Thirty men can do a piece of work in 16 days working 8 hrs a day. How many men are needed to complete another work, which is twice the first one, in 10 days working 12 hrs a day? The following are the steps involved in solving the above problem. Arrange them in sequential order. A) M 2 = \(\rm\frac{30 \times 16 \times 8 \times 2x}{x \times 12 \times 10}\) B) \(\rm\frac{30 \times 16 \times 8}{x}=\frac{M_2 \times 12 \times 10}{2 x}\) C) \(\rm\frac{M_1 D_1 H_1}{ W_1}=\frac{M_2 D_2 H_2}{W_2}\) D) M 2 = 64
CBAD
Understanding Work and Time Problems
This problem is a classic example of work and time, specifically involving the relationship between the number of workers, the time taken (days and hours per day), and the amount of work done. When solving such problems, we use a formula that relates these quantities.
The General Formula for Work, Men, Days, and Hours
The fundamental principle is that the total work done is proportional to the number of men, the number of days, and the number of hours worked per day. This relationship can be expressed using the formula:
\(\rm\frac{M_1 D_1 H_1}{ W_1}=\frac{M_2 D_2 H_2}{W_2}\)
Where:
- \(M_1\) and \(M_2\) are the number of men in the first and second cases, respectively.
- \(D_1\) and \(D_2\) are the number of days taken in the first and second cases, respectively.
- \(H_1\) and \(H_2\) are the number of hours worked per day in the first and second cases, respectively.
- \(W_1\) and \(W_2\) are the amount of work done in the first and second cases, respectively.
Step C provides this general formula.
Applying the Work and Time Formula
Let's identify the values given in the problem:
Case 1:
- Number of men (\(M_1\)) = 30
- Number of days (\(D_1\)) = 16
- Hours per day (\(H_1\)) = 8 hrs
- Amount of work (\(W_1\)) = Assume this is 'x'
Case 2:
- Number of men (\(M_2\)) = ? (This is what we need to find)
- Number of days (\(D_2\)) = 10
- Hours per day (\(H_2\)) = 12 hrs
- Amount of work (\(W_2\)) = Twice the first work, so \(2x\)
Now, we substitute these values into the general formula \(\rm\frac{M_1 D_1 H_1}{ W_1}=\frac{M_2 D_2 H_2}{W_2}\):
\(\rm\frac{30 \times 16 \times 8}{x}=\frac{M_2 \times 10 \times 12}{2 x}\)
This corresponds to Step B.
Solving for the Number of Men (\(M_2\))
We now have the equation from Step B. To find \(M_2\), we need to rearrange this equation:
\(\rm\frac{30 \times 16 \times 8}{x}=\frac{M_2 \times 10 \times 12}{2 x}\)
Multiply both sides by \(2x\) to isolate the terms with \(M_2\):
\(\rm \left(\frac{30 \times 16 \times 8}{x}\right) \times 2x = M_2 \times 10 \times 12\)
\(\rm 30 \times 16 \times 8 \times 2 = M_2 \times 10 \times 12\)
Now, divide by \(10 \times 12\) to solve for \(M_2\):
\(\rm M_2 = \frac{30 \times 16 \times 8 \times 2}{10 \times 12}\)
This calculation setup matches Step A (where the terms were already grouped as presented in the step).
Calculating the Final Answer for \(M_2\)
Let's perform the calculation shown in Step A:
\(\rm M_2 = \frac{30 \times 16 \times 8 \times 2}{10 \times 12}\)
We can simplify this expression:
\(\rm M_2 = \frac{30 \times 16 \times 16}{120}\)
Divide 30 by 10, which is 3:
\(\rm M_2 = \frac{3 \times 16 \times 16}{12}\)
Divide 12 by 3, which is 4:
\(\rm M_2 = \frac{16 \times 16}{4}\)
Divide 16 by 4, which is 4:
\(\rm M_2 = 4 \times 16\)
\(\rm M_2 = 64\)
This gives us the final number of men needed, which is 64. This result matches Step D.
Sequential Order of Steps
Based on our analysis, the steps follow this logical order:
- State the general formula (Step C).
- Substitute the given values into the formula (Step B).
- Rearrange the formula to solve for the unknown (\(M_2\)) (Step A).
- Calculate the final numerical result (Step D).
Therefore, the sequential order of the steps is C, B, A, D.
Revision Table: Work and Time Formula
| Component | Symbol | Represents | Relationship to Work |
|---|---|---|---|
| Men | M | Number of workers | Directly proportional |
| Days | D | Number of days worked | Directly proportional |
| Hours per Day | H | Hours worked each day | Directly proportional |
| Work Done | W | Amount of work completed | Directly proportional to M, D, H product |
Additional Information: Variations of Work and Time Problems
Work and time problems can involve different scenarios. Some common variations include:
- Problems involving efficiency: Workers with different efficiencies working together.
- Problems involving pipes and cisterns: Filling or emptying tanks (similar to work done).
- Problems where work is done by a varying number of people over time.
The core principle remains the same: the total amount of work done is the key factor in relating the variables.
Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
