Question

2 men and 7 boys can do a piece of work in 4 days.The same work is done in 3 days by 4 men and 4 boys.How long would it take one man alone and one boy alone to do it?

Answer 1

Hi !

Time taken by one man = x days , and that by 1 boy = y days

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

work done in 1 day by one boy= 1/y

work done by 2 men in one day = 2/x

work done by 7 boys in one day = 7/y

so ,

2/x + 7/y = 1/4      -----> [ 1]

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Similarly , 
work done by 4 men in one day = 4/x
work done by 4 boys in one day = 4/y

4/x + 4/y = 1/3       ----> [2]

Multiplying  equation 1 by  , 2 ,  

2(2/x + 7/y = 1/4   )
 = 4/x + 14/y = 1/2           ---> [3]

solving equations 2 and 3



 4/x + 4/y = 1/3  
- 4/x + 14/y = 1/2  
==============
-10/y = 2-3/6
-10/y = -1/6

-1/6y = -10
y = -10 × 6/-1
   = 60 

2/x + 7/y = 1/4
2/x + 7/60 = 1/4
7/60 - 1/4 = -2/x
-8/60 = -2/x

-120 = -8x
x = -120/-8
x = 15

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Time taken by 1 man to do the  whole job = 15 days
time taken by 1 boy to do the whole  job = 60 days

time taken by one boy and one man to do one day's work = 1/60 + 1/15
                                          = 1/60 + 4/60
                                          = 5/60  = 1/12

∴ Together they can do it in 12 days !

Answer 2

(2 men + 7 boys ) do a piece of work in 4 days .

same work done by ( 4 men + 4 boys) in 3 days .

use, M₁D₁ = M₂D₂
where M₁ , M₂ are the number of men and men and D₁, D₂ are the number of days .

(2 men + 7 boys )*4 = (4 men + 4 boys )*3
16 boys = 4 men
1 men = 4 boys

so, (2*4 boys + 7boys ) do work in 4 days
15 boys do work in 4 days
1 boys does work in 4*15 = 60 days

again,
(4men + 4*(1/4)men ) do work in 3 days
(4 men + 1 man ) do work in 3 days
5 men do work in 3 days
1 man does work in 3*5 = 15days