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Find the sum:

Last updated date: 29th May 2024
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Hint: We solve this question by using the basic concepts of fractional addition. We are given the two fractions $\dfrac{5}{3}$ and $\dfrac{3}{5}$ , and we are required to add them. We do this by taking the LCM of the two denominators. Then we multiply individually the numerator and denominator by a factor to make the denominators of the two fractions equal to the LCM. Then we add both the numerators and represent it in the fraction form. We can also represent this in mixed fraction format if it is given in the improper format and in the decimal format.

Complete step-by-step solution:
In order to answer this question, let us consider the equation given above which involves the sum of the two fractions $\dfrac{5}{3}$ and $\dfrac{3}{5}$ .
$\Rightarrow \dfrac{5}{3}+\dfrac{3}{5}$
To add the above two fractions, we need to consider the denominators of the two fractions and find their LCM. We are required to find the LCM of least common multiple of 3 and 5 which is given as
$\Rightarrow LCM\left( 3,5 \right)=3\times 5=15$
Since the above two numbers are prime numbers, their LCM is simply given by their product.
We are required to convert the given denominators in the above expression to 15 and in order to do so, we multiply the numerator and denominator of the first term by 5 and multiply the numerator and denominator of the second term by 3. This can be given as,
$\Rightarrow \dfrac{5\times 5}{3\times 5}+\dfrac{3\times 3}{5\times 3}$
Taking a product of the respective terms we simplify it as,
$\Rightarrow \dfrac{25}{15}+\dfrac{9}{15}$
Since the denominator is the same now, we take the denominator common and add the two numerators.
$\Rightarrow \dfrac{25+9}{15}$
Adding the two numerators, we get an improper fraction as follows,
$\Rightarrow \dfrac{34}{15}$
This can be represented in a mixed fraction form by dividing the two numbers as,
$\Rightarrow 15\overset{2}{\overline{\left){\begin{align}
  & \text{ }34 \\
 & -30 \\
 & \overline{\text{ 4}} \\
We can represent this in improper fraction as
$\Rightarrow 2\dfrac{4}{15}$
We can also represent this in decimal format by further dividing the above number using decimal division as,
$\Rightarrow 15\overset{2.26}{\overline{\left){\begin{align}
  & \text{ }34 \\
 & -30 \\
 & \overline{\begin{align}
  & \text{ }40 \\
 & \text{ }-30 \\
 & \overline{\begin{align}
  & \text{ }100 \\
 & \text{ }-90 \\
 & \overline{\text{ 10}} \\
\end{align}} \\
\end{align}} \\
By using the above decimal division, we get a decimal number for the above fraction as $2.26.$
Hence, the sum of the two fractions is equal to $\dfrac{34}{15}$ or 2.26.

Note: We need to be careful while dividing using the decimal division and have to include a 0 at the end of the remainder before dividing with the divisor after including the decimal point. We also need to note that the above fraction on repeated division leads to the same reminder and the result is given as $2.\overline{66}.$ The bar above the terms after the decimal point suggest that they are recurring decimals.