The female- male ratio of a village decreases continuously at the rate proportional to their ratio at any time. If the ratio of female : male of the villages was 980:1000 in 2001 and 920 : 1000 in 2011. What will be the ratio in 2021 ?

Options

A. 864:1000
B. 864 : 100
C. 1000 : 864
D. 100 : 864

Answer

To solve the problem of the continuously decreasing female-male ratio in a village, we will follow these steps:

### Step 1: Define the variables and the differential equation
Let \( R(t) = \frac{F(t)}{M(t)} \) be the female to male ratio at time \( t \). According to the problem, the rate of change of this ratio is proportional to the ratio itself. We can express this as:
\[ \frac{dR}{dt} = -kR \]
where \( k \) is a positive constant.

### Step 2: Solve the differential equation
This is a separable differential equation. We can rearrange it:
\[ \frac{dR}{R} = -k \, dt \]
Integrating both sides gives:
\[ \ln |R| = -kt + C \]
Exponentiating both sides results in:
\[ R = e^{-kt + C} = e^C e^{-kt} \]
Let \( C_1 = e^C \), so we have:
\[ R(t) = C_1 e^{-kt} \]

### Step 3: Use initial conditions to find constants
From the problem, we know:
- In 2001 (let's set \( t = 0 \)), \( R(0) = \frac{980}{1000} = 0.98 \).
- In 2011 (which is \( t = 10 \)), \( R(10) = \frac{920}{1000} = 0.92 \).

Using the initial condition:
\[ R(0) = C_1 e^{0} = C_1 = 0.98 \]
Thus, we have:
\[ R(t) = 0.98 e^{-kt} \]

### Step 4: Use the second condition to find \( k \)
Using \( R(10) = 0.92 \):
\[ 0.92 = 0.98 e^{-10k} \]
Dividing both sides by 0.98 gives:
\[ \frac{0.92}{0.98} = e^{-10k} \]
Taking the natural logarithm:
\[ \ln\left(\frac{0.92}{0.98}\right) = -10k \]
Thus,
\[ k = -\frac{1}{10} \ln\left(\frac{0.92}{0.98}\right) \]

### Step 5: Find the ratio in 2021
Now we need to find \( R(20) \):
\[ R(20) = 0.98 e^{-20k} \]
Substituting \( k \):
\[ R(20) = 0.98 e^{-20 \left(-\frac{1}{10} \ln\left(\frac{0.92}{0.98}\right)\right)} = 0.98 \left(\frac{0.92}{0.98}\right)^2 \]
Calculating:
\[ R(20) = 0.98 \cdot \frac{0.92^2}{0.98^2} = \frac{0.98 \cdot 0.8464}{0.9604} \]
Calculating the final ratio:
\[ R(20) \approx \frac{0.828032}{0.9604} \approx 0.861 \]
Thus, the ratio of female to male in 2021 is approximately \( \frac{861}{1000} \).

### Final Answer:
The ratio of females to males in 2021 is approximately \( 861:1000 \).