Question

Given, A, B, and C can reap a field in $15\dfrac{3}{4}$ days, B, C, and D in 14 days, C, D, and A in 18 days, and D, A, and B in 21 days. In what time A, B, C, and D together reap it?

Answer 1

Hint: To solve this question we consider that the total amount of work is 100%. Then, we will find the percentage of work done by A, B, and C in one day, the percentage of work done by B, C, and D in one day, the percentage of work done by C, D, and A in one day, and the percentage of work done by D, A and B in one day. Then, with this information, we will find the percentage of work done in one day when A works alone and then find the number of days required to complete the work.

Complete step by step answer:
Given: - A, B, and C can reap a field in $15\dfrac{3}{4}$ days.
B, C, and D can reap a field in 14 days,
C, D, and A can reap a field in 18 days.
D, A, and B can reap a field in 21 days.
A, B, and C require $15\dfrac{3}{4}$ days to complete 100 % work. So, the percentage of work completed by A, B, and C in one day can found out as:
$ \Rightarrow A + B + C = \dfrac{{100\% }}{{\dfrac{{63}}{4}}}$
Cancel out the common factor,
$ \Rightarrow A + B + C = 6.35\% $ ….. (1)
B, C, and D required 14 days to complete 100 % work. So, the percentage of work completed by B, C, and D in one day can found out as:
$ \Rightarrow B + C + D = \dfrac{{100\% }}{{14}}$
Cancel out the common factor,
$ \Rightarrow B + C + D = 7.14\% $ ….. (2)
C, D, and A required 18 days to complete 100 % work. So, the percentage of work completed by C, D, and A in one day can found out as:
$ \Rightarrow C + D + A = \dfrac{{100\% }}{{18}}$
Cancel out the common factor,
$ \Rightarrow C + D + A = 5.55\% $ ….. (3)
D, A, and B required 21 days to complete 100 % work. So, the percentage of work completed by D, A, and B in one day can found out as:
$ \Rightarrow D + A + B = \dfrac{{100\% }}{{21}}$
Cancel out the common factor,
$ \Rightarrow D + A + B = 4.76\% $ ….. (4)
Now add equation (1), (2), (3), and (4) and find the value of A + B + C + D.
$ \Rightarrow 3A + 3B + 3C = 6.35\% + 7.14\% + 5.55\% + 4.76\% $
Add the terms on the right side,
$ \Rightarrow 3\left( {A + B + C + D} \right) = 23.8\% $
Divide both sides by 3,
$ \Rightarrow A + B + C + D = 7.93\% $
So, 7.93% of work is completed in one day if A, B, C, and D work together.
Therefore, the number of days to complete the work is,
$\therefore \dfrac{{100}}{{7.93}} = 12.6 = 12\dfrac{3}{5}$
Hence, the number of days to complete the work is $12\dfrac{3}{5}$ days.

Note:
This question can be solved in another way.
Step by step answer: -
Given: - A, B, and C can reap a field in $15\dfrac{3}{4}$ days.
B, C, and D can reap a field in 14 days,
C, D, and A can reap a field in 18 days.
D, A, and B can reap a field in 21 days.
A, B, and C require $15\dfrac{3}{4}$ days to complete the work. So, the work completed by A, B, and C in one day can found out as:
$ \Rightarrow \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{1}{{\dfrac{{63}}{4}}}$
Multiply 4 in the numerator,
$ \Rightarrow \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} = \dfrac{4}{{63}}$ ….. (1)
B, C, and D required 14 days to complete the work. So, the work completed by B, C, and D in one day can found out as:
$ \Rightarrow \dfrac{1}{B} + \dfrac{1}{C} + \dfrac{1}{D} = \dfrac{1}{{14}}$ ….. (2)
C, D, and A required 18 days to complete the work. So, the work completed by C, D, and A in one day can found out as:
$ \Rightarrow \dfrac{1}{C} + \dfrac{1}{D} + \dfrac{1}{A} = \dfrac{1}{{18}}$ ….. (3)
D, A, and B required 21 days to complete the work. So, the work completed by D, A, and B in one day can found out as:
$ \Rightarrow \dfrac{1}{D} + \dfrac{1}{A} + \dfrac{1}{B} = \dfrac{1}{{21}}$ ….. (4)
Now add equation (1), (2), (3), and (4) and find the value of A + B + C + D.
$\dfrac{3}{A} + \dfrac{3}{B} + \dfrac{3}{C} + \dfrac{3}{D} = \dfrac{4}{{63}} + \dfrac{1}{{14}} + \dfrac{1}{{18}} + \dfrac{1}{{21}}$
Take LCM on the right side,
$ \Rightarrow 3\left( {\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} + \dfrac{1}{D}} \right) = \dfrac{{8 + 9 + 7 + 6}}{{126}}$
Add the terms on the right side,
$ \Rightarrow 3\left( {\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} + \dfrac{1}{D}} \right) = \dfrac{{30}}{{126}}$
Cancel out the common factors,
$ \Rightarrow 3\left( {\dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} + \dfrac{1}{D}} \right) = \dfrac{5}{{21}}$
Divide both sides by 3,
$ \Rightarrow \dfrac{1}{A} + \dfrac{1}{B} + \dfrac{1}{C} + \dfrac{1}{D} = \dfrac{5}{{63}}$
So, $\dfrac{5}{{63}}$ of work is completed in one day if A, B, C, and D work together.
Therefore, the number of days to complete the work is,
$\therefore \dfrac{1}{{\dfrac{5}{{63}}}} = \dfrac{{63}}{5} = 12\dfrac{3}{5}$
Hence, the number of days to complete the work is $12\dfrac{3}{5}$ days.