Question

How do you simplify \[\dfrac{7}{8}+\dfrac{1}{5}\]?

Answer 1

Hint: 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 8 and 5. Now, in the numerator divide this obtained L.C.M. by 8 and multiply the obtained number with 7, now divide the L.C.M. by 5 and take its product with 1. Finally, add the numbers obtained by the two products to get the answer.

Complete step by step solution:
Here, we have been provided with the expression \[\dfrac{7}{8}+\dfrac{1}{5}\] and we are asked to simplify it. That means we have to add the two fractions and convert them into a single fraction.
Now, let us assume the given expression as ‘E’, so we have,
\[\Rightarrow E=\dfrac{7}{8}+\dfrac{1}{5}\]
To take the sum of these fractions first we need to find the L.C.M. of the denominators 8 and 5. Let us use the prime factorization method to find the L.C.M., so we have,
\[\begin{align}
  & \Rightarrow 8=2\times 2\times 2={{2}^{3}} \\
 & \Rightarrow 5={{5}^{1}} \\
\end{align}\]
Therefore, the L.C.M. can be given as the product of highest power of each prime factor taken with their bases, so we have,
\[\Rightarrow \] L.C.M. (8, 5) = \[{{2}^{3}}\times {{5}^{1}}\]
\[\Rightarrow \] L.C.M. (8, 5) = 40
Now, the denominator of the resultant fraction will become 40. In the denominator we have got 40, now in the numerator we will have the expression: - \[7\times \left( \dfrac{L.C.M.}{8} \right)+1\times \left( \dfrac{L.C.M.}{5} \right)\]. Therefore, the expression ‘E’ becomes,
\[\begin{align}
  & \Rightarrow E=\dfrac{7\times \left( \dfrac{40}{8} \right)+1\times \left( \dfrac{40}{5} \right)}{40} \\
 & \Rightarrow E=\dfrac{\left( 7\times 5 \right)+\left( 1\times 8 \right)}{40} \\
\end{align}\]
\[\begin{align}
  & \Rightarrow E=\dfrac{35+8}{40} \\
 & \Rightarrow E=\dfrac{43}{40} \\
\end{align}\]
Hence, our answer is \[\dfrac{43}{40}\].

Note: One must know how to find the sum of two fractions by using the prime factorization of the denominators by which we calculate 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 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.