The total population of a village is 5000. The number of males and females increases by 10% and 15% respectively and consequently the population of the village becomes 5600. What was the number of males in the village ?

Options

A. 2000
B. 2500
C. 3000
D. 4000

Answer

To solve the problem step by step, we need to find the number of males in the village given the total population and the percentage increase in the male and female populations.

### Step 1: Define Variables
Let:
- \( M \) = Number of males in the village
- \( F \) = Number of females in the village

From the problem, we know that:
\[ M + F = 5000 \] (Equation 1)

### Step 2: Calculate the Increased Population
The male population increases by 10%, and the female population increases by 15%. Therefore, the new populations can be expressed as:
- New male population = \( M + 0.10M = 1.10M \)
- New female population = \( F + 0.15F = 1.15F \)

The total new population is given as 5600:
\[ 1.10M + 1.15F = 5600 \] (Equation 2)

### Step 3: Solve the Equations
Now we have two equations:
1. \( M + F = 5000 \) (Equation 1)
2. \( 1.10M + 1.15F = 5600 \) (Equation 2)

From Equation 1, we can express \( F \) in terms of \( M \):
\[ F = 5000 - M \]

### Step 4: Substitute \( F \) in Equation 2
Substituting \( F \) in Equation 2:
\[ 1.10M + 1.15(5000 - M) = 5600 \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ 1.10M + 5750 - 1.15M = 5600 \]
Combine like terms:
\[ -0.05M + 5750 = 5600 \]

### Step 6: Isolate \( M \)
Now, isolate \( M \):
\[ -0.05M = 5600 - 5750 \]
\[ -0.05M = -150 \]
\[ M = \frac{-150}{-0.05} \]
\[ M = 3000 \]

### Step 7: Conclusion
Thus, the number of males in the village is:
\[ \boxed{3000} \]

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