}$

**Identifying the Denominators**

The denominators are 1 and 8, which are different.

**Finding the LCM of the Denominators **

The LCM of 1 and 8 is 8

**Adding the Numerators**

${\dfrac{5\times 8}{1\times 8}+\dfrac{3\times 1}{8\times 1}}$

= ${\dfrac{40}{8}+\dfrac{3}{8}}$

= ${\dfrac{40+3}{8}}$

= ${\dfrac{43}{8}}$

= ${5\dfrac{3}{8}}$

However, a simple way to add a whole number to a proper fraction is to combine and express them as a mixed number.

For example, ${5+\dfrac{3}{8}}$ is written as ${5\dfrac{3}{8}}$

To add fractions with mixed numbers, we convert the mixed numbers into improper fractions and then add them.

If ${\dfrac{5}{8}+1\dfrac{3}{10}}$

**Converting to Improper Fraction**

Converting ${1\dfrac{3}{10}}$ to improper fraction,

${1\dfrac{3}{10}}$ = ${\dfrac{\left( 1\times 10\right) +3}{10}}$ = ${\dfrac{13}{10}}$

**Identifying the Denominators**

Here, the denominators are 8 and 10

**Finding the LCM of the Denominators **

The LCM of 8 and 10 is 40

**Making the** **Denominators the Same and Adding the Numerators**

${\dfrac{5\times 5}{8\times 5}+\dfrac{13\times 4}{10\times 4}}$

= ${\dfrac{25}{40}+\dfrac{52}{40}}$

= ${\dfrac{25+52}{40}}$

= ${\dfrac{77}{40}}$

= ${1\dfrac{37}{40}}$

Thus, the sum is ${1\dfrac{37}{40}}$

**Find the sum of ${2\dfrac{1}{4}+\dfrac{1}{5}}$**

Solution:

Here, the denominators are 4 and 5

The LCM of 4 and 5 is 20

Also, ${2\dfrac{1}{4}}$ = ${\dfrac{\left( 2\times 4\right) +1}{4}}$ = ${\dfrac{9}{4}}$

Now, ${2\dfrac{1}{4}+\dfrac{1}{5}}$

= ${\dfrac{9}{4}+\dfrac{1}{5}}$

= ${\dfrac{9\times 5}{4\times 5}+\dfrac{1\times 4}{5\times 4}}$

= ${\dfrac{45}{20}+\dfrac{4}{20}}$

= ${\dfrac{45+4}{20}}$

= ${\dfrac{49}{20}}$

= ${2\dfrac{9}{20}}$

Thus, the sum is ${2\dfrac{9}{20}}$

Now, let us add fractions, including variables.

If ${\dfrac{p}{2}+\dfrac{2p}{5}}$

**Identifying the Denominators**

The denominators are 2 and 5

**Finding the LCM of the Denominators **

The LCM of 2 and 5 is 10

**Making the** **Denominators the Same and Adding the Numerators**

${\dfrac{p\times 5}{2\times 5}+\dfrac{2p\times 2}{5\times 2}}$

= ${\dfrac{5p}{10}+\dfrac{4p}{10}}$

= ${\dfrac{5p+4p}{10}}$

= ${\dfrac{9p}{10}}$

Thus, the sum is ${\dfrac{9p}{10}}$

Last modified on August 28th, 2024