Question

How do you evaluate \[\dfrac{3}{9}+\dfrac{4}{5}\]?

Answer 1

Hint: First of all simplify the fraction $\dfrac{3}{9}$ by cancelling the common factor 3 to get $\dfrac{1}{3}$. Assume the sum of given fractions as ‘E’. Now, convert this sum of fractions into a single fraction by taking their sum. Take the L.C.M. of the denominators of the two fractions by finding the first common multiple of the two numbers 3 and 5. Now, in the numerator divide the obtained L.C.M. by 3 and multiply the obtained number with 1, now divide the L.C.M. by 5 and take its product with 4. Finally, add the numbers obtained by the two products to get the answer.

Complete step-by-step answer:
Here, we have been provided with the expression \[\dfrac{3}{9}+\dfrac{4}{5}\] and we are asked to simplify it. That means we have to add the two fractions and convert them into a single fraction.
Now, we can see that the fraction $\dfrac{3}{9}$ can be further simplified by cancelling the common factor 3, so get the simplified form of this fraction as $\dfrac{1}{3}$.
Now, let us assume the new expression as ‘E’, so we have,
\[\Rightarrow E=\dfrac{1}{3}+\dfrac{4}{5}\]
To take the sum of these fractions first we need to find the L.C.M. of the denominators 3 and 5. Let us see some multiples of 3 and 5 to find the L.C.M, so we have,
\[\begin{align}
  & \Rightarrow 3=3,6,9,12,15,18,21,.... \\
 & \Rightarrow 5=5,10,15,20,25,..... \\
\end{align}\]
Clearly, we can see that the first common multiple is 15. Therefore, the L.C.M of 3 and 5 is 15.
Now, the denominator of the resultant fraction will become 15. In the denominator we have got 15, now in the numerator we will have the expression: - \[1\times \left( \dfrac{L.C.M.}{3} \right)+4\times \left( \dfrac{L.C.M.}{5} \right)\]. Therefore, the expression ‘E’ becomes,
\[\begin{align}
  & \Rightarrow E=\dfrac{1\times \left( \dfrac{15}{3} \right)+4\times \left( \dfrac{15}{5} \right)}{15} \\
 & \Rightarrow E=\dfrac{\left( 1\times 5 \right)+\left( 4\times 3 \right)}{15} \\
 & \Rightarrow E=\dfrac{5+12}{15} \\
 & \Rightarrow E=\dfrac{17}{15} \\
\end{align}\]
Hence, our answer is \[\dfrac{17}{15}\].

Note: One may note that here the numbers in the denominator of the two fractions were small and that is why did not use the prime factorization method to find the L.C.M. You can also remember the general method to find the sum of two fractions given as: - \[\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{b\times d}\]. This generalized form is used when the denominator contains large numbers and they are generally prime numbers. This is because in such cases we find L.C.M. by directly taking the product of denominators without using the prime factorization method.