Answer

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**Hint:**We first have to form the proportionality equation for the variables. Take an arbitrary constant and use the given values of the variables to find the value of the constant. Finally, substitute the constant’s value to find the equation.

**Complete step-by-step solution:**

We have been given the relation between two variables where we assume number of men as r and number of days as t.

The inversely proportional number is actually directly proportional to the inverse of the given number. The relation between r and t is inverse relation.

It’s given r varies inversely as t which gives $r\propto \dfrac{1}{t}$.

To get rid of the proportionality we use the proportionality constant which gives $r=\dfrac{k}{t}\Rightarrow rt=k$.

Here, the number k is the proportionality constant. It’s given $r=36$ when $t=18$.

We put the values in the equation $rt=k$ to find the value of k.

So, $36\times 18=k$.

Therefore, the equation becomes with the value of k as $rt=36\times 18$.

Now we simplify the equation to get the value of t for number of men being 27

\[\begin{align}

& 27\times t=36\times 18 \\

& \Rightarrow t=\dfrac{36\times 18}{27}=4\times 6=24 \\

\end{align}\]

Therefore, the number of days required to complete the work is $24$.

Option (d) is the correct answer.

**Note:**In a direct proportion, the ratio between matching quantities stays the same if they are divided. They form equivalent fractions. In an indirect (or inverse) proportion, as one quantity increases, the other decreases. In an inverse proportion, the product of the matching quantities stays the same.

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