Question

What is the least common denominator for adding the fractions $\dfrac{4}{15},\dfrac{1}{12}$ and $\dfrac{3}{8}$ ?

Answer 1

Hint: For solving this question you should know about the least common denominator or lowest common denominator of a fraction. The least common denominator is the value of denominator which can be divided by all the denominators of the fractions and this denominator is also known as lowest common denominator because we take the smallest value in it as a denominator.

Complete step-by-step solution:
According to our question we have to calculate the least common denominator to make sure the addition of $\dfrac{4}{15},\dfrac{1}{12}$ and $\dfrac{3}{8}$ fractions.
Let us consider $\dfrac{4}{15}$. If we multiply the numerator and denominator by the same number then the fraction remains the same. Similarly, if we multiply numerator and denominator in the fraction then too it will be the same number as before. If we notice then it is clear that we can add, subtract or compare these three fractions, only if their denominators are the same. Hence only the purpose of finding the least common denominator or lowest common denominator is to identify a common denominator, to which all the denominators can be raised.
We can increase 15 to : 30, 45, 60, 75, 90, 105, 120, 135… By multiplying it by 2, 3, 4, 5… and so on.
Similarly, 12 can be increased to 24, 36, 48, 60, 72, 84, 96, 108, 120, 132…
And 8 can be increased to 16, 24, 32…112, 120, 128…
In all these the common minimum term is 120. So, the addition of these fractions will be as:
$\begin{align}
  & \dfrac{4}{15}=\dfrac{4}{15}\times \dfrac{8}{8}=\dfrac{32}{120} \\
 & \dfrac{1}{12}=\dfrac{1}{12}\times \dfrac{10}{10}=\dfrac{10}{120} \\
 & \dfrac{3}{8}=\dfrac{3}{8}\times \dfrac{15}{15}=\dfrac{45}{120} \\
\end{align}$
So, we can write the addition of $\dfrac{4}{15}+\dfrac{1}{12}+\dfrac{3}{8}$ as addition of $\dfrac{32}{120}+\dfrac{10}{120}+\dfrac{45}{120}$. So,
$\begin{align}
  & \Rightarrow \dfrac{4}{15}+\dfrac{1}{12}+\dfrac{3}{8}=\dfrac{32}{120}+\dfrac{10}{120}+\dfrac{45}{120} \\
 & \Rightarrow \dfrac{4}{15}+\dfrac{1}{12}+\dfrac{3}{8}=\dfrac{87}{120} \\
\end{align}$
So, the least common denominator is 120.

Note: During calculating the least common denominator and lowest common denominator it is important to make the denominator as a number which is divided by all denominators of the fraction. And it will be the lowest number for that operation which will be divided by all denominators. Also, the possible mistake one can do is misreading any fractional term and then ending up with the wrong answer. So, we have to be careful.