If x men working x hours per day can do x units of work in x days, then y men working y hours per day in y days would be able to do k units of work. What is the value of k?
y 3x -2
Understanding the Work and Rate Problem
This problem involves calculating the amount of work done based on the number of workers, the hours they work, and the duration. We are given a scenario with 'x' workers and asked to find the work done ('k') under different conditions with 'y' workers. The key is to understand the relationship between the amount of work done and the factors involved.
Key Relationship in Work Problems
The total amount of work done is directly proportional to the number of men working, the number of hours they work each day, and the number of days they work. We can express this relationship using the following formula:
\[ \frac{W_1}{M_1 \times H_1 \times D_1} = \frac{W_2}{M_2 \times H_2 \times D_2} \]Where:
- \(W_1\) is the work done in the first case.
- \(M_1\) is the number of men in the first case.
- \(H_1\) is the hours per day in the first case.
- \(D_1\) is the number of days in the first case.
- \(W_2\) is the work done in the second case.
- \(M_2\) is the number of men in the second case.
- \(H_2\) is the hours per day in the second case.
- \(D_2\) is the number of days in the second case.
Applying the Formula to the Given Scenario
Let's identify the values from the problem:
- First Case:
- Work done (\(W_1\)): \(x\) units
- Number of men (\(M_1\)): \(x\)
- Hours per day (\(H_1\)): \(x\)
- Number of days (\(D_1\)): \(x\)
- Second Case:
- Work done (\(W_2\)): \(k\) units
- Number of men (\(M_2\)): \(y\)
- Hours per day (\(H_2\)): \(y\)
- Number of days (\(D_2\)): \(y\)
Calculating the Unknown Work (k)
Now, we plug these values into the formula:
\[ \frac{x}{x \times x \times x} = \frac{k}{y \times y \times y} \]Simplify the denominators:
\[ \frac{x}{x^3} = \frac{k}{y^3} \]Using exponent rules (\( \frac{a^m}{a^n} = a^{m-n} \)), we can simplify the left side:
\[ x^{1-3} = \frac{k}{y^3} \] \[ x^{-2} = \frac{k}{y^3} \]To find the value of \(k\), we need to isolate it. Multiply both sides of the equation by \(y^3\):
\[ k = y^3 \times x^{-2} \]This can also be written as:
\[ k = \frac{y^3}{x^2} \]Conclusion
Therefore, the value of \(k\), representing the units of work done by \(y\) men working \(y\) hours per day in \(y\) days, is \(y^3 x^{-2}\). This matches the fourth option provided.
Similar Questions
A, B and C can complete a work in x, 1.5x and 2x days respectively. If they complete the work together, in what ratio should they be paid ?
X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to ?
If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, C and A can finish the same work in 15 days; then in how many days can A, B and C together finish half of the work?
Important Questions from Time and Work
Mohan can do a piece of work in 10 days and Sohan in 15 days. They started working together, but after 3 days Mohan left the work . What time will Sohan take to finish the work?
A tank is filled in 8 hours by three taps A, B and C. The tap C is thrice as fast as B and B is twice as fast as A. How much time will pipe B alone take to fill the tank?
Had been one menless, then the number of days required to do a piece of work would have been one more. If the number of Man. Days required to complete the work is 56, how many workers were there?
A can do a piece of work in 16 hours, B and C can do it in 8 hours while A and C can do it 12 hours. How long will B alone take to do it?
- lf 12 men can do a work in 20 days, in how many days will the work be done by 15 men-
