Let's solve this step by step:
1. Understanding the work rates:
- Let the amount of work B can do in one day be [tex]\(1\)[/tex] unit.
- A does [tex]\(\frac{1}{4}\)[/tex] as much work as B, so the work done by A in one day is [tex]\(\frac{1}{4}\)[/tex] unit.
- The combined work done by A and B in one day is:
[tex]\[
\text{Work by A} + \text{Work by B} = \frac{1}{4} + 1 = \frac{5}{4} \text{ units}
\][/tex]
- C does [tex]\(\frac{1}{2}\)[/tex] as much work as A and B together, hence the work done by C in one day is:
[tex]\[
\frac{1}{2} \times \frac{5}{4} = \frac{5}{8} \text{ units}
\][/tex]
2. Total work calculation:
- C alone can complete the entire work in 60 days. Therefore, the total work [tex]\(W\)[/tex] can be expressed in terms of C's daily work:
[tex]\[
\text{Total work} = 60 \times \frac{5}{8} = 37.5 \text{ units}
\][/tex]
3. Combined work rate of A, B, and C:
- The work rate when A, B, and C work together is:
[tex]\[
\text{Work by A} + \text{Work by B} + \text{Work by C} = \frac{1}{4} + 1 + \frac{5}{8}
\][/tex]
- Converting each term to a common denominator ([tex]\(8\)[/tex]):
[tex]\[
\frac{1}{4} = \frac{2}{8}, \quad 1 = \frac{8}{8}, \quad \frac{5}{8} = \frac{5}{8}
\][/tex]
Adding these together:
[tex]\[
\frac{2}{8} + \frac{8}{8} + \frac{5}{8} = \frac{15}{8}
\][/tex]
4. Calculating the number of days:
- Let [tex]\(D\)[/tex] be the number of days required for A, B, and C working together to finish the work. The total work divided by the combined work rate gives:
[tex]\[
D = \frac{\text{Total work}}{\text{Combined work rate}} = \frac{37.5}{\frac{15}{8}}
\][/tex]
- Simplifying this:
[tex]\[
D = \frac{37.5 \times 8}{15} = 20 \text{ days}
\][/tex]
Thus, together, A, B, and C will complete the work in 20 days.
The correct answer is:
[tex]\[ \boxed{20 \text{ days}} \][/tex]
