X 5625 are divided among A, B and C, so that A receives 1/2 as much as B and C together receive and B receives 1/4 as much as A and C together receive. Find the sum of shares of A and B.

Options

A. X 5000
B. X 3000
C. X 15000
D. X 9000

Answer

To solve the problem step by step, we will define the shares of A, B, and C based on the given conditions and then find the sum of the shares of A and B.

### Step 1: Define the shares based on the conditions
Let the shares of A, B, and C be represented as A, B, and C respectively. According to the problem:
- A receives \( \frac{1}{2} \) as much as B and C together.
- B receives \( \frac{1}{4} \) as much as A and C together.

From the first condition, we can write:
\[ A = \frac{1}{2}(B + C) \]
Multiplying both sides by 2 gives:
\[ 2A = B + C \] (Equation 1)

From the second condition, we can write:
\[ B = \frac{1}{4}(A + C) \]
Multiplying both sides by 4 gives:
\[ 4B = A + C \] (Equation 2)

### Step 2: Express C in terms of A and B
From Equation 1, we can express C as:
\[ C = 2A - B \] (Substituting C in terms of A and B)

### Step 3: Substitute C in Equation 2
Now, substitute C in Equation 2:
\[ 4B = A + (2A - B) \]
This simplifies to:
\[ 4B = 3A - B \]
Adding B to both sides gives:
\[ 5B = 3A \]
Thus, we can express B in terms of A:
\[ B = \frac{3}{5}A \] (Equation 3)

### Step 4: Substitute B back into Equation 1
Now substitute B from Equation 3 back into Equation 1:
\[ 2A = \frac{3}{5}A + C \]
Rearranging gives:
\[ C = 2A - \frac{3}{5}A \]
This simplifies to:
\[ C = \frac{10}{5}A - \frac{3}{5}A = \frac{7}{5}A \] (Equation 4)

### Step 5: Find the total sum of A, B, and C
Now we know:
- \( A = A \)
- \( B = \frac{3}{5}A \)
- \( C = \frac{7}{5}A \)

The total sum of A, B, and C is:
\[ A + B + C = A + \frac{3}{5}A + \frac{7}{5}A = A + \frac{10}{5}A = A + 2A = 3A \]

### Step 6: Set the total equal to 5625
According to the problem, the total amount is 5625:
\[ 3A = 5625 \]
Dividing both sides by 3 gives:
\[ A = \frac{5625}{3} = 1875 \]

### Step 7: Find B and C
Using Equation 3 to find B:
\[ B = \frac{3}{5}A = \frac{3}{5} \times 1875 = 1125 \]

Using Equation 4 to find C:
\[ C = \frac{7}{5}A = \frac{7}{5} \times 1875 = 2625 \]

### Step 8: Find the sum of shares of A and B
Now, we can find the sum of the shares of A and B:
\[ A + B = 1875 + 1125 = 3000 \]

Thus, the sum of the shares of A and B is **3000**.