Sum of Fractions

πŸ†Practice addition of fractions

To add fractions, we must find the common denominator simplifying, expanding, or multiplying the denominators.
Then, you only need to add the numerators to get the result.

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Test yourself on addition of fractions!

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Solve the following exercise:

\( \frac{3}{9}+\frac{1}{9}=\text{?} \)

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Sum of Fractions

In this article, you will learn the easiest ways to add fractions, this will allow you to add all kinds of fractions without any inconvenience.
Shall we start?

The first step to solving a fraction addition is to find the common denominator.
Getting a common denominator means that we end up with two fractions with the same denominator. We will do this by simplifying, expanding, or multiplying the denominators.
After finding the common denominator, we will continue with the second step.
The second step to solving a fraction addition is to add the numerators.
We may encounter different cases of additions that we will study next:


First case:

One of the denominators that appear in the initial exercise will be the common denominator.

Sometimes we will encounter quite simple fractions where we won't need to expand or simplify both, but only one of the fractions.
Let's see an example:

32+14=\frac{3}{2}+\frac{1}{4}=

Upon observing these denominators, we will immediately realize that, if we multiply the denominator 22 by 22, we will reach the denominator 44.
This way, we will reach the common denominator and will be able to solve the exercise easily.

Observe -> When multiplying the denominator to transform it into the common denominator, we must also multiply the numerator by the same number so that the value of the fraction does not change.

We will do this by expanding by 22 and we will obtain:

64+14=\frac{6}{4}+\frac{1}{4}=

Now let's move to the second step and add the numerators.
Attention –> We do not add the denominators. Once we reach the common denominator, only the numerators are added, which, in fact, have the same denominator.



Let's see it in an exercise:

64+14=74\frac{6}{4}+\frac{1}{4}=\frac{7}{4}

We add 1+61+6 and leave the denominator only once.

If we wish, we can simplify the fraction and write it as follows: 1341\frac{3}{4}

Another exercise:

13+26=\frac{1}{3}+\frac{2}{6}=

Solution:
We will realize that, if we multiply 33 by 22 we will reach 66 and, this will be the common denominator.

We will obtain:

​​26+26=​​\frac{2}{6}+\frac{2}{6}=

Let's add the numerators and we will obtain:
46\frac{4}{6}
We can simplify and arrive at: 23\frac{2}{3}


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Second case:

The common denominator will be the product of the given denominators.

Sometimes we will encounter slightly more complicated exercises in which it will not be enough to expand a single fraction to obtain the common denominator, but rather, we must intervene in both fractions.
Do not worry, the way to act in such a case is simply to multiply the first fraction by the denominator of the second and multiply the second fraction by the denominator of the first.

See how simple this is:

Let's multiply the denominators:
We will multiply the 25\frac{2}{5} by 44 (the denominator of the second fraction) and the 14\frac{1}{4}Β by 55 (the denominator of the first fraction).
We will obtain:
820+520=\frac{8}{20}+\frac{5}{20}=

Let's add the numerators and we will arrive at the solution:
1320\frac{13}{20}
Did you see how simple it was? This is a technical method that does not require us to think about how to reach the common denominator.
Therefore, we recommend using it in all fraction addition exercises.


Third case:

Sum of 33 fractions

In case there are 33 fractions with different denominators in the exercise, we will first find a common denominator for 22 of them (the simplest ones), then we will find the common denominator between the one we obtained and the third given fraction.

Let's see an example and you'll see it's very simple:
210+23+45=\frac{2}{10}+\frac{2}{3}+\frac{4}{5}=
Let's look at the denominators and ask ourselves - Among the three denominators, which pair of them is easier to find a common denominator for?
The answer is 55 and 1010, since 1010 is the common denominator for both.
Therefore, we will multiply 45\frac{4}{5} by 22 and we will get:
210+23+810=\frac{2}{10}+\frac{2}{3}+\frac{8}{10}=
Now we can add the numerators that already have a common denominator to arrive at a clearer and more orderly exercise (this step is not mandatory, but it will help us later.

Now we just have to find the common denominator between 1010 -> the new denominator we found, and 33 the third denominator of the exercise.
We will do this with the method of multiplying denominators and we will get:
3030+2030=\frac{30}{30}+\frac{20}{30}=
Let's add the numerators and we will get:
5030\frac{50}{30}
We can simplify and arrive at:
53=123\frac{5}{3}=1\frac{2}{3}


Examples and exercises with solutions for adding fractions

Exercise #1

48+410= \frac{4}{8}+\frac{4}{10}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common multiple between 8 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 8 and 10

In this case, the lowest common multiple is 40

Now, let's multiply each number in the appropriate multiples to reach the number 40

We will multiply the first number by 5

We will multiply the second number by 4

4Γ—58Γ—5+4Γ—410Γ—4=2040+1640 \frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}

Now let's calculate:

20+1640=3640 \frac{20+16}{40}=\frac{36}{40}

Answer

3640 \frac{36}{40}

Exercise #2

Solve the following exercise:

39+19=? \frac{3}{9}+\frac{1}{9}=\text{?}

Video Solution

Answer

49 \frac{4}{9}

Exercise #3

Solve the following exercise:

15+35=? \frac{1}{5}+\frac{3}{5}=\text{?}

Video Solution

Answer

45 \frac{4}{5}

Exercise #4

Solve the following exercise:

13+13=? \frac{1}{3}+\frac{1}{3}=\text{?}

Video Solution

Answer

23 \frac{2}{3}

Exercise #5

Solve the following exercise:

25+35=? \frac{2}{5}+\frac{3}{5}=\text{?}

Video Solution

Answer

1

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