The ages of A and B are in the ratio `3:1`. Fifteen years hence, the ratio will be `2:1`. Their present age (in years) are

Options

A. 45, 15
B. 21, 7
C. 30, 10
D. 60, 20

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the present ages of A and B.
Let the present age of A be \( 3x \) and the present age of B be \( x \), based on the given ratio of their ages \( 3:1 \).

### Step 2: Set up the equation for their ages in 15 years.
In 15 years, A's age will be \( 3x + 15 \) and B's age will be \( x + 15 \). According to the problem, the ratio of their ages in 15 years will be \( 2:1 \).

### Step 3: Write the equation based on the future ratio.
We can express the future age ratio as:
\[ \frac{3x + 15}{x + 15} = \frac{2}{1} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives us:
\[ 3x + 15 = 2(x + 15) \]

### Step 5: Expand and simplify the equation.
Expanding the right side:
\[ 3x + 15 = 2x + 30 \]

### Step 6: Rearrange the equation to isolate \( x \).
Subtract \( 2x \) from both sides:
\[ 3x - 2x + 15 = 30 \]
This simplifies to:
\[ x + 15 = 30 \]

### Step 7: Solve for \( x \).
Subtract 15 from both sides:
\[ x = 30 - 15 \]
Thus, \( x = 15 \).

### Step 8: Calculate the present ages of A and B.
Now that we have \( x \), we can find the present ages:
- A's age: \( 3x = 3 \times 15 = 45 \)
- B's age: \( x = 15 \)

### Final Answer:
The present ages of A and B are:
- A: 45 years
- B: 15 years

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