Question

8 men and 12 boys can finish a piece of work in 10 days.while 6 men and 8 boys can finish it in 14 days. find the time taken by one man alone and one boy alone to finish the work. ​

Answer 1

Solution :

$\bf{\purple{\underline{\bf{Given\::}}}}$

8 men and 12 boys can finish a piece of work in 10 days. While 6 men and 8 boys can finish it in 14 days.

$\bf{\purple{\underline{\bf{To\:find\::}}}}$

The time taken by one man alone and one boy alone to finish the work.

$\bf{\purple{\underline{\bf{Explanation\::}}}}$

Let the work done by 1 man in 1 day be r

Let the work done by 1 boy in 1 day be m

A/q

$\bullet\sf{8r+12m=\dfrac{1}{10} ..................(1)}\\\\\bullet\sf{6r+8m=\dfrac{1}{14} ...................(2)}$

$\underline{\bigstar\:{\boldsymbol{Using\:elimination\:method\::}}}$

Multiplying in equation (1) by 3 & multiplying in equation (2) by 4;

$\dashrightarrow\sf{3\bigg[8r+12m=\dfrac{1}{10} \bigg]}\\\\\\\dashrightarrow\sf{24r+36m=\dfrac{1}{10} .....................(3)}$

&

$\dashrightarrow\sf{4\bigg[6r+8m=\dfrac{1}{14} \bigg]}\\\\\\\dashrightarrow\sf{24r+32m=\dfrac{4}{14} .....................(4)}$

Subtracting equation (3) & equation (4),we get;

$\dashrightarrow\sf{\cancel{24r-24r} +36m-32m=\dfrac{3}{10} -\dfrac{4}{14} }\\\\\\\dashrightarrow\sf{4m=\dfrac{21-20}{70} }\\\\\\\dashrightarrow\sf{4m=\dfrac{1}{70} }\\\\\\\dashrightarrow\sf{280m=1}\\\\\\\dashrightarrow\sf{\orange{\underline{m=\dfrac{1}{280}}} }$

Putting the value of m in equation (2),we get;

$\dashrightarrow\sf{6r+8\bigg(\dfrac{1}{280} \bigg)=\dfrac{1}{14} }\\\\\\\dashrightarrow\sf{6r+\cancel{\dfrac{8}{280}} =\dfrac{1}{14} }\\\\\\\\\dashrightarrow\sf{6r+\dfrac{1}{35} =\dfrac{1}{14} }$

$\dashrightarrow\sf{6r=\dfrac{1}{14} -\dfrac{1}{35} }\\\\\\\dashrightarrow\sf{6r=\dfrac{5-2}{70} }\\\\\\\dashrightarrow\sf{6r=\dfrac{3}{70} }\\\\\\\dashrightarrow\sf{420r=3}\\\\\\\dashrightarrow\sf{r=\cancel{\dfrac{3}{420} }}\\\\\\\dashrightarrow\sf{\orange{\underline{r=\dfrac{1}{140} }}}$

Thus;

1 man do 1/140 of the work in 1 day,so

$\star\underline{\bf{Total\:time=140\:days\:of\:man.}}}$

1 boy do 1/280 of the work in 1 day,so

$\star\underline{\bf{Total\:time=280\:days\:of\:boy.}}}$

Answer 2

Answer:

Step-by-step explanation:

the time taken by man be x days

time taken by boys be y days

work done by 1 man in one day=1/x

work done by 1 boy in one day=1/y

8/x+12/y = 1/10............eq 1

6/x+ 8/y = 1/14..........eq2

Let u=1/x & v=1/y

8u+ 12y=1/10.........eq3

6u+8v= 1/14...........eq4

Multiplying eq3 by 2 & eq4 by 3

16u+ 24y=2/10.........eq5

18u+24v= 3/14.........eq6

subtract eq 5 & 6

-2u= -3/14+2/10

-2u = (-3×10 +2×14)/140

-2u=( -30+28)/140

-2u=-2/140

u=1/140

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