Question

How do you evaluate $\dfrac{1}{3} + \dfrac{1}{2} + \dfrac{2}{3}$ ?

Answer 1

Hint: In this question, we are given an expression and we have to tell how we can evaluate it. Along with the answer, you have to write the steps also. Start with taking the LCM of the denominator. Then, make the denominator the same of all the terms by multiplying with the required number in denominator and numerator. Next step is to add all the terms and simplify them.

Complete step-by-step solution:
We are given an expression and we have been asked to evaluate it.
$ \Rightarrow \dfrac{1}{3} + \dfrac{1}{2} + \dfrac{2}{3}$ ……... (given)
Step 1: This step involves finding the common denominator. Since there is nothing in common between $2$ and $3$, we will find their LCM.
LCM of $2$ and $3$= $2 \times 3 = 6$
Step 2: Under this step, we have to make the denominator common. Since the LCM is $6$, we will make the denominator $6$.
$ \Rightarrow \dfrac{{1 \times 2}}{{3 \times 2}} + \dfrac{{1 \times 3}}{{2 \times 3}} + \dfrac{{2 \times 2}}{{3 \times 2}}$
On simplifying, we will get,
$ \Rightarrow \dfrac{2}{6} + \dfrac{3}{6} + \dfrac{4}{6}$
Step 3: Now, simply add all the terms.
$ \Rightarrow \dfrac{2}{6} + \dfrac{3}{6} + \dfrac{4}{6} = \dfrac{9}{6}$
Our answer is in improper fraction. We will convert it into a mixed fraction.
$ \Rightarrow \dfrac{9}{6} = \dfrac{3}{2} = 1\dfrac{1}{2}$

Hence, the value of $\dfrac{1}{3} + \dfrac{1}{2} + \dfrac{2}{3}$ is $1\dfrac{1}{2}$.

Note: How to convert an improper fraction into a mixed fraction?
1) Divide the numerator by the denominator.
For example: $\dfrac{9}{6}$$ \Rightarrow 9 = 6 \times 1 + 3$
2) Now, write down the quotient as the whole number.
Comparing $ \Rightarrow 9 = 6 \times 1 + 3$ with $D = D \times Q + R$
We get, $Q = 1$. This is our whole number.
3) This step involves writing the remainder as numerator and divisor as denominator (basically, the divisor remains the same.)
Again, we will compare $ \Rightarrow 9 = 6 \times 1 + 3$ with $D = D \times Q + R$.
We get, $R = 3$ and $D = 6$.
Our answer is –
$ \Rightarrow Q\dfrac{R}{D} = 1\dfrac{3}{6} = 1\dfrac{1}{2}$
Hence, this is how we convert an improper fraction into a mixed fraction.