A and B separately can build a wall in 12 and 16 days, respectively. If they work for 1 day alternatively, starting with A, in how many days will the wall be built?

Let the total work = W

Number of days required for A to build the wall = 12 days

$$=$$>  Work done by A in 1 day = $$\frac{W}{12}$$

Number of days required for B to build the wall = 16 days

$$=$$>  Work done by B in 1 day = $$\frac{W}{16}$$

Work done by both A and B in 2 days working alternatively for 1 day = $$\frac{W}{12}+\frac{W}{16}=\frac{7W}{48}$$

$$=$$>  Work done by both A and B in 12 days working alternatively for 1 day = $$\frac{7W}{48}\times\frac{12}{2}=\frac{7W}{8}$$

Remaining work after 12 days = $$W-\frac{7W}{8}=\frac{W}{8}$$

Remaining work after A working on the 13th day = $$\frac{W}{8}-\frac{W}{12}=\frac{W}{24}$$

$$=$$>  The remaining work $$\frac{W}{24}$$ will be completed by B

Number of days required for B to complete $$\frac{W}{24}$$ = $$\frac{\frac{W}{24}}{\frac{W}{16}}=\frac{2}{3}$$ day

$$\therefore\ $$Number of days required to complete the total work = 12 + 1 + $$\frac{2}{3}$$ = $$13 \frac{2}{3}$$ days

Hence, the correct answer is Option C