If `B_1, B_2` and `B_3` are three kinds of birds on a tree in the ratio `3 : 7 : 5` and number of `B_2` birds are more than `B_1` birds by a multiple of `9` and `7` both, then minimum number of birds on the tree are:

Options

A. 630
B. 238
C. 942
D. 945

Answer

To solve the problem step by step, we will follow the given information and derive the minimum number of birds on the tree.

### Step 1: Understand the Ratios
We are given that the number of birds of types B1, B2, and B3 are in the ratio of 3:7:5.
Let the number of birds be represented as:
- B1 = 3X
- B2 = 7X
- B3 = 5X

### Step 2: Set Up the Condition
We know that the number of B2 birds is more than the number of B1 birds by a multiple of both 9 and 7. The least common multiple (LCM) of 9 and 7 is 63. Therefore, we can express this condition as:
\[ B2 - B1 = 63k \]
for some integer \( k \).

### Step 3: Substitute the Values
Substituting the expressions for B1 and B2:
\[ 7X - 3X = 63k \]
This simplifies to:
\[ 4X = 63k \]

### Step 4: Solve for X
From the equation \( 4X = 63k \), we can express \( X \) as:
\[ X = \frac{63k}{4} \]
To ensure \( X \) is an integer, \( k \) must be a multiple of 4. Let's set \( k = 4m \) for some integer \( m \):
\[ X = \frac{63(4m)}{4} = 63m \]

### Step 5: Calculate the Number of Birds
Now substituting \( X \) back into the expressions for the number of birds:
- B1 = \( 3X = 3(63m) = 189m \)
- B2 = \( 7X = 7(63m) = 441m \)
- B3 = \( 5X = 5(63m) = 315m \)

### Step 6: Total Number of Birds
The total number of birds on the tree is:
\[ B1 + B2 + B3 = 189m + 441m + 315m = 945m \]

### Step 7: Find the Minimum Number of Birds
To find the minimum number of birds, we set \( m = 1 \):
\[ \text{Total birds} = 945 \times 1 = 945 \]

Thus, the minimum number of birds on the tree is **945**.

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