Determine B's Work Rate and Completion Time in a Collaborative Task

Suppose A and B can complete a piece of work together in 30 days. If A works for 16 days and then B finishes the remaining work in 44 days, how long would it take for B to complete the work alone?

Defining Variables

Let A and B be the work rates (amount of work per day) for individuals A and B, respectively.

Step-by-Step Solution

1. Combined Work Rate: A and B together can finish the work in 30 days, meaning their combined work rate is:

text{Work rate of A and B} frac{1 text{ work}}{30 text{ days}} frac{1}{30} text{ work/day}

2. Work Done by A: A works for 16 days. If we let a be the work rate of A and b be the work rate of B, we know that:

a cdot 16 frac{1}{30} cdot 16 cdot text{work}

3. Remaining Work: After A has worked for 16 days, the remaining work is:

text{Remaining work} 1 - 16a

B finishes this remaining work in 44 days, so the work done by B in that time is:

44b 1 - 16a

Substitute and Solve for a and b

From the equation a cdot b frac{1}{30}, we can express b in terms of a:

b frac{1}{30a}

Substitute this into the equation for the remaining work:

1 - 16a 44 left(frac{1}{30a} - a right )

Expand and simplify the equation:

1 - 16a frac{44}{30} - 44a

Rearrange the equation to solve for a:

1 - frac{44}{30} -44a 16a

-frac{14}{30} -28a

Solving for a:

a frac{7}{420} frac{1}{60}

Now, substituting a back into the equation for b:

b frac{1}{30} - a frac{1}{30} - frac{1}{60} frac{2}{60} - frac{1}{60} frac{1}{60}

Days Taken by B Alone

Since B's work rate is b frac{1}{60}, it means B can complete 1 entire work in:

1 text{ work} 60 text{ days}

Therefore, B can finish the whole work alone in 60 days.