Question

4 men and 6 boys can finish a piece of work in 5 days, while 3 men and 4 boys can finish it in
7 days. Find the time taken by 1 man alone or that by 1 boy alone.

Answer 1

Answer :-

→ 1 man = 35 days and 1 boy = 70 days .

Step-by-step explanation :-

Suppose 1 man alone can finish the work in x days and 1 boy alone can finish it in y days .

Then, 1 man's days work = 1/x .

And, 1 boy's 1 day'y work = 1/y .

4 men and 6 boys can finish a piece of work in 5 days .

⇒ ( 4 men's 1 day's work ) + ( 6 boy's 1 day's work ) = 1/5 .

⇒ 4/x + 6/y = 1/5 .

⇒ 4u + 6v = 1/5 .........(1) , where 1/x = u and 1/y = v .

Again, 3 men and 4 boys can finish it in

7 days.

⇒ ( 3 men's 1 day's work ) + (4 boy's 1 day's work ) = 1/7 .

⇒ 3/x + 4/y = 1/7 .

⇒ 3u + 4v = 1/7 .........(2) .

Now, on multiplying equation (1) by 3 and, equation (2) by 4 .

⇒ 12u + 18v = 3/5 ...........(3) .

And,

⇒ 12u + 16v = 4/7 ...........(4) .

Now, on substracting the result , we get

$\sf \implies2v = \frac{3}{5} - \frac{4}{7} . \\ \\ \sf \implies2v = \frac{21 - 20}{35} . \\ \\ \sf \implies2v = \frac{1}{35} . \\ \\ \sf \implies v = \frac{1}{70} . \\ \\ \sf \implies \frac{1}{y} = \frac{1}{70} . \\ \\ \large \pink{\it \therefore y = 70.}$

On putting v = 1/70 in equation (2) , we get

$\sf \implies3u = \bigg( \frac{1}{7} - \frac{4}{70} \bigg). \\ \\ \sf \implies3u = \frac{10 - 4}{70}. \\ \\ \sf \implies3u = \frac{6}{70} . \\ \\ \sf \implies \frac{1}{x} = \frac{6}{70} \times \frac{1}{3} . \\ \\ \large \orange{ \it \therefore x = 35.}$

∴ one man alone can finish the work in 35 days .

And, one boy alone can finish the work in 70 days .

Hence, it is solved .

Answer 2

Answer :-

→ 1 man = 35 days and 1 boy = 70 days .

Step-by-step explanation :-

Suppose 1 man alone can finish the work in x days and 1 boy alone can finish it in y days .

Then, 1 man's days work = 1/x .

And, 1 boy's 1 day'y work = 1/y .

4 men and 6 boys can finish a piece of work in 5 days .

⇒ ( 4 men's 1 day's work ) + ( 6 boy's 1 day's work ) = 1/5 .

⇒ 4/x + 6/y = 1/5 .

⇒ 4u + 6v = 1/5 .........(1) , where 1/x = u and 1/y = v .

Again, 3 men and 4 boys can finish it in

7 days.

⇒ ( 3 men's 1 day's work ) + (4 boy's 1 day's work ) = 1/7 .

⇒ 3/x + 4/y = 1/7 .

⇒ 3u + 4v = 1/7 .........(2) .

Now, on multiplying equation (1) by 3 and, equation (2) by 4 .

⇒ 12u + 18v = 3/5 ...........(3) .

And,

⇒ 12u + 16v = 4/7 ...........(4) .

Now, on substracting the result , we get

$\begin{lgathered}\sf \implies2v = \frac{3}{5} - \frac{4}{7} . \\ \\ \sf \implies2v = \frac{21 - 20}{35} . \\ \\ \sf \implies2v = \frac{1}{35} . \\ \\ \sf \implies v = \frac{1}{70} . \\ \\ \sf \implies \frac{1}{y} = \frac{1}{70} . \\ \\ \large \pink{\it \therefore y = 70.}\end{lgathered}$

On putting v = 1/70 in equation (2) , we get

$\begin{lgathered}\sf \implies3u = \bigg( \frac{1}{7} - \frac{4}{70} \bigg). \\ \\ \sf \implies3u = \frac{10 - 4}{70}. \\ \\ \sf \implies3u = \frac{6}{70} . \\ \\ \sf \implies \frac{1}{x} = \frac{6}{70} \times \frac{1}{3} . \\ \\ \large \orange{ \it \therefore x = 35.}\end{lgathered}$

∴ one man alone can finish the work in 35 days .

And, one boy alone can finish the work in 70 days .

Hence, it is solved .