Answer: Option C. -> 8 days

(A + B + C)'s 1 day's work = \(\frac{1}{6}\)

(A + B)'s 1 day's work = \(\frac{1}{8}\)

(B + C)'s 1 day's work = \(\frac{1}{12}\)

So , (A + C)'s 1 day's work  = \(\left(2\times \frac{1}{6}\right)-\left(\frac{1}{8}+\frac{1}{12}\right)\)

= \(\left(\frac{1}{3}-\frac{5}{24}\right)\)

= \(\frac{3}{24}\)

= \(\frac{1}{8}\)

So, A and C together will do the work in 8 days.

Answer: Option C. -> 10 days

(B + C)'s 1 day's work = \(\left(\frac{1}{9}+\frac{1}{12}\right)=\frac{7}{36} \)

Work done by B and C in 3 days = \(\left(\frac{7}{36}\times3\right)=\frac{7}{12} \)

Remaining work =  \(\left(1-\frac{7}{12}\right)= \frac{5}{12}\)

Now,   \(\frac{1}{24} \)    work is done by A in 1 day.

So,  \(\frac{5}{12}\)   work is done by A in \(\left(24\times\frac{5}{12}\right)= 10 days.\)

Answer: Option A. -> \(13\frac{1}{3}days\)

Work done by X in 8 days = \(\left(\frac{1}{40}\times8\right)=\frac{1}{5}\)

Remaining work = \(\left(1-\frac{1}{5}\right)=\frac{4}{5}\)

Now,   \(\frac{4}{5}\)  work is done by Y in 16 days.

Whole work will be done by Y in \(\left(16\times\frac{5}{4}\right)= 20 days.\)

So, X's 1 day's work =  \(\frac{1}{40}\) , Y's 1 day's work = \(\frac{1}{20}\)  

(X + Y)'s 1 day's work = \(\left(\frac{1}{40}+\frac{1}{20}\right) = \frac{3}{40}\)

Hence, X and Y will together complete the work in \(\left(\frac{40}{3}\right) = 13\frac{1}{3}days.\)

Answer: Option B. -> 11 days

(A's 1 day's work) : (B's 1 day's work) = \(\frac{7}{4}:1 = 7:4\)

Let A's and B's 1 day's work be 7x and 4x respectively.

Then, 7x + 4x =   \(\frac{1}{7}\Rightarrow 11x =\frac{1}{7}\Rightarrow x=\frac{1}{77}\)

So, A's 1 day's work =   \(\left(\frac{1}{77}\times7\right) = \frac{1}{11}\)

Answer: Option C. -> 60 days

Let A's 1 day's work = x and B's 1 day's work = y.

Then , x+y =   \(\frac{1}{30}\)   and 16x + 44y = 1.

lving these two equations, we get: x =    \(\frac{1}{60}\) and y =  \(\frac{1}{60}\)

Therefore , B's 1 day's work = \(\frac{1}{60}\)

Hence, B alone shall finish the whole work in 60 days.

Answer: Option C. -> Rs 6
Let the daily wage of a man, a woman, and a boy be m, w, and b, respectively.
From the given statement, we can form the following equations:
5m = w (because 5 men are equal to as many women)w = 8b (because as many women are equal to 8 boys)
Substituting the second equation in the first, we get:
5m = 8b
Or, m = (8/5)b
Now, we can write the total daily wage as:
5m + w + 8b = 90
Substituting the value of w from the second equation, we get:
5m + 8b + 8b = 90
Or, 5m + 16b = 90
Substituting the value of m from the earlier equation, we get:
5(8/5)b + 16b = 90
Or, 8b + 16b = 90
Or, 24b = 90
Or, b = 90/24 = 15/4
Substituting the value of b in the equation for m, we get:
m = (8/5) x (15/4) = 6
Therefore, the daily wage of a man is Rs 6, which is option (C).

Answer: Option C. -> Rs 2700, Rs 1800

Answer: Option B. -> 16 day

Answer: Option C. -> Rs 255 , Rs 150 , Rs 75

Answer: Option B. -> Rs 115

Let's assume the daily wage of the worker to be x.
Total wage for n days = n*x
Given, the worker is engaged for a certain number of days for Rs 1725, therefore we have:
n*x = 1725 ...(1)
Now, the worker was absent for 7 days, so the total number of days worked would be (n-7).
Given, the worker was paid Rs 920, so we have:
(n-7)*x = 920 ...(2)
On solving equations (1) and (2), we get:
n = 23 and x = 75
Therefore, the daily wage of the worker is Rs 75.

Explanation:
Let's assume the daily wage of the worker to be x.
The total wage paid for n days is given as Rs 1725. Therefore, we get the equation:
n*x = 1725
The worker was absent for 7 days. Therefore, the total number of days worked by the worker is (n-7). The amount paid to the worker is given as Rs 920. Therefore, we get the equation:
(n-7)*x = 920
On solving the above two equations, we get:
n = 23 and x = 75
Therefore, the daily wage of the worker is Rs 75.
Some relevant definitions and formulas used in the solution are:
Linear equation: An equation of the form "ax + b = c", where "a", "b", and "c" are constants and "x" is a variable.
System of linear equations: Two or more linear equations with the same variables.
Substitution method: A method for solving a system of linear equations by solving one equation for one variable and substituting the result into the other equation(s).
Variable: A symbol or letter used to represent an unknown quantity in an equation or expression.

If you think the solution is wrong then please provide your own solution below in the comments section .