Understanding the Work Problem

This question is a classic example of a time and work problem involving multiple groups of workers with different efficiencies (men and women). The key is to figure out the individual work rate of one man and one woman per day and then use that to calculate the total time for a different group.

Setting Up Equations for Work Rate

Let's assume:

• The amount of work one man can do in one day is \(m\).
• The amount of work one woman can do in one day is \(w\).

The total work is considered as 1 unit.

According to the first statement, "Two men and 7 women can complete a work in 28 days". This means the total work done by (2 men + 7 women) in one day is \(\frac{1}{28}\) of the total work. We can write this as an equation:

\(2m + 7w = \frac{1}{28}\) (Equation 1)

According to the second statement, "6 men and 16 women can do the same work in 11 days". This means the total work done by (6 men + 16 women) in one day is \(\frac{1}{11}\) of the total work. We can write this as another equation:

\(6m + 16w = \frac{1}{11}\) (Equation 2)

Solving for Individual Work Rates

Now we have a system of two linear equations with two variables (m and w). We need to solve these equations simultaneously to find the values of \(m\) and \(w\).

Equation 1: \(2m + 7w = \frac{1}{28}\)

Equation 2: \(6m + 16w = \frac{1}{11}\)

We can multiply Equation 1 by 3 to make the coefficient of \(m\) the same as in Equation 2:

\(3 \times (2m + 7w) = 3 \times \frac{1}{28}\)

\(6m + 21w = \frac{3}{28}\) (Equation 3)

Now, subtract Equation 2 from Equation 3:

\((6m + 21w) - (6m + 16w) = \frac{3}{28} - \frac{1}{11}\)

\(5w = \frac{3 \times 11 - 1 \times 28}{28 \times 11}\)

\(5w = \frac{33 - 28}{308}\)

\(5w = \frac{5}{308}\)

Dividing both sides by 5:

\(w = \frac{5}{308 \times 5} = \frac{1}{308}\)

So, one woman can complete \(\frac{1}{308}\) of the work in one day.

Now substitute the value of \(w\) into Equation 1 to find \(m\):

\(2m + 7w = \frac{1}{28}\)

\(2m + 7 \times \frac{1}{308} = \frac{1}{28}\)

\(2m + \frac{7}{308} = \frac{1}{28}\)

Simplify the fraction \(\frac{7}{308}\): \(308 \div 7 = 44\). So, \(\frac{7}{308} = \frac{1}{44}\).

\(2m + \frac{1}{44} = \frac{1}{28}\)

Subtract \(\frac{1}{44}\) from both sides:

\(2m = \frac{1}{28} - \frac{1}{44}\)

Find a common denominator for 28 and 44. The least common multiple is 308. \(28 \times 11 = 308\), \(44 \times 7 = 308\).

\(2m = \frac{1 \times 11}{28 \times 11} - \frac{1 \times 7}{44 \times 7}\)

\(2m = \frac{11}{308} - \frac{7}{308}\)

\(2m = \frac{11 - 7}{308} = \frac{4}{308}\)

Simplify the fraction \(\frac{4}{308}\): \(308 \div 4 = 77\). So, \(\frac{4}{308} = \frac{1}{77}\).

\(2m = \frac{1}{77}\)

Divide both sides by 2:

\(m = \frac{1}{77 \times 2} = \frac{1}{154}\)

So, one man can complete \(\frac{1}{154}\) of the work in one day.

Calculating Work by 7 Men per Day

We need to find out how many days it takes for 7 men to complete the same work. First, let's find the total work done by 7 men in one day.

Work done by 1 man in 1 day = \(\frac{1}{154}\)

Work done by 7 men in 1 day = \(7 \times m = 7 \times \frac{1}{154}\)

Simplify the fraction: \(154 \div 7 = 22\).

Work done by 7 men in 1 day = \(\frac{7}{154} = \frac{1}{22}\)

So, 7 men can complete \(\frac{1}{22}\) of the total work in one day.

Finding the Time Taken by 7 Men

If 7 men complete \(\frac{1}{22}\) of the work in one day, the total number of days required to complete the entire work (1 unit) is the reciprocal of the work done per day by 7 men.

\text{Time taken} = \frac{\text{Total Work}}{\text{Work done per day}}

\text{Time taken by 7 men} = \frac{1}{\frac{1}{22}} = 1 \times 22 = 22\)

Therefore, 7 men can complete the same work in 22 days.

Summary of Steps

Here's a quick summary of the steps taken:

1. Defined variables \(m\) and \(w\) for the daily work rate of a man and a woman.
2. Set up two linear equations based on the given information.
3. Solved the system of equations to find the individual work rates (\(m\) and \(w\)).
4. Calculated the combined work rate of 7 men per day.
5. Determined the total time required for 7 men to complete the work.

Revision Table: Key Concepts in Time and Work

Additional Information on Solving Work Problems

Time and work problems often involve scenarios where individuals or groups work at different rates. The core idea is to standardize the work done per unit of time (usually per day). Here are some common variations and approaches:

• Individual vs. Group Work: Problems might give times for individuals or groups and ask for times for different combinations. Always find the individual work rate first if possible.
• Working Together: When people work together, their daily work rates add up. If A does \(1/a\) work per day and B does \(1/b\) per day, together they do \((1/a + 1/b)\) work per day.
• Working Alternatively: Sometimes, people work one after another. Calculate the work done in one cycle (e.g., A works day 1, B works day 2) and the number of days per cycle.
• Leaving or Joining: If workers leave or join mid-task, calculate the work done before the change and the remaining work. Then, calculate the time for the remaining work with the new group composition.
• Efficiency Ratios: Problems might give efficiency ratios (e.g., A is twice as efficient as B). This directly relates their work rates (Rate of A = 2 × Rate of B).

Solving these problems relies on converting the given information into daily work rates and then manipulating these rates to find the required time or number of workers.