To find out how many days [tex]\( B \)[/tex] alone will take to complete the work, let's denote:
- [tex]\( x \)[/tex] as the number of days [tex]\( B \)[/tex] takes to complete the work alone.
- Since [tex]\( A \)[/tex] takes three times the number of days taken by [tex]\( B \)[/tex], [tex]\( A \)[/tex] will take [tex]\( 3x \)[/tex] days to complete the same work alone.
Now, we need to think about their work rates:
1. The work rate of [tex]\( B \)[/tex] = [tex]\(\frac{1}{x}\)[/tex] of the work per day.
2. The work rate of [tex]\( A \)[/tex] = [tex]\(\frac{1}{3x}\)[/tex] of the work per day.
When [tex]\( A \)[/tex] and [tex]\( B \)[/tex] work together, their combined work rate is the sum of their individual work rates.
So, combined work rate of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ \frac{1}{x} + \frac{1}{3x} \][/tex]
Combining the fractions, we get:
[tex]\[ \frac{1 + \frac{1}{3}}{x} = \frac{4}{3x} \][/tex]
Given that [tex]\( A \)[/tex] and [tex]\( B \)[/tex] together complete the work in 36 days, we know that:
[tex]\[ \text{Combined work rate} \times \text{Number of days} = 1 \text{ (since it's one whole piece of work)} \][/tex]
So:
[tex]\[ \frac{4}{3x} \times 36 = 1 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{4 \times 36}{3x} = 1 \][/tex]
[tex]\[ \frac{144}{3x} = 1 \][/tex]
[tex]\[ 144 = 3x \][/tex]
[tex]\[ x = 48 \][/tex]
Thus, [tex]\( B \)[/tex] alone completes the work in 48 days.
