Study the following information carefully to answer the given questions
The total population of three villages A, B and C together is 80,000 and the ratio of their population is `5:4:7`.
Out of the total population of village A, the age of `16%`of villagers is equal to 61 years or more the age of `36%` of villagers is equal to or more than 31 years but less than 61 years and the age of the remaining villagers is less than 31 years.
Out of the total population of village B, the age of one- fifth of villagers is equal to 61 years or more. The age of `12/25` of the remaining villagers is equal to or more than 31 years but less than 61 years and the age of the remaining villagers is less than 31 years
In village C, the number of villagers whose age is equal to 61 years or more is `50%` more than the difference between the number of villagers in village A whose age is less than31 years and that in village B. The number of villagers whose age is equal to or more than 31 years but less than 61 years is `80%` more than that in village A in the same age group. The age of the remaining villagers is less than 31 years
By what per cent is the population of village B whose age is equal to or greater than 31 years less than the population of village A in .the same age group?

Options

A. `13 3/7`
B. `14 2/7`
C. `11 2/13`
D. `10 2/13`

Answer

To solve the problem step by step, we will first determine the populations of villages A, B, and C based on the given ratio and total population. Then, we will analyze the age distribution in each village and finally calculate the required percentage.

### Step 1: Calculate the populations of villages A, B, and C
The total population of the three villages is 80,000 and the ratio of their populations is 5:4:7.

Let the populations of villages A, B, and C be represented as:
- A = 5x
- B = 4x
- C = 7x

From the equation:
\[ 5x + 4x + 7x = 80000 \]
\[ 16x = 80000 \]
\[ x = 5000 \]

Now we can find the populations:
- Population of A = \( 5x = 5 \times 5000 = 25000 \)
- Population of B = \( 4x = 4 \times 5000 = 20000 \)
- Population of C = \( 7x = 7 \times 5000 = 35000 \)

### Step 2: Analyze the age distribution in village A
In village A:
- 16% are 61 years or more:
\[ 0.16 \times 25000 = 4000 \]
- 36% are 31 years or more but less than 61 years:
\[ 0.36 \times 25000 = 9000 \]
- Remaining villagers (less than 31 years):
\[ 25000 - (4000 + 9000) = 25000 - 13000 = 12000 \]

### Step 3: Analyze the age distribution in village B
In village B:
- One-fifth are 61 years or more:
\[ \frac{1}{5} \times 20000 = 4000 \]
- The remaining population is:
\[ 20000 - 4000 = 16000 \]
- \( \frac{12}{25} \) of the remaining villagers are 31 years or more but less than 61 years:
\[ \frac{12}{25} \times 16000 = 7680 \]
- Remaining villagers (less than 31 years):
\[ 16000 - 7680 = 8320 \]

### Step 4: Calculate the populations of villagers in age groups for village B
- Population of B (age 31 or more):
\[ 4000 + 7680 = 11680 \]

### Step 5: Calculate the populations of villagers in age groups for village A
- Population of A (age 31 or more):
\[ 9000 + 4000 = 13000 \]

### Step 6: Determine the percentage difference
We need to find by what percent the population of village B (age 31 or more) is less than the population of village A (age 31 or more).

Let the difference be:
\[ \text{Difference} = 13000 - 11680 = 1320 \]

Now, calculate the percentage:
\[ \text{Percentage} = \left( \frac{\text{Difference}}{\text{Population of A (age 31 or more)}} \right) \times 100 \]
\[ \text{Percentage} = \left( \frac{1320}{13000} \right) \times 100 \]
\[ \text{Percentage} = 10.15\% \]

### Final Answer
The population of village B whose age is equal to or greater than 31 years is approximately **10.15%** less than the population of village A in the same age group.