There are three sections A,B and C in a class .Every section has some boy and some girl students in it .Probability of a girl being selected when one student is selected randomly from section A is `(2)/(5)` , that from section B is `(4)/(9)` and that from section C is `(5)/(9)`.
If the number of girls in sections A is same as the number of boys in section C , then what is the ratio of number of boys in section A to the number of boys in section C ?

Options

A. A)`2 :3`
B. B)`3 :4`
C. C)`3 :2`
D. D)`4 :3`

Answer

To solve the problem step by step, we will use the given probabilities to find the number of boys and girls in each section and then determine the required ratio.

### Step 1: Understand the probabilities
The probability of selecting a girl from each section is given as follows:
- Section A: \( P(\text{girl}) = \frac{2}{5} \)
- Section B: \( P(\text{girl}) = \frac{4}{9} \)
- Section C: \( P(\text{girl}) = \frac{5}{9} \)

### Step 2: Set up equations for the number of boys and girls
Let:
- \( G_A \) = number of girls in section A
- \( B_A \) = number of boys in section A
- \( G_B \) = number of girls in section B
- \( B_B \) = number of boys in section B
- \( G_C \) = number of girls in section C
- \( B_C \) = number of boys in section C

From the probabilities, we can express the number of girls and boys in each section:
1. For Section A:
\[ \frac{G_A}{G_A + B_A} = \frac{2}{5} \]
This implies:
\[ 5G_A = 2(G_A + B_A) \implies 5G_A = 2G_A + 2B_A \implies 3G_A = 2B_A \implies B_A = \frac{3}{2}G_A \]

2. For Section B:
\[ \frac{G_B}{G_B + B_B} = \frac{4}{9} \]
This implies:
\[ 9G_B = 4(G_B + B_B) \implies 9G_B = 4G_B + 4B_B \implies 5G_B = 4B_B \implies B_B = \frac{5}{4}G_B \]

3. For Section C:
\[ \frac{G_C}{G_C + B_C} = \frac{5}{9} \]
This implies:
\[ 9G_C = 5(G_C + B_C) \implies 9G_C = 5G_C + 5B_C \implies 4G_C = 5B_C \implies B_C = \frac{4}{5}G_C \]

### Step 3: Use the condition given in the problem
We know that the number of girls in section A is the same as the number of boys in section C:
\[ G_A = B_C \]
Substituting \( B_C \) from the equation we derived:
\[ G_A = \frac{4}{5}G_C \]

### Step 4: Express \( G_C \) in terms of \( G_A \)
From the equation \( G_A = \frac{4}{5}G_C \):
\[ G_C = \frac{5}{4}G_A \]

### Step 5: Substitute \( G_C \) back into the equation for \( B_C \)
Now substituting \( G_C \) into the equation for \( B_C \):
\[ B_C = \frac{4}{5}G_C = \frac{4}{5} \cdot \frac{5}{4}G_A = G_A \]

### Step 6: Find the ratio of boys in section A to boys in section C
Now we have:
- From Section A: \( B_A = \frac{3}{2}G_A \)
- From Section C: \( B_C = G_A \)

Now we can find the ratio:
\[ \text{Ratio of } B_A \text{ to } B_C = \frac{B_A}{B_C} = \frac{\frac{3}{2}G_A}{G_A} = \frac{3}{2} \]

### Final Answer
Thus, the ratio of the number of boys in section A to the number of boys in section C is:
\[ \boxed{3:2} \]