A certain number of men, twice as many women and thrice as many boys earn in 6 days Rs. 5100. A woman earns one and a half times as a boy and a man as much as a woman and a boy together per day. How many women were there, if a boy earned Rs. 25 daily.

Options

A. 4
B. 7
C. 12
D. 36

Answer

To solve the problem step by step, let's break it down systematically.

### Step 1: Define Variables
Let:
- \( X \) = number of men
- \( Y \) = number of women
- \( Z \) = number of boys

According to the problem:
- Number of women = \( 2Y \)
- Number of boys = \( 3Z \)

### Step 2: Earnings of Each Type
Given that a boy earns Rs. 25 daily, we can find the earnings of women and men:
- Earnings of a boy per day = Rs. 25
- Earnings of a woman per day = \( 1.5 \times 25 = Rs. 37.5 \)
- Earnings of a man per day = Earnings of a woman + Earnings of a boy = \( 37.5 + 25 = Rs. 62.5 \)

### Step 3: Total Earnings in 6 Days
The total earnings in 6 days is Rs. 5100. The total earnings can be expressed as:
\[ \text{Total Earnings} = \text{(Number of Men)} \times \text{(Earnings of a Man)} \times 6 + \text{(Number of Women)} \times \text{(Earnings of a Woman)} \times 6 + \text{(Number of Boys)} \times \text{(Earnings of a Boy)} \times 6 \]
Substituting the values we have:
\[ 5100 = X \times 62.5 \times 6 + 2Y \times 37.5 \times 6 + 3Z \times 25 \times 6 \]

### Step 4: Simplify the Equation
Dividing the entire equation by 6:
\[ 850 = X \times 62.5 + 2Y \times 37.5 + 3Z \times 25 \]

### Step 5: Substitute Relationships
From the problem, we know:
- \( Z = 3Y \) (since there are thrice as many boys as women)

Now substituting \( Z \) in terms of \( Y \):
\[ 850 = X \times 62.5 + 2Y \times 37.5 + 3(3Y) \times 25 \]
\[ 850 = X \times 62.5 + 2Y \times 37.5 + 9Y \times 25 \]

### Step 6: Combine Like Terms
Now, let's combine the terms involving \( Y \):
\[ 850 = X \times 62.5 + (2 \times 37.5 + 9 \times 25)Y \]
Calculating the coefficients:
\[ 850 = X \times 62.5 + (75 + 225)Y \]
\[ 850 = X \times 62.5 + 300Y \]

### Step 7: Rearranging the Equation
Rearranging gives us:
\[ X \times 62.5 + 300Y = 850 \]

### Step 8: Express \( X \) in terms of \( Y \)
From the equation, we can express \( X \):
\[ X \times 62.5 = 850 - 300Y \]
\[ X = \frac{850 - 300Y}{62.5} \]

### Step 9: Finding Integer Solutions
To find the number of women \( Y \), we need \( X \) to be a non-negative integer. We can try different values for \( Y \) until \( X \) is a non-negative integer.

### Step 10: Testing Values for \( Y \)
Let's test \( Y = 1, 2, 3, \ldots \) until we find a suitable integer for \( X \):
- For \( Y = 1 \):
\[ X = \frac{850 - 300 \times 1}{62.5} = \frac{550}{62.5} = 8.8 \quad \text{(not an integer)} \]
- For \( Y = 2 \):
\[ X = \frac{850 - 300 \times 2}{62.5} = \frac{250}{62.5} = 4 \quad \text{(integer)} \]
- For \( Y = 3 \):
\[ X = \frac{850 - 300 \times 3}{62.5} = \frac{-50}{62.5} \quad \text{(negative)} \]

### Conclusion
Thus, the only valid solution is \( Y = 2 \). Therefore, the number of women is **2**.