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The sum of two numbers is 40 and their product is 375. What will be the sum of their reciprocals?
(a) $\dfrac{1}{40}$
(b) $\dfrac{8}{75}$
(c) $\dfrac{75}{4}$
(d) $\dfrac{75}{8}$

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Last updated date: 23rd Jul 2024
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Answer
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Hint: Firstly, we will assume the numbers as x and y. Then, we will form mathematical equations using the given statements. The sum of these numbers can be written as $x+y=40$ and the product can be written as $xy=375$ . Now, we have to add the reciprocals of x and y. The reciprocal of a number is obtained by exchanging their numerator and denominator. We have to take the LCM and simplify the result. Then, we have to substitute the values of the sum of x and y and their product in the resultant expression.

Complete step by step solution:
Let us assume the numbers to be x and y. We are given that the sum of these numbers is 40. We can write this mathematically as
$\Rightarrow x+y=40...\left( i \right)$
We are also given that the product of these numbers is 375. We can write this mathematically as
$\Rightarrow xy=375...\left( ii \right)$
We have to find the sum of the reciprocal of the numbers. We know that the reciprocal of a number is obtained by exchanging their numerator and denominator. We can write the number x as $\dfrac{x}{1}$ and y as $\dfrac{y}{1}$ . Therefore, their reciprocals will be $\dfrac{1}{x}$ and $\dfrac{1}{y}$ respectively.
Now, we have to add these reciprocals.
$\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}$
We have to take the LCM of the denominator. We can see that the denominators are different variables. Therefore, the LCM will be their product, that is, xy.
\[\begin{align}
  & \Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1\times y}{x\times y}+\dfrac{1\times x}{y\times x} \\
 & \Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{y}{xy}+\dfrac{x}{xy} \\
\end{align}\]
Let us add the terms on the RHS.
\[\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{y+x}{xy}\]
We have to rearrange the terms in the numerator.
\[\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{x+y}{xy}\]
Let us substitute (i) and (ii) in the above expression.
\[\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{40}{375}\]
Let us cancel the common factor 5 from the numerator and the denominator.
\[\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{{{\require{cancel}\cancel{40}}^{8}}}{{{\require{cancel}\cancel{375}}^{75}}}\]
We can write the result as
\[\Rightarrow \dfrac{1}{x}+\dfrac{1}{y}=\dfrac{8}{75}\]
Therefore, the sum of the reciprocals of the two numbers is $\dfrac{8}{75}$

So, the correct answer is “Option b”.

Note: Students must deeply learn the concept of algebra and also to create algebraic expressions and equations from the statements. They must be thorough with algebraic operations. Students must know how to take algebraic LCM. They must also know to take the reciprocal of the numbers.