[tex]\large\underline{\sf{Solution-}}[/tex]

Let assume that

1 man can alone take x days.

1 boy can alone take y days.

So,

1 day work of 1 man = [tex]\dfrac{1}{x} [/tex]

1 day work of 1 boy = [tex]\dfrac{1}{y} [/tex]

Given that,

2 men & 3 boys together can do a piece of work in 8 days.

Thus,

1 day work of 2 men = [tex]\dfrac{2}{x} [/tex]

1 day work of 3 boys = [tex]\dfrac{3}{y} [/tex]

So,

[tex]\sf \: 8\bigg(\dfrac{2}{x} + \dfrac{3}{y} \bigg) = 1 \\ \\ [/tex]

[tex]\sf \: \dfrac{2}{x} + \dfrac{3}{y} = \frac{1}{8} - - - (1)\\ \\ [/tex]

Further given that, 3 men & 2 boys together can do a piece of work in 6 days.

Thus,

1 day work of 3 men = [tex]\dfrac{3}{x} [/tex]

1 day work of 2 boys = [tex]\dfrac{2}{y} [/tex]

So,

[tex]\sf \: 6\bigg(\dfrac{3}{x} + \dfrac{2}{y} \bigg) = 1 \\ \\ [/tex]

[tex]\sf \: \dfrac{3}{x} + \dfrac{2}{y} = \frac{1}{6} - - - (2)\\ \\ [/tex]

On multiply equation (1) by 2 and equation (2) by (3),

[tex]\sf \: \dfrac{4}{x} + \dfrac{6}{y} = \frac{1}{4} - - - (3)\\ \\ [/tex]

[tex]\sf \: \dfrac{9}{x} + \dfrac{6}{y} = \frac{1}{2} - - - (4)\\ \\ [/tex]

On Subtracting equation (3) from (4), we get

[tex]\sf \: \dfrac{5}{x} = \dfrac{1}{2} - \frac{1}{4}\\ \\ [/tex]

[tex]\sf \: \dfrac{5}{x} = \dfrac{2- 1}{4} \\ \\ [/tex]

[tex]\sf \: \dfrac{5}{x} = \dfrac{1}{4} \\ \\ [/tex]

[tex]\bf\implies \:x = 20 \\ \\ [/tex]

On substituting x in equation (1), we get

[tex]\sf \: \dfrac{2}{20} + \dfrac{3}{y} = \frac{1}{8} \\ \\ [/tex]

[tex]\sf \: \dfrac{1}{10} + \dfrac{3}{y} = \frac{1}{8} \\ \\ [/tex]

[tex]\sf \: \dfrac{3}{y} = \frac{1}{8} - \frac{1}{10} \\ \\ [/tex]

[tex]\sf \: \dfrac{3}{y} = \frac{5 - 4}{40} \\ \\ [/tex]

[tex]\sf \: \dfrac{3}{y} = \frac{1}{40} \\ \\ [/tex]

[tex]\bf\implies \:y = 120 \\ \\ [/tex]

Hence,

1 man can alone take 20 days.

1 boy can alone take 120 days.

Step-by-step explanation:

Sol-

Case 1-

2 men and 3 boys can do a piece of work in = 8 days

(2 × 8) men and (3 × 8) boys can do the work in = 1 day

16 men and 24 boys can do the work in = 1 day

Case 2-

3 men and 2 boys can do a piece of work in = 6 days

(3× 6) men and (6 × 2) boys can do the work in = 1 day

18 men and 12 boys can do the work in = 1 day

Case 1 = Case 2 as the time taken is equal.

According to the time take

16 men + 24 boys = 18 men + 12 boys

24 boys - 12 boys = 18 men - 16 men

12 boys = 2 men

1 man = 12/2 = 6 boys Or 1 Boy = 1/6 men

We can find the number of days by using any of the cases

Case 1

16 men and 24 boys can do the work in = 1 day

(16 × 6) boys and 24 boys can do the work in = 1 day

96 boys and 24 boys can do the work in = 1 day

120 boys can do the work in = 1 days

1 boy can do the work in = 1 × 120 = 120 days

16 men and 24 boys can do the work in = 1 day

16 men and (24/6) men can do the work in = 1 day

16 men and 4 men can do the work in = 1 day

20 men can do the work in = 1 day

1 man can do the work in = 1 × 20 = 20 days

Case 2

18 men and 12 boys can do the work in = 1 day

(18 × 6) boys and 12 boys can do the work in = 1 day

108 boys and 12 boys can do the work in = 1 day

120 boys can do the work in = 1 day

So, 1 boy can do the work in = 1 × 120 days

Now,

18 men and 12 boys can do the work in = 1 day

18 men and (12/6) men can do the work in = 1 day

18 men and 2 men can do the work in =1 day

20 men can do the work in = 1 day

1 man can do the work in = 1 × 20 = 20 days

So,

1 boy alone can do the work in 120 days

1 man alone can do the work in 20 days

[tex] \huge{ \red{ \mathfrak{That's \: your \: required \: answer}}}[/tex]