Question

Find the value of the sum ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}$.

Answer 1

Hint: Here, we will be proceeding by finding out the LCM of the numbers (3,4 and 6) which are present in the denominator of the fractions whose sum we need to find.

Complete step-by-step answer:
Let x be the value of the sum ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}$
i.e., $x={\displaystyle \frac{1}{3}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}$

In order to solve the above sum of different fractions, we will take the LCM. Since, the LCM of 3,4 and 6 can be calculated by multiplying all the existing prime factors when these numbers are represented in a form of multiplication of prime factors.

Number 3 is itself a prime number, number 4 can be represented as the multiplication of 2 with 2 where 2 is the prime number and number 6 can be represented as the multiplication of 2 with 3 where both 2 and 3 are prime factors of 6.

So, $3=3$, $4=2\times 2$ and $6=2\times 3$

Clearly, all the existing prime factors are 2,2 and 3. So, LCM of 3,4 and 6 is the multiplication of prime factors 2,2 and 3.
i.e., LCM of 3,4 and 6 $=2\times 2\times 3=12$

Now, the sum whose value is required can be written as $x={\displaystyle \frac{1}{3}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}={\displaystyle \frac{\left(4\times 1\right)+\left(3\times 3\right)+\left(2\times 5\right)}{12}}={\displaystyle \frac{4+9+10}{12}}={\displaystyle \frac{23}{12}}$

Hence, the value of the sum ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}$ is ${\displaystyle \frac{23}{12}}$.

Note: In this particular problem, we calculated the LCM of 3,4 and 6 (which are the denominators of the fractions whose sum is required) and then that LCM is divided with the denominators of the fractions and the quotient obtained is multiplied with the respective numerator of that fraction and this is done for each fraction.