Sum of ages of A and B is 12 years more than twice the age of C and Sum of ages of A & D is twice the age of C. If the average age of B&D is 50 years and average age of all four is also 50 years, then find the difference between the age of A and C?

Options

A. A)6 years
B. B)1 years
C. C)3 years
D. D)4 years

Answer

To solve the problem step by step, we will define the ages of A, B, C, and D as follows:

Let:
- Age of A = A
- Age of B = B
- Age of C = C
- Age of D = D

### Step 1: Set up the equations based on the problem statement.
1. The sum of ages of A and B is 12 years more than twice the age of C:
\[ A + B = 2C + 12 \quad \text{(Equation 1)} \]
2. The sum of ages of A and D is twice the age of C:
\[ A + D = 2C \quad \text{(Equation 2)} \]
3. The average age of B and D is 50 years:
\[ \frac{B + D}{2} = 50 \implies B + D = 100 \quad \text{(Equation 3)} \]
4. The average age of all four is also 50 years:
\[ \frac{A + B + C + D}{4} = 50 \implies A + B + C + D = 200 \quad \text{(Equation 4)} \]

### Step 2: Use Equations 3 and 4 to find relationships.
From Equation 3:
\[ B + D = 100 \]

From Equation 4:
\[ A + B + C + D = 200 \]
We can substitute \(B + D\) from Equation 3 into Equation 4:
\[ A + 100 + C = 200 \]
This simplifies to:
\[ A + C = 100 \quad \text{(Equation 5)} \]

### Step 3: Substitute Equation 5 into Equation 2.
From Equation 2:
\[ A + D = 2C \]
Now, substitute \(A\) from Equation 5:
\[ (100 - C) + D = 2C \]
This simplifies to:
\[ 100 - C + D = 2C \]
Rearranging gives:
\[ D = 3C - 100 \quad \text{(Equation 6)} \]

### Step 4: Substitute Equation 6 into Equation 3.
Now we substitute \(D\) from Equation 6 into Equation 3:
\[ B + (3C - 100) = 100 \]
This simplifies to:
\[ B + 3C - 100 = 100 \]
Rearranging gives:
\[ B + 3C = 200 \quad \text{(Equation 7)} \]

### Step 5: Solve Equations 1 and 7 simultaneously.
Now we have:
1. \(A + B = 2C + 12\) (Equation 1)
2. \(B + 3C = 200\) (Equation 7)

From Equation 5, we know \(A = 100 - C\). Substitute \(A\) into Equation 1:
\[ (100 - C) + B = 2C + 12 \]
This simplifies to:
\[ 100 - C + B = 2C + 12 \]
Rearranging gives:
\[ B = 3C - 88 \quad \text{(Equation 8)} \]

### Step 6: Substitute Equation 8 into Equation 7.
Now substitute \(B\) from Equation 8 into Equation 7:
\[ (3C - 88) + 3C = 200 \]
This simplifies to:
\[ 6C - 88 = 200 \]
Adding 88 to both sides gives:
\[ 6C = 288 \]
Dividing by 6 gives:
\[ C = 48 \]

### Step 7: Find the age of A.
Substituting \(C = 48\) back into Equation 5:
\[ A + 48 = 100 \implies A = 100 - 48 = 52 \]

### Step 8: Calculate the difference between ages of A and C.
The difference between the ages of A and C is:
\[ A - C = 52 - 48 = 4 \]

### Final Answer:
The difference between the age of A and C is **4 years**.