If `a+b+c+d = 24,a+b=c+d,a+c = 2(b+d)` and a = b , what is the value of C?

Options

A. 2
B. 6
C. 8
D. 10

Answer

To solve the problem step by step, we will use the given equations systematically.

### Given:
1. \( a + b + c + d = 24 \)
2. \( a + b = c + d \)
3. \( a + c = 2(b + d) \)
4. \( a = b \)

### Step 1: Substitute \( a = b \) in the equations
Since \( a = b \), we can replace \( b \) with \( a \) in the equations.

### Step 2: Rewrite the first equation
Substituting \( b \) in the first equation:
\[ a + a + c + d = 24 \implies 2a + c + d = 24 \]

### Step 3: Rewrite the second equation
Substituting \( b \) in the second equation:
\[ a + a = c + d \implies 2a = c + d \implies c + d = 2a \]

### Step 4: Substitute \( c + d = 2a \) into the first equation
From \( 2a + c + d = 24 \):
\[ 2a + 2a = 24 \implies 4a = 24 \implies a = \frac{24}{4} = 6 \]

### Step 5: Find the value of \( b \)
Since \( a = b \), we have:
\[ b = 6 \]

### Step 6: Find \( c + d \)
Using \( c + d = 2a \):
\[ c + d = 2 \times 6 = 12 \]

### Step 7: Rewrite the third equation
Substituting \( a \) and \( b \) in the third equation:
\[ a + c = 2(b + d) \implies 6 + c = 2(6 + d) \]

### Step 8: Substitute \( d = 12 - c \) into the third equation
We know \( d = 12 - c \). Substitute this into the equation:
\[ 6 + c = 2(6 + (12 - c)) \]
This simplifies to:
\[ 6 + c = 2(18 - c) \implies 6 + c = 36 - 2c \]

### Step 9: Solve for \( c \)
Rearranging gives:
\[ c + 2c = 36 - 6 \implies 3c = 30 \implies c = \frac{30}{3} = 10 \]

### Conclusion
Thus, the value of \( c \) is \( 10 \).