The population of a town is 53,000. If in a year the number of males was to increases by 6% and that of the females by 4%, the population will grow to 55,630. Find the difference between the number of males and females in town at present

Options

A. 3000
B. 4000
C. 2000
D. 5000

Answer

To solve the problem, we will set up a system of equations based on the information provided.

### Step 1: Define Variables
Let:
- \( X \) = number of males in the town
- \( Y \) = number of females in the town

### Step 2: Set Up the First Equation
According to the problem, the total population of the town is 53,000. Therefore, we can write the first equation as:
\[ X + Y = 53,000 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that the number of males increases by 6% and the number of females increases by 4%, leading to a new total population of 55,630. We can express this as:
\[ X + 0.06X + Y + 0.04Y = 55,630 \]
This simplifies to:
\[ 1.06X + 1.04Y = 55,630 \quad \text{(Equation 2)} \]

### Step 4: Eliminate Decimals
To eliminate the decimals in Equation 2, we can multiply the entire equation by 100:
\[ 106X + 104Y = 5,563,000 \quad \text{(Equation 3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( X + Y = 53,000 \) (Equation 1)
2. \( 106X + 104Y = 5,563,000 \) (Equation 3)

We can express \( Y \) from Equation 1:
\[ Y = 53,000 - X \]

### Step 6: Substitute \( Y \) in Equation 3
Substituting \( Y \) in Equation 3:
\[ 106X + 104(53,000 - X) = 5,563,000 \]
Expanding this gives:
\[ 106X + 5,512,000 - 104X = 5,563,000 \]
Combining like terms:
\[ 2X + 5,512,000 = 5,563,000 \]
Subtracting \( 5,512,000 \) from both sides:
\[ 2X = 51,000 \]
Dividing by 2:
\[ X = 25,500 \]

### Step 7: Find \( Y \)
Now substituting \( X \) back into Equation 1 to find \( Y \):
\[ Y = 53,000 - 25,500 = 27,500 \]

### Step 8: Find the Difference
Finally, we find the difference between the number of males and females:
\[ |X - Y| = |25,500 - 27,500| = 2,000 \]

### Final Answer
The difference between the number of males and females in the town at present is \( 2,000 \).

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