Question

How do you simplify $\dfrac{1}{4} + \dfrac{1}{5}$?

Answer 1

Hint: Here, we are given two unlike fractions. To add them, we will find the L.C.M of the denominators. Then, we will multiply the numerators of the unlike fractions by the factors that we multiply the denominators with. Finally, we will add the numerators and reduce them to the simplest form if necessary.

Complete step by step solution:
We are required to add the fractions $\dfrac{1}{4}$ and $\dfrac{1}{5}$.
We observe that the denominators of these fractions are not the same. Such fractions are called fractions.
To add unlike fractions, we have to make the denominators of both the fractions one and the same.
For this, we will first find the L.C.M of the denominators i.e., we will find the L.C.M of 4 and 5.
We see that both 4 and 5 do not have any common factors. In this case, the L.C.M of 4 and 5 will be their product.
Hence, L.C.M $(4,5) = 4 \times 5 = 20$
Now that we have the L.C.M, we must convert the denominators of the fractions, each to 20.
The denominator of the first fraction is 4. To make it 20, we must multiply both numerator and numerator by 5. Thus, we get the first fraction as
$\dfrac{1}{4} = \dfrac{{1 \times 5}}{{4 \times 5}} = \dfrac{5}{{20}}$
In the second fraction, the denominator is 5. To make it 20, we must multiply both numerator and numerator by 4. Thus, we get the second fraction as
$\dfrac{1}{5} = \dfrac{{1 \times 4}}{{5 \times 4}} = \dfrac{4}{{20}}$
Now that the denominators are equal, i.e., the fractions are like fractions, so we can perform the addition.
$\dfrac{1}{4} + \dfrac{1}{5} = \dfrac{5}{{20}} + \dfrac{4}{{20}}$
When we add the fractions, we will add only the numerators, since the denominators are equal. The numerators are 5 and 4.
$ \Rightarrow \dfrac{1}{4} + \dfrac{1}{5} = \dfrac{{5 + 4}}{{20}} = \dfrac{9}{{20}}$

Therefore, the sum of $\dfrac{1}{4}$ and $\dfrac{1}{5}$ is $\dfrac{9}{{20}}$.

Since 9 and 20 do not have any common factor, we cannot reduce it further and so, it is in the simplest form.

Note:
The denominators of the given unlike fractions do not have any common factors. So, to add them, we can make the denominators equal by cross multiplication. The numerator of the first fraction is multiplied by 5, which is the denominator of the second fraction. The numerator of the second fraction is multiplied by 4, which is the denominator of the first fraction. Hence,
$\dfrac{1}{4} + \dfrac{1}{5} = \dfrac{{(1 \times 5) + (1 \times 4)}}{{20}} = \dfrac{9}{{20}}$.