Question

40 men can complete a work in 40 days. They start working together. After every 10 days. 5 men left the work and process is continued till end. Then find in how many days total work will be finished ? ​

Answer 1

Answer: 56.66 days

Explanation:

Let us consider the part of work every man completes per day = x

$40x = \frac{1}{40}$

Since 5 men leave every 10 days,

$10\times40x+10\times35x+10\times30x ... = 1$

$\Rightarrow 10\times40x + 10\times\frac{35}{40}\times40x + 10\times\frac{30}{40}\times40x ... = 1$

$\Rightarrow 10\times\frac{1}{40} + 10\times\frac{35}{40\times40} + 10\times\frac{30}{40\times40} ... = 1$

Here we only need to find the number of terms this continues until it reaches 1, and multiply by 10 since the value of each term is due to 10 days of work. There may a term at the end which is due to less than 10 days of work which we will call yz for now. y is the number of days and z is the last term

$\frac{10}{40\times40}(40+35+30+25...)+yz=1$

$\frac{10}{40\times40}(40+35+30+25...)\leq 1$

$\Rightarrow 40+35+30+25...\leq160$

In the LHS of the equation we see that the sum of an arithmetic progression with the first term as 40 and the common difference as -5 is less than or equal to 160. We will now find the highest number of positive terms that will keep the expression under or equal to 160

$\frac{n}{2}\{2a+d(n-1)\}=160$

$\Rightarrow \frac{n}{2}\{2\times40-5(n-1)\}=160$

$\Rightarrow 80n-5n^2+5n=320$

$\Rightarrow -5n^2 + 85n - 320=0$

$\Rightarrow n^2-17n+64=0$

Discriminant = b² - 4ac = 17² - 4×1×64 = 289 - 256 = 33

$n=\frac{-b\pm\sqrt{D}}{2a}$

$\Rightarrow n=\frac{-(-17)+\sqrt{33}}{2},\frac{-(-17)-\sqrt{33}}{2}$

$\Rightarrow n=\frac{17+5.7445}{2},\frac{17-5.7445}{2}$

$= 11.37225,5.62775$

We will choose the latter as the former would involve negative terms which is not possible. Rounding down 5.62775 we get 5, so we find that the first 5 terms would suffice our equation, consequently, z = sixth term = 40 - 5(6 - 1) = 40 - 25 = 15

$yz=1-\frac{10}{40\times40}(40+35+30+25+20)$

$\Rightarrow \frac{y}{40\times40}15=1-\frac{10}{40\times40}(40+35+30+25+20)$

$\Rightarrow 15y = 1600 - 400 - 350 - 300 - 250 - 200 = 100$

$\Rightarrow y=\frac{100}{15}$

Finally the total days needed to finish the work

= 10×5 + 100/15

= 50 + 6.66 = 56.66

Hope it helped. Take care