Q 4. Three men and two women can do a piece of work in four days, while two men and three women can do the same work in five days. If the workers are paid for the units of work
completed and the daily wage of a woman is Rs. 480, then what will be the daily wage of a man?

A. Rs.960 
B. Rs. 1,440 
C. Rs. 1,680 
D. Rs.1,540 
E. Rs.1,760

Let the daily work done by one man be \( M \) and the daily work done by one woman be \( W \).

### Step 1: Set up the work equations
- Three men and two women can finish the work in 4 days:
\[
4 \times (3M + 2W) = 1
\]
This means in one day:
\[
3M + 2W = \frac{1}{4}
\]

- Two men and three women can finish the work in 5 days:
\[
5 \times (2M + 3W) = 1
\]
This means in one day:
\[
2M + 3W = \frac{1}{5}
\]

Now we have the system of equations:
1. \( 3M + 2W = \frac{1}{4} \)
2. \( 2M + 3W = \frac{1}{5} \)

### Step 2: Solve the system of equations
We can use either substitution or elimination. Let's use elimination:

- Multiply the first equation by 3:
\[
9M + 6W = \frac{3}{4}
\]
- Multiply the second equation by 2:
\[
4M + 6W = \frac{2}{5}
\]

Now subtract the second equation from the first:
\[
(9M + 6W) - (4M + 6W) = \frac{3}{4} - \frac{2}{5}
\]
\[
5M = \frac{15}{20} - \frac{8}{20} = \frac{7}{20}
\]
\[
M = \frac{7}{100}
\]

### Step 3: Find \( W \)
Substitute \( M = \frac{7}{100} \) into one of the original equations. Using \( 3M + 2W = \frac{1}{4} \):
\[
3 \times \frac{7}{100} + 2W = \frac{1}{4}
\]
\[
\frac{21}{100} + 2W = \frac{1}{4}
\]
\[
2W = \frac{1}{4} - \frac{21}{100} = \frac{25}{100} - \frac{21}{100} = \frac{4}{100}
\]
\[
W = \frac{2}{100} = \frac{1}{50}
\]

### Step 4: Determine the daily wage of a man
We are told that the daily wage of a woman is Rs. 480, so the wage of a woman is proportional to the work done:
\[
W = \frac{1}{50} \implies \text{Daily wage of woman} = 480 \text{ Rs.}
\]
Since \( M = \frac{7}{100} \), the daily wage of a man is:
\[
\frac{M}{W} = \frac{\frac{7}{100}}{\frac{1}{50}} = \frac{7}{2}
\]
Thus, the daily wage of a man is:
\[
\frac{7}{2} \times 480 = 7 \times 240 = 1680 \text{ Rs.}
\]

### Final Answer:
The daily wage of a man is Rs. 1,680.

**Answer: C. Rs. 1,680**

---
Would you like further details on any step? Here are 5 related questions to explore:

1. How is the proportionality between work done and wages determined?
2. What methods can be used to solve systems of equations?
3. How can this problem be solved using matrix methods?
4. Can the method of substitution be applied more effectively in this case?
5. How does the number of days affect the system of work equations?

**Tip:** Always double-check units (like wages or time) when solving real-life problems involving rates.

Three men and two women can do a piece of work in four days, while two men and three women can do the same work in five days. If the workers are paid for the units of work completed and the daily wage of a woman is Rs. 480, then what will be the daily wage of a man? A. Rs.960 B. Rs. 1,440 C. Rs. 1,680 D. Rs.1,540 E. Rs.1,760

Learn how to calculate the daily wage of a man when given the work rates of three men and two women, along with their payment structure. This problem involves solving a system of linear equations using the elimination method and understanding proportional wages. Suitable for Grades 10-12, this step-by-step solution covers key algebraic concepts.

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