Three printing machines, P, Q and R, when each is working alone, can print one lakh books in, respectively, 8 hours, 16 hours and 24 hours. These machines print books in turns of one hour each, starting with P, followed by Q and then by R. However, after a total of 9 hours, P stops functioning, but Q and R continue working alternately for an hour each, with Q working during the 10th hour, till the printing of all the one lakh books is completed. In how many hours in all did machines Q and R complete printing the remaining books?

`. Let's write down the MathML blocks carefully without LaTeX, without display="block", using raw unicode characters. Let's check MathML elements: Work done by P in 1 hour: `$\frac{1}{8}$` Work done by Q in 1 hour: `$\frac{1}{16}$` Work done by R in 1 hour: `$\frac{1}{24}$` Let's construct clean XML outputs: ```xml 6

The correct answer is 6.

Step 1: Calculate the work done by each machine in 1 hour.
Let the total work of printing one lakh books be equal to 1 unit.

Work done by machine P in 1 hour = $\frac{1}{8}$

Work done by machine Q in 1 hour = $\frac{1}{16}$

Work done by machine R in 1 hour = $\frac{1}{24}$

Step 2: Calculate the work completed in the first 9 hours.
The machines work in turns of 1 hour each in the order P, Q, R, P, Q, R, P, Q, R. Thus, in 9 hours, each machine works for 3 hours.

Work done by P in 3 hours = $3\times \frac{1}{8}=\frac{3}{8}$

Work done by Q in 3 hours = $3\times \frac{1}{16}=\frac{3}{16}$

Work done by R in 3 hours = $3\times \frac{1}{24}=\frac{3}{24}=\frac{1}{8}$

Total work completed in the first 9 hours = $\frac{3}{8}+\frac{3}{16}+\frac{1}{8}=\frac{4}{8}+\frac{3}{16}=\frac{1}{2}+\frac{3}{16}=\frac{11}{16}$

Step 3: Calculate the remaining work.

Remaining work = $1-\frac{11}{16}=\frac{5}{16}$

Step 4: Calculate the time needed by Q and R to finish the remaining work.
After 9 hours, P stops working, and Q and R work alternately for 1 hour each (Q in 10th hour, R in 11th hour, and so on).

In a 2-hour cycle (1 hour of Q + 1 hour of R), the work done is:
$\frac{1}{16}+\frac{1}{24}=\frac{3+2}{48}=\frac{5}{48}$

Let the number of such 2-hour cycles required be $n$.
$n\times \frac{5}{48}=\frac{5}{16}$

$n=\frac{5}{16}\times \frac{48}{5}=3$

Since each cycle consists of 2 hours, 3 cycles will take:
$3\times 2=6$ hours.

Therefore, machines Q and R completed printing the remaining books in 6 hours.