If $B_{1}$, $B_{2}$ and $B_{3}$ are three kinds of birds on a tree in the ratio 3 : 7 : 5 and the number of $B_{2}$ birds are more than $B_{1}$ birds by a multiple of 9 and 7 both, then the minimum number of birds on the tree are

Question

Accepted Answer

Here's a detailed explanation for why Option 4 is the correct answer and why the other options are incorrect:

**Step-by-step Derivation of the Correct Answer (Option 4):**

1.  **Represent the number of birds using the given ratio:**
    Let the common multiple for the ratio be $x$.
    *   Number of $B_1$ birds = $3x$
    *   Number of $B_2$ birds = $7x$
    *   Number of $B_3$ birds = $5x$
    Since the number of birds must be a whole number, $x$ must be a positive integer.

2.  **Formulate the condition based on the difference between $B_2$ and $B_1$:**
    The problem states that "the number of $B_2$ birds are more than $B_1$ birds by a multiple of 9 and 7 both".
    This means the difference $(B_2 - B_1)$ must be a multiple of both 9 and 7.
    The least common multiple (LCM) of 9 and 7 is $9 	imes 7 = 63$ (since 9 and 7 are coprime).
    So, $B_2 - B_1$ must be a multiple of 63.
    Let $B_2 - B_1 = 63k$, where $k$ is a positive integer (since $B_2$ is "more than" $B_1$).

3.  **Substitute the expressions for $B_1$ and $B_2$ into the condition:**
    $7x - 3x = 63k$
    $4x = 63k$

4.  **Find the minimum value of $x$:**
    For $x$ to be an integer, $63k$ must be divisible by 4. Since 63 is not divisible by 4, $k$ must be divisible by 4.
    To find the *minimum* number of birds, we need the *minimum* positive integer value for $x$. This occurs when $k$ is the smallest positive integer that makes $x$ an integer.
    The smallest positive integer value for $k$ that is a multiple of 4 is $k=4$.
    Substitute $k=4$ into the equation $4x = 63k$:
    $4x = 63 	imes 4$
    $x = 63$

5.  **Calculate the minimum total number of birds:**
    The total number of birds on the tree is the sum of $B_1$, $B_2$, and $B_3$:
    Total birds = $3x + 7x + 5x = 15x$
    Substitute the minimum value of $x=63$:
    Total birds = $15 	imes 63$
    Total birds = $945$

Therefore, the minimum number of birds on the tree is 945.

**Justification for Option 4 (945) being correct:**
Our step-by-step calculation, based on the given conditions, directly leads to 945 as the minimum total number of birds. This value satisfies all the criteria:
*   The ratio $B_1:B_2:B_3$ is $3 	imes 63 : 7 	imes 63 : 5 	imes 63 = 189 : 441 : 315$, which simplifies to $3:7:5$.
*   The difference $B_2 - B_1 = 441 - 189 = 252$.
*   $252$ is a multiple of 9 ($252 = 9 	imes 28$).
*   $252$ is a multiple of 7 ($252 = 7 	imes 36$).
*   Since $252 = 4 	imes 63$, it is indeed a multiple of both 9 and 7 (i.e., a multiple of 63).
*   This is the minimum possible total because we used the smallest positive integer value for $k$ (which was 4) to ensure $x$ is an integer.

**Explanation for why other options are incorrect:**

1.  **Option 1: 630**
    *   If the total number of birds is 630, then $15x = 630 \implies x = 630 / 15 = 42$.
    *   Number of $B_1$ birds = $3 	imes 42 = 126$.
    *   Number of $B_2$ birds = $7 	imes 42 = 294$.
    *   The difference $B_2 - B_1 = 294 - 126 = 168$.
    *   Now, check if 168 is a multiple of 9 and 7 (i.e., 63).
        *   $168 \div 9 = 18.66...$ (Not a multiple of 9).
        *   $168 \div 7 = 24$ (Is a multiple of 7).
    *   Since 168 is not a multiple of 9, this option does not satisfy the given condition.

2.  **Option 2: 238**
    *   If the total number of birds is 238, then $15x = 238 \implies x = 238 / 15$.
    *   $238 / 15$ is not an integer ($238/15 \approx 15.86$). Since the number of birds must be an integer, $x$ must be an integer. Therefore, this option is incorrect.

3.  **Option 3: 942**
    *   If the total number of birds is 942, then $15x = 942 \implies x = 942 / 15$.
    *   $942 / 15 = 314 / 5 = 62.8$. This is not an integer. Since the number of birds must be an integer, $x$ must be an integer. Therefore, this option is incorrect.

Thus, only Option 4 (945) correctly satisfies all the conditions given in the problem.