Question

8 men and12 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 that by one boy alone to finish the work.​

Answer 1

SOLUTION

Let one man alone can finish the work in x days and one boy alone can finish the work in y days. Then,

One day work of one man= 1/x

One day work of one boy= 1/y

Therefore,

One day work of 8 men= 8/x

One day work of 12 boys= 12/y

Since 8 men & 12 boys can finish the work in 10 days

$= > 10( \frac{8}{x} + \frac{12}{y} ) = 1 = > \frac{80}{x} + \frac{120}{y} = 1............(1)$

Again,6 men & 8 boys can finish the work in 14 days

$= > 14( \frac{6}{x} + \frac{8}{y} ) = 1 = > \frac{84}{x} + \frac{112}{y} = 1.............(2) \\ \\ Put \: \frac{1}{x} = u \: and \: \frac{1}{y} = v \: in \: equation \: (1) \: and \: (2) \: we \: get$

=)80u +120v-1=0

=)84u +112v-1=0

By using crosa multiplication, we have

$= > \frac{u}{120 \times - 1 - 112 \times - 1} = \frac{ - v}{80 \times - 1 - 84 \times - 1} = \frac{1}{80 \times 112 - 84 \times 120} \\ \\ = > \frac{u}{ - 120 + 112} = \frac{ - v}{ - 80 + 84} = \frac{1}{80 \times 112 - 84 \times 120} \\ \\ = > \frac{u}{ - 8} = \frac{ - v}{4} = \frac{1}{ - 1120} \\ \\ = > u = \frac{ - 8}{ - 1120} = > \frac{1}{140} \: \: \: and \: \: \: v = \frac{ - 4}{ - 1120} = > \frac{1}{280} \\ \\ We \: have \: u = \frac{1}{140} = > \frac{1}{x} = \frac{1}{140} \\ \\ = > x = 140 \\ \\ and \\ = > v = \frac{1}{280} = > \frac{1}{y} = \frac{1}{280} \\ \\ = > y = 280$

Hence, one man alone can finish the work in 140 days.

& one boy alone can finish the work in 280 days.

Hope it helps ☺️

Answer 2

Step-by-step explanation:

Let 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

put this value in eq 5

16u+ 24y=2/10

16×1/140 +24y =1/5

4/35+24y= 1/5

24y= 1/5-4/35

24y= (1×7 -4)/35

24y = (7-4)/35

24y= 3/35

y= 3/35×24

y= 1/280

u=1/x

1/140 = 1/x

x=140

v=1/y

1/280 = 1/y

y=280

A man complete the work in 140 days

A boy can complete the work in 280 days