Sudha is four times as old as Rita. After 12 years, Sudha will be twice as old as Rita. Find their present ages.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let Rita's present age be \( x \).

### Step 2: Express Sudha's age in terms of Rita's age
Since Sudha is four times as old as Rita, we can express Sudha's age as:
\[ \text{Sudha's age} = 4x \]

### Step 3: Write the expressions for their ages after 12 years
After 12 years, Rita's age will be:
\[ \text{Rita's age after 12 years} = x + 12 \]
Sudha's age after 12 years will be:
\[ \text{Sudha's age after 12 years} = 4x + 12 \]

### Step 4: Set up the equation based on the given information
According to the problem, after 12 years, Sudha will be twice as old as Rita. This gives us the equation:
\[ 4x + 12 = 2(x + 12) \]

### Step 5: Simplify the equation
Expanding the right side of the equation:
\[ 4x + 12 = 2x + 24 \]

### Step 6: Rearrange the equation to isolate \( x \)
Subtract \( 2x \) from both sides:
\[ 4x - 2x + 12 = 24 \]
This simplifies to:
\[ 2x + 12 = 24 \]

Next, subtract 12 from both sides:
\[ 2x = 24 - 12 \]
This simplifies to:
\[ 2x = 12 \]

### Step 7: Solve for \( x \)
Divide both sides by 2:
\[ x = \frac{12}{2} = 6 \]

### Step 8: Find the present ages
Now that we have \( x \), we can find Rita's and Sudha's present ages:
- Rita's present age:
\[ \text{Rita's age} = x = 6 \text{ years} \]
- Sudha's present age:
\[ \text{Sudha's age} = 4x = 4 \times 6 = 24 \text{ years} \]

### Final Answer
- Rita's present age is 6 years.
- Sudha's present age is 24 years.

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