Question

How do you simplify \[1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}\]?

Answer 1

Hint: In this problem we have to simply the given expression \[1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}\] to its reduced form. We can first take the similar terms where the denominator is the same and add it. Then we can take the other terms, cross multiply it to get the simplified form and finally add every term and convert the fraction form to get the simplified form.

Complete step by step answer:
We know that the given expression to be simplified is,
\[1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}\]…… (1)
Here, we can take the first two terms \[1+\dfrac{1}{2}\] from the above expression and cross multiply to get,
\[\begin{align}
  & \Rightarrow 1+\dfrac{1}{2}=\dfrac{2+1}{2} \\
 & \Rightarrow \dfrac{3}{2} \\
\end{align}\]
Now we can add this result in the expression (1), instead of first two terms.
\[\Rightarrow \dfrac{3}{2}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}\] ….. (2)
Now, we can take the first two terms \[\dfrac{3}{2}+\dfrac{1}{3}\] in the above expression (2) and cross multiply it, we get,
\[\begin{align}
  & \Rightarrow \dfrac{3}{2}+\dfrac{1}{3}=\dfrac{9+1}{6} \\
 & \Rightarrow \dfrac{10}{6} \\
\end{align}\]
Now we can add the above result in the expression (2) instead of first two terms in the expression (2), we get
\[\Rightarrow \dfrac{10}{6}+\dfrac{1}{6}+\dfrac{1}{6}\]
Here, we have same denominator and we can add the numerator alone to get the answer,
\[\begin{align}
  & \Rightarrow \dfrac{10+1+1}{6} \\
 & \Rightarrow \dfrac{12}{6} \\
\end{align}\]
We can cancel the above fraction using multiplication, we get
\[\Rightarrow 2\]

Therefore, the simplified form of \[1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}\] is 2.

Note: Students make mistakes while multiplying in cross multiplication, which should be concentrated. We should know that if we have the same denominator, we can add or subtract the numerator keeping the common denominator. We should know the multiplication tables to solve these types of problems.