Explanations:In work rate problems, the work rate is the fraction of the job completed per unit of time. If a person can complete a job in \( T \) days, their work rate is \( \frac{1}{T} \). When multiple people work together, their combined work rate is the sum of their individual work rates. The time taken to complete the job together is the reciprocal of the combined work rate.
Let's apply this concept to the given problem. Amul can complete the work in 5 days, so his work rate is \( \frac{1}{5} \). Parle can complete the work in 4 days, so his work rate is \( \frac{1}{4} \). Together with Goodday, they can complete the work in 2 days, so their combined work rate is \( \frac{1}{2} \).
Let \( G \) be the number of days Goodday takes to complete the work alone. Therefore, Goodday's work rate is \( \frac{1}{G} \). The combined work rate of Amul, Parle, and Goodday is the sum of their individual work rates:
\[
\frac{1}{5} + \frac{1}{4} + \frac{1}{G} = \frac{1}{2}
\]
To find \( G \), we need to solve this equation. First, we find a common denominator for the fractions on the left-hand side. The least common multiple of 5, 4, and \( G \) is \( 20G \). Rewriting the equation with this common denominator, we get:
\[
\frac{4G}{20G} + \frac{5G}{20G} + \frac{20}{20G} = \frac{1}{2}
\]
Combining the fractions on the left-hand side, we have:
\[
\frac{4G + 5G + 20}{20G} = \frac{1}{2}
\]
Simplifying the numerator, we get:
\[
\frac{9G + 20}{20G} = \frac{1}{2}
\]
To solve for \( G \), we cross-multiply:
\[
2(9G + 20) = 20G
\]
\[
18G + 40 = 20G
\]
Subtracting \( 18G \) from both sides, we get:
\[
40 = 2G
\]
\[
G = 20
\]
Therefore, Goodday alone can complete the work in 20 days. This matches the correct answer, Option A.