Question

8 men and 6 boys can finish a piece of work in one day while 2 men and 3 boys can finish it in 3 days. Find the time taken by one man alone and that by one boy alone to finish the work.

Answer 1

Appropriate Question:

• 8 men and 6 boys can finish a piece of work in one day while 2 men and 3 boys can finish it in 3 days. Find the time taken by one man alone and that by one boy alone to finish the work.

Solution:

Let one man alone take x days to finish a piece of work and one boy alone can finish it in y days.

$\sf{one \: man's \: one \: day's \: work = \frac{1}{x} \: \: and} \sf{one \: boy's \: one \: day's \: work = \frac{1}{y} }$

Now,

$\sf{8 \: man's \: 1 \: day's \: work = \frac{8}{x} \: and \: 6 \: boy's \: 1 \: day's \: work = \frac{6}{y} }$

According To The Statement,

$\sf{ \frac{8}{x} + \frac{6}{y} = 1 } \: \: \: \: \: ....(i)$

$\sf{Now, \: 2 \: Man's \: \: 1 d ay's \: work = \frac{2}{x} \: and \: 3 \: boy's \: 1 \: day's \: work = \frac{3}{y} }$

$: \implies \sf{{ \frac{2}{x} + \frac{3}{y} = \frac{1}{3} }} \: \: \: \: \: \: \: \: ....(ii)$

$\sf{Let \: \frac{1}{x} = a \: and \: \frac{1}{y} = b }$

Then, 8a + 6b = 1 ⠀⠀....(iii)

$\longmapsto\sf{2a + 3b = \frac{1}{3} } \\ \\ \longmapsto\sf{6a + 9b = 1} \: \: \: \: \: ...(iv)$

Multiplying (iii) by 3 and (iv) by 2, we get

$\longmapsto\sf{24a + 18b \: = 3} \: \: \: \: \: \: \: ...(v) \\ \\ \longmapsto\sf{12a + 18b = 2} \: \: \: \: ...(vi)$

Subtracting (vi) From (v), we get,

$\longmapsto\sf{12a = 1} \\ \\ \longmapsto\sf{a = \frac{1}{12} }$

$\longmapsto\sf{Putting \: the \: value \: of \: a = \frac{1}{12} in \: (iv), \: we \: get, } \\ \\ \longmapsto\sf{6\bigg[ \frac{1}{12} \bigg]+9b=1 } \\ \\ \longmapsto\sf{ \frac{1}{2 } + 9b = 1} \\ \\ \longmapsto\sf{9b = 1 - \frac{1}{2} = \frac{1}{2} } \\ \\ \longmapsto\sf{b = \frac{1}{18} }$

Thus,

$\longmapsto\sf{a = \frac{1}{12} } \\ \\ \longmapsto\sf{ \frac{1}{x} = \frac{1}{12} } \\ \\ \longmapsto\sf{x = 12 \: \: \: \: \: and \: b = \frac{1}{18} } \\ \\ \longmapsto\sf{ \frac{1}{y} = \frac{1}{18} } \\ \\ \longmapsto\sf{y = 18}$

Hence, one man alone can finish the piece of work in 12 days and one boy alone can finish the work in 18 days.