The average age of A and B is 20 years. If A is to be replaced by C, the average would be 19 years. The average of C and A is 21 years. The ages of A, B and C in order (in years) are :

Answer

To solve the problem step by step, let's denote the ages of A, B, and C as \( A \), \( B \), and \( C \) respectively.

### Step 1: Find the sum of the ages of A and B
The average age of A and B is given as 20 years. The formula for average is:

\[ \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \]

For A and B, we have:

\[ 20 = \frac{A + B}{2} \]

Multiplying both sides by 2 gives:

\[ A + B = 40 \quad \text{(Equation 1)} \]

### Step 2: Find the sum of the ages of B and C
When A is replaced by C, the new average age becomes 19 years. Thus:

\[ 19 = \frac{B + C}{2} \]

Multiplying both sides by 2 gives:

\[ B + C = 38 \quad \text{(Equation 2)} \]

### Step 3: Find the sum of the ages of A and C
The average age of A and C is given as 21 years. Therefore:

\[ 21 = \frac{A + C}{2} \]

Multiplying both sides by 2 gives:

\[ A + C = 42 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have a system of three equations:

1. \( A + B = 40 \) (Equation 1)
2. \( B + C = 38 \) (Equation 2)
3. \( A + C = 42 \) (Equation 3)

We can solve these equations step by step.

#### Step 4.1: Express B in terms of A
From Equation 1:

\[ B = 40 - A \quad \text{(Equation 4)} \]

#### Step 4.2: Substitute B in Equation 2
Substituting Equation 4 into Equation 2:

\[ (40 - A) + C = 38 \]

This simplifies to:

\[ C = A - 2 \quad \text{(Equation 5)} \]

#### Step 4.3: Substitute C in Equation 3
Now substitute Equation 5 into Equation 3:

\[ A + (A - 2) = 42 \]

This simplifies to:

\[ 2A - 2 = 42 \]

Adding 2 to both sides gives:

\[ 2A = 44 \]

Dividing by 2:

\[ A = 22 \]

### Step 5: Find B and C
Now, substitute \( A = 22 \) back into Equation 4 to find B:

\[ B = 40 - 22 = 18 \]

Next, substitute \( A = 22 \) into Equation 5 to find C:

\[ C = 22 - 2 = 20 \]

### Final Answer
The ages of A, B, and C are:

- A = 22 years
- B = 18 years
- C = 20 years

Thus, the final answer is:

**A = 22 years, B = 18 years, C = 20 years.**

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