Question

8 men and 4 boys can finish a piece of work in 5 days. 6 women and 8 boys can
finish the same work in 5 days.Also,4 men and 6 women can finish the same
work in 5 days .In how many days can 2 men, 3 women and 10 boys finish the
same work at their double efficiency?​

Answer 1

Step-by-step explanation:

Let W denotes the whole given work.

Let a, b & c denote the numbers of days in which 1 man, 1 woman & 1 boy respectively can complete the whole work W by working alone.

Hence in 1 day by working alone, 1 man, 1 woman & 1 boy can complete the amounts of work respectively W/a, W/b & W/c.

It is mentioned that

(i) 4 men and 2 boys together can complete the whole work W in 5 days. So we get the following relation,

5*[4*(W/a) + 2*(W/c)] = W or 4/a + 2/c = 1/5 …… (1a)

(ii) 3 women and 4 boys together can complete the whole work W in 5 days. So we get the following relation,

5*[3*(W/b) + 4*(W/c)] = W or 3/b + 4/c = 1/5 …… (1b)

(iii) 2 men and 3 women together can complete the whole work W in 5 days. So we get the following relation,

5*[2*(W/a) + 3*(W/b)] = W or 2/a + 3/b = 1/5 …… (1c)

From (1a) + (1b) + (1c), we get,

6/a + 6/b + 6/c = 3/5 or 1/a + 1/b + 1/c = 1/10 …… (1d)

Let D denotes the time (in days) that 1 man, 1 woman and 1 boy together will take to complete the whole work W. So we get the following relation,

D*[1*(W/a) + 1*(W/b) + 1*(W/c)] = W

D*(1/a + 1/b + 1/c) = 1 or D/10 = 1 [from (1d)]

D = 10 (days) [Ans]

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Answer 2

Answer:

2 men,10 boys and 3 women will take 2.5 days to complete the work at their double efficiency

Step-by-step explanation:

Let the work done by 1 man,1 boy and 1 woman in 1 day be the following respectively

$\frac{1}{p} , \frac{1}{q} , \frac{1}{r}$

Given that 8 men and 4 boys can finish a piece of work in 5 days.Therefore,work done in 1 day can be written as

$\frac{8}{p} + \frac{4}{q} = \frac{1}{5} \\ = > \frac{2}{p} + \frac{1}{q} = \frac{1}{20} - - > ( 1)$

Similarly 6 women and 8 boys can finish the work in 5 days.Hence

$= > \frac{6}{r} + \frac{8}{q} = \frac{1}{5} \\ = > \frac{3}{r} + \frac{4}{q} = \frac{1}{10} - - > (2)$

4 men and 6 women can finish the work in 5 days.Hence,

$\frac{4}{p} + \frac{6}{r} = \frac{1}{5} \\ = > \frac{2}{p} + \frac{3}{r} = \frac{1}{10} - - > (3)$

Subtracting equation 1 from equation 3,we get

$(3) - (1) = > ( \frac{2}{p} + \frac{3}{r} ) - ( \frac{2}{p} + \frac{1}{q} ) \\ = \frac{1}{10} - \frac{1}{20} = \frac{1}{20} \\ = > \frac{3}{r} - \frac{1}{q} = \frac{1}{20} - - > (4)$

Subtracting equation 4 from 2,we get

$(2) - (4) = > ( \frac{ 3}{r} + \frac{4}{q} ) - ( \frac{3}{r} - \frac{1}{q} ) \\ = \frac{1}{10} - \frac{1}{20} = \frac{1}{20} \\ = > \frac{5}{q} = \frac{1}{20} \\ = > q = 100$

It means that 1 boy completes work in 100 days

Substituting value of q in equation 1,we get

$\frac{2}{p} + \frac{1}{100} = \frac{1}{20} \\ = > \frac{2}{p} = \frac{1}{25} = > p = 50$

It means that 1 man can complete the work in 50 days.Substituting q in equation 2,we get

$\frac{3}{r} + \frac{4}{100} = \frac{1}{10} \\ = > \frac{3}{r} = \frac{6}{100} = > r = 50$

It means that 1 woman can complete the work in 50 days.Now as the efficiency is doubled,the days of work are halved hence

$p'=25\\q'=50 \\r'=25$

Now 2 men,3 women and 10 boys can complete the work in

$\frac{1}{\frac{2}{p'} + \frac{10}{q'} + \frac{3}{r'} } \\ = \frac{1}{ \frac{2}{25} + \frac{10}{50} + \frac{3}{25} } = \frac{1}{\frac{4 + 10 + 6}{50}} = \frac{50}{20} \\ = 2.5 \: days$

Therefore,2 men,10 boys and 3 women will take 2.5 days to complete the work at their double efficiency

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