Question

Find the sum of the fractions $\dfrac{4}{15}\text{ and }\dfrac{17}{20}$ ?

Answer 1

Hint: We know that to add two fractions, the denominators of both the fractions must be the same. If the denominators are different, we need to convert these denominators into their Lowest Common Multiples (LCM), and then add the corresponding fractions, to get our result.

Complete step by step solution:
Let us assume a variable $x$ such that
$x=\dfrac{4}{15}+\dfrac{17}{20}...\left( i \right)$
and we need to find this variable $x$ to get the result.
Here, we can see that the denominators of both the fractions are different. So, we can not add them directly. We need to convert these fractions such that they have the same denominators.
For this, we need to find the Lowest Common Multiple (LCM) of the two denominators, 15 and 20.
Let us use the method of continued division to find the LCM.
$\begin{align}
  & 2\left| \!{\underline {\,
  15,20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  15,10 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15,5\text{ } \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,5\text{ } \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1\text{ } \,}} \right. \\
\end{align}$
$\therefore LCM=2\times 2\times 3\times 5$
$\Rightarrow LCM=60$
The LCM of 15 and 20 is 60. So, we need to convert the denominators 15 and 20 into 60.
For the fraction $\dfrac{4}{15}$ , we need to convert the denominator 15 into 60.
To achieve this, we have to multiply 15 by 4. But, the value of the fraction must not change. So, we have to multiply 4 in both numerator and denominator.
$\dfrac{4}{15}=\dfrac{4\times 4}{15\times 4}$
$\Rightarrow \dfrac{4}{15}=\dfrac{16}{60}...\left( ii \right)$
For the fraction $\dfrac{17}{20}$ , we need to convert the denominator 20 into 60.
To achieve this, we have to multiply 20 by 3. But, the value of the fraction must not change. So, we have to multiply 3 in both numerator and denominator.
$\dfrac{17}{20}=\dfrac{17\times 3}{20\times 3}$
$\Rightarrow \dfrac{17}{20}=\dfrac{51}{60}...\left( iii \right)$
Putting the values from equation (ii) and equation (iii) in equation (i), we get
$x=\dfrac{16}{60}+\dfrac{51}{60}$
$\Rightarrow x=\dfrac{67}{60}$
Here, this fraction is in its simplest form.
Hence, the sum of $\dfrac{4}{15}\text{ and }\dfrac{17}{20}\text{ is }\dfrac{67}{60}$ .

Note: We must keep in mind that some texts use the term Lowest Common Denominator (LCD), which is nothing but the LCM of denominators. For example, in our question, 60 is the LCD of $\dfrac{4}{15}\text{ and }\dfrac{17}{20}$ . In such problems, we must not forget to simplify the result in its simplest form.