Question

Add the following rational numbers: $\dfrac{-5}{16}$ and $\dfrac{7}{24}$.

Answer 1

Hint: We need to find the sum of two rational numbers. Firstly, we find the least common factor of the denominators of two numbers. Then, we express the given numbers with the same denominator and simplify to get the result.

Complete step by step solution:
We are given two rational numbers and need to find their sum. We will be solving the given question by expressing the two rational numbers with their LCM as the common denominator.
Fractions, in mathematics, are used to represent the portion or the part of the entire thing. They are generally represented as,
$\Rightarrow \dfrac{a}{b}$
Here,
$a$ is the numerator of the fraction
$b$ is the denominator of the fraction
For Example:
$\Rightarrow \dfrac{1}{2},\dfrac{2}{5}$
A rational number is a number that can be expressed in the form of fractions. It is usually represented as follows,
$\Rightarrow \dfrac{p}{q}$ where $q\ne 0$
The given rational numbers are $\dfrac{-5}{16}$ and $\dfrac{7}{24}$
We need to find the LCM of the denominators $16\;$ and $24\;$
Finding the prime factors of the numbers, we get,
Prime Factorization of $16\;$ : $2\times 2\times 2\times 2$
Prime Factorization of $24\;$ : $2\times 2\times 2\times 3$
Both the numbers have a number $8$ in common. So, we have to remove $8$ from one of the numbers.
Removing $\left( 2\times 2\times 2 \right)$ or $8$ from the number $24\;$ , we are left with $2\times 2\times 2\times 2\times 3$
Multiplying the remaining numbers, we get,
$\Rightarrow 2\times 2\times 2\times 2\times 3=48$
$\therefore LCM=48$
Now, we have to express the given numbers with this LCM as their common denominator.
$\Rightarrow \dfrac{-5\times 3}{16\times 3}=\dfrac{-15}{48}$
$\Rightarrow \dfrac{7\times 2}{24\times 2}=\dfrac{14}{48}$
Adding the given numbers,
$\Rightarrow \dfrac{-5}{16}+\dfrac{7}{24}$
Substituting the above values, we get,
$\Rightarrow \dfrac{-15}{48}+\dfrac{14}{48}$
$\Rightarrow \dfrac{-15+14}{48}$
$\Rightarrow \dfrac{-1}{48}$
$\therefore$ The result of the addition is $\dfrac{-1}{48}$

Note: The addition and subtraction of fractions are only carried out if they have the same denominator. We can equalize the denominators of the fractions of unlike fractions as follows,
$\Rightarrow \dfrac{a}{b}+\dfrac{c}{d}=\dfrac{\left( a\times d \right)+\left( b\times c \right)}{\left( b\times d \right)}$ .