A says to B, “I was four times as old as you were when I was as old as you are. “If the sum of their present ages is 33, then the present ages of A and B respectively are :

Options

A. 18 years, 15 years
B. 21 years, 12 years
C. 24 years, 9 years
D. 27 years, 6 years

Explanation

To solve the problem, we need to find the present ages of A and B based on the information provided. Let's break it down step by step.

### Step 1: Set Up the Variables
Let:
- A's present age = A
- B's present age = B

From the problem, we know that:
\[ A + B = 33 \] (Equation 1)

### Step 2: Analyze the Statement
A says, “I was four times as old as you were when I was as old as you are.”

This means that at some point in the past, A's age was equal to B's current age. Let's denote the number of years ago this was as \( x \).

At that time:
- A's age = \( A - x \)
- B's age = \( B - x \)

According to A's statement:
\[ A - x = 4(B - x) \] (Equation 2)

### Step 3: Express \( x \) in Terms of A and B
From Equation 2, we can rearrange it:
\[ A - x = 4B - 4x \]
\[ A + 3x = 4B \]
\[ 3x = 4B - A \]
\[ x = \frac{4B - A}{3} \] (Equation 3)

### Step 4: Substitute \( x \) Back
Now, substitute \( x \) back into the context of the problem. Since we know \( A + B = 33 \), we can express \( A \) in terms of \( B \):
\[ A = 33 - B \]

### Step 5: Substitute into Equation 3
Now substitute \( A \) into Equation 3:
\[ x = \frac{4B - (33 - B)}{3} \]
\[ x = \frac{4B - 33 + B}{3} \]
\[ x = \frac{5B - 33}{3} \]

### Step 6: Substitute \( x \) into the Age Condition
Since \( x \) must be a non-negative integer, we need \( 5B - 33 \) to be non-negative:
\[ 5B - 33 \geq 0 \]
\[ 5B \geq 33 \]
\[ B \geq \frac{33}{5} \]
\[ B \geq 6.6 \]

Since \( B \) must be a whole number, the smallest integer \( B \) can be is 7.

### Step 7: Calculate Possible Ages
Now we can check possible values for \( B \):
1. If \( B = 7 \):
\[ A = 33 - 7 = 26 \]

2. If \( B = 8 \):
\[ A = 33 - 8 = 25 \]

3. If \( B = 9 \):
\[ A = 33 - 9 = 24 \]

4. If \( B = 10 \):
\[ A = 33 - 10 = 23 \]

5. If \( B = 11 \):
\[ A = 33 - 11 = 22 \]

6. If \( B = 12 \):
\[ A = 33 - 12 = 21 \]

### Step 8: Check the Condition
Now we need to check which of these pairs satisfies A's statement:
- For \( A = 21 \) and \( B = 12 \):
- Years ago when A was as old as B is now:
- \( x = 21 - 12 = 9 \)
- A's age 9 years ago = \( 21 - 9 = 12 \)
- B's age 9 years ago = \( 12 - 9 = 3 \)
- Check: \( 12 = 4 \times 3 \) (True)

Thus, the present ages of A and B are:
- A = 21 years
- B = 12 years

### Final Answer
The present ages of A and B respectively are **21 years and 12 years**.