Question

How do you simplify \[\dfrac{1}{3} + \dfrac{5}{{12}}\]?

Answer 1

Hint: Here, we will first make the denominator of all the fractions the same. To make the denominator of the fractions the same, we will find the LCM of the numbers present in the denominator of the fractions. Then we will multiply the numerator and denominator of the fraction with the number such that all the denominators will become equal. Then we will find the sum by adding the numerator of the fractions keeping the denominators the same to find the required value.

Complete step by step solution:
Here we need to find the sum of these two given fractions i.e. \[\dfrac{1}{3} + \dfrac{5}{{12}}\]
For that, we will first make the denominator of all the fractions the same. To make the denominator of the fractions the same, we will find the LCM of the numbers present in the denominator of the fractions.
Therefore, we will find the LCM of 3 and 12.
To find the LCM, we will find the factors of these numbers.
\[\begin{array}{l}3 = 1 \times 3\\12 = 1 \times 2 \times 2 \times 3\end{array}\]
We can see that the LCM of 3 and 12 \[ = 2 \times 2 \times 3 = 12\].
We have to make the denominator of every fraction equal to 12.
So we will multiply the numerator and denominator of the first fraction, \[\dfrac{1}{3}\], by 4.
 \[\dfrac{{1 \times 4}}{{3 \times 4}} = \dfrac{4}{{12}}\]
Now, we will multiply the numerator and denominator of the second fraction, \[\dfrac{5}{{12}}\], by 1.
\[\dfrac{{5 \times 1}}{{12 \times 1}} = \dfrac{5}{{12}}\]
Now, we will substitute these two fractions in place of the given fractions.
\[ \Rightarrow \dfrac{1}{3} + \dfrac{5}{{12}} = \dfrac{4}{{12}} + \dfrac{5}{{12}}\]
Now we will find the sum by adding the numerator of the fractions keeping the denominators the same.
\[ \Rightarrow \dfrac{1}{3} + \dfrac{5}{{12}} = \dfrac{{4 + 5}}{{12}}\]
On adding the numerators, we get
\[ \Rightarrow \dfrac{1}{3} + \dfrac{5}{{12}} = \dfrac{9}{{12}}\]
We can observe that there is a common factor between the numerator and denominator of the above fraction.
So, dividing both numerator and denominator by 3, we get
\[ \Rightarrow \dfrac{1}{3} + \dfrac{5}{{12}} = \dfrac{3}{4}\]

Hence, the value of the sum of the fractions is equal to \[\dfrac{3}{4}\].

Note:
Here we have obtained the sum of the fractions given in the question. We need to keep in mind that the fractions which have different denominators, cannot be added directly. We have to first find the LCM and then we have to make the denominators the same and then only we add the fractions. If the fraction is a mixed fraction then we have to first convert it into an improper fraction, then make the denominator the same (if it is not the same) to perform the addition operation.