P and Q together can complete a job in 16 days. Q and R together can complete it in 24 days, and R and P together can do it in 32 days. In how many days can P alone complete the job?

The correct answer is $38\frac{2}{5}$ days.

Let us break down the problem step-by-step to find how many days P alone will take to complete the job.

Step 1: Express the one-day work for each pair.

Let the work done by P, Q, and R in 1 day be P, Q, and R respectively.

Given that P and Q together complete the job in 16 days, their combined 1-day work is:

$P+Q=\frac{1}{16}$

Given that Q and R together complete the job in 24 days, their combined 1-day work is:

$Q+R=\frac{1}{24}$

Given that R and P together complete the job in 32 days, their combined 1-day work is:

$R+P=\frac{1}{32}$

Step 2: Find the combined one-day work of P, Q, and R together.

Adding all three equations:

$(P+Q)+(Q+R)+(R+P)=\frac{1}{16}+\frac{1}{24}+\frac{1}{32}$

$2(P+Q+R)=\frac{1}{16}+\frac{1}{24}+\frac{1}{32}$

To add the fractions, find the Least Common Multiple (LCM) of 16, 24, and 32, which is 96:

$2(P+Q+R)=\frac{6+4+3}{96}$

$2(P+Q+R)=\frac{13}{96}$

$P+Q+R=\frac{13}{192}$

Step 3: Calculate the one-day work of P alone.

Subtract the combined 1-day work of (Q + R) from the combined 1-day work of (P + Q + R):

$P=(P+Q+R)-(Q+R)$

$P=\frac{13}{192}-\frac{1}{24}$

Convert $\frac{1}{24}$ to a fraction with a denominator of 192 ($24\times 8=192$):

$P=\frac{13}{192}-\frac{8}{192}$

$P=\frac{5}{192}$

Step 4: Determine total days taken by P alone.

The time taken by P alone to complete the job is the reciprocal of P's 1-day work:

$\text{Time taken by P}=\frac{192}{5}\text{days}$

Converting $\frac{192}{5}$ into a mixed fraction:

$192÷5=38\text{with a remainder of}2$

$\text{Time taken}=38\frac{2}{5}\text{days}$