Let's solve this step-by-step:
1. Determine the work rates:
- Ram's work rate: Ram alone can complete the work in 20 days. Hence, in one day, Ram can complete [tex]\(\frac{1}{20}\)[/tex] of the work.
- Shyam's work rate: Shyam alone can complete the work in 30 days. Hence, in one day, Shyam can complete [tex]\(\frac{1}{30}\)[/tex] of the work.
2. Combine the work rates:
- When working together, their combined work rate per day is the sum of their individual rates.
- Combined work rate = [tex]\(\frac{1}{20} + \frac{1}{30}\)[/tex].
3. Find a common denominator to add the fractions:
- The least common multiple (LCM) of 20 and 30 is 60.
- [tex]\(\frac{1}{20}\)[/tex] can be converted to [tex]\(\frac{3}{60}\)[/tex] (since [tex]\(20 \times 3 = 60\)[/tex]).
- [tex]\(\frac{1}{30}\)[/tex] can be converted to [tex]\(\frac{2}{60}\)[/tex] (since [tex]\(30 \times 2 = 60\)[/tex]).
- Hence, [tex]\(\frac{1}{20} + \frac{1}{30} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12}\)[/tex].
4. Calculate the work done in 6 days:
- If Ram and Shyam work together, they can complete [tex]\(\frac{1}{12}\)[/tex] of the work in one day.
- In 6 days, the total work done is [tex]\(6 \times \frac{1}{12} = \frac{6}{12} = \frac{1}{2}\)[/tex].
Therefore, the amount of work Ram and Shyam can do together in 6 days is [tex]\(\frac{1}{2}\)[/tex].
The correct answer is:
(iii) [tex]\(\frac{1}{2}\)[/tex]
