20 people (men,woman,children) earn 20 coins …

Ans. let the number of men, women and children be denoted by M, W and C respectively.

Then

M + W + C = 20    ....... (1)

3M + 1.5 W + 0.5 C = 20  ........ (2)

Multiply equation (2) by (2) 

6M + 3W + C = 40    ........(3)

From (1) and (2)

W + C = 20 - M ..........(4)

3 W + C = 40 - 6M     ...... (5)

The above equations will be solved for W and C in terms of M

 W = 10 - {tex}{5M\over 2} {/tex}  and  C = 10  +  {tex} {3M\over 2}{/tex}

Because W and C have to be positive integers greater than 1 we conclude 

(a) M is an even number.

(b) W {tex}\ge{/tex} 1 implies {tex}{5M\over 2 } \le 9{/tex}

it is seen W = 5 and C = 13 when M = 2 . Also {tex}{5M\over 2} = 5 \lt 9{/tex}

The only solution is M = 2, W =5 and C = 13

So Number of Men = 2

Number of Women = 5

Number of Children = 13