In acertain year, the population of certain town was 9000. If in the next year the population of males increase by 5% and that of the females by 8% and the total population increase to 9600. then what was the ratio of population of males and females in that given year?

Options

A. `4:5`
B. `5:4`
C. `2:3`
D. `4:6`

Explanation

To solve the problem step by step, we need to find the ratio of the population of males and females in the given year based on the information provided.

### Step 1: Understand the problem
The total population in a certain year is 9000. The next year, the male population increases by 5% and the female population increases by 8%, resulting in a total population of 9600.

### Step 2: Calculate the total increase in population
The increase in population from the first year to the second year is:
\[ \text{Total increase} = 9600 - 9000 = 600 \]

### Step 3: Let the population of males and females be represented
Let:
- \( M \) = population of males in the first year
- \( F \) = population of females in the first year

From the problem, we know:
\[ M + F = 9000 \]

### Step 4: Calculate the population after increases
After the increases, the population becomes:
- Males: \( M + 0.05M = 1.05M \)
- Females: \( F + 0.08F = 1.08F \)

The total population after the increases is:
\[ 1.05M + 1.08F = 9600 \]

### Step 5: Set up the equations
Now we have two equations:
1. \( M + F = 9000 \)
2. \( 1.05M + 1.08F = 9600 \)

### Step 6: Solve the first equation for one variable
From the first equation, we can express \( F \) in terms of \( M \):
\[ F = 9000 - M \]

### Step 7: Substitute into the second equation
Substituting \( F \) in the second equation:
\[ 1.05M + 1.08(9000 - M) = 9600 \]

### Step 8: Simplify the equation
Expanding and simplifying:
\[ 1.05M + 9720 - 1.08M = 9600 \]
\[ -0.03M + 9720 = 9600 \]
\[ -0.03M = 9600 - 9720 \]
\[ -0.03M = -120 \]

### Step 9: Solve for \( M \)
Dividing both sides by -0.03:
\[ M = \frac{-120}{-0.03} = 4000 \]

### Step 10: Find \( F \)
Now substitute \( M \) back to find \( F \):
\[ F = 9000 - 4000 = 5000 \]

### Step 11: Find the ratio of males to females
Now we can find the ratio of males to females:
\[ \text{Ratio of males to females} = \frac{M}{F} = \frac{4000}{5000} = \frac{4}{5} \]

### Final Answer
The ratio of the population of males to females in that given year is \( 4:5 \).

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