The average age of A and B is 20 years. If C were to replace A, the average would be 19 and if C were to replace B, the averge would be 21. What are the age of A, B and C?

Options

A. A) 22, 18, 20
B. B) 20, 20, 18
C. C) 18, 22, 20
D. D) None of these

Answer

To solve the problem step by step, we will derive the ages of A, B, and C using the information provided in the question.

### Step 1: Set Up the Equations
The average age of A and B is given as 20 years. We can express this mathematically:
\[ \frac{A + B}{2} = 20 \]
Multiplying both sides by 2 gives us:
\[ A + B = 40 \quad \text{(Equation 1)} \]

### Step 2: Replace A with C
Next, we are told that if C replaces A, the average age becomes 19 years:
\[ \frac{C + B}{2} = 19 \]
Multiplying both sides by 2 gives us:
\[ C + B = 38 \quad \text{(Equation 2)} \]

### Step 3: Replace B with C
We also know that if C replaces B, the average age becomes 21 years:
\[ \frac{A + C}{2} = 21 \]
Multiplying both sides by 2 gives us:
\[ A + C = 42 \quad \text{(Equation 3)} \]

### Step 4: Add the Equations
Now, we will add all three equations together:
\[ (A + B) + (C + B) + (A + C) = 40 + 38 + 42 \]
This simplifies to:
\[ 2A + 2B + 2C = 120 \]
Dividing the entire equation by 2 gives us:
\[ A + B + C = 60 \quad \text{(Equation 4)} \]

### Step 5: Solve for A
Now, we can find the value of A using Equation 1:
\[ A + B = 40 \]
From Equation 4, we know:
\[ B + C = 38 \]
Substituting \(B\) from Equation 1 into Equation 4:
\[ A + (40 - A) + C = 60 \]
This simplifies to:
\[ 40 + C = 60 \implies C = 20 \]

### Step 6: Solve for B
Now, we can find B using Equation 2:
\[ C + B = 38 \]
Substituting \(C = 20\):
\[ 20 + B = 38 \implies B = 18 \]

### Step 7: Solve for A
Finally, we can find A using Equation 1:
\[ A + B = 40 \]
Substituting \(B = 18\):
\[ A + 18 = 40 \implies A = 22 \]

### Final Ages
Thus, the ages of A, B, and C are:
- A = 22 years
- B = 18 years
- C = 20 years

### Summary of the Solution
- A = 22
- B = 18
- C = 20