15 years ago, Kanika's father was five times as old as Kanika. After 15 years, her father will be thrice as old as her. The present age of Kanika is _____

Options

A. 30 years
B. 10 years
C. 45 years
D. 25 years

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( y \) = present age of Kanika
- \( x \) = present age of Kanika's father

### Step 2: Set Up the First Equation
According to the problem, 15 years ago, Kanika's father was five times as old as Kanika. This can be expressed as:
\[ x - 15 = 5(y - 15) \]
Expanding this equation gives:
\[ x - 15 = 5y - 75 \]
Rearranging this, we get:
\[ x - 5y = -60 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that after 15 years, Kanika's father will be three times as old as Kanika. This can be expressed as:
\[ x + 15 = 3(y + 15) \]
Expanding this equation gives:
\[ x + 15 = 3y + 45 \]
Rearranging this, we get:
\[ x - 3y = 30 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x - 5y = -60 \)
2. \( x - 3y = 30 \)

We can solve these equations simultaneously. Let's subtract Equation 1 from Equation 2:
\[ (x - 3y) - (x - 5y) = 30 - (-60) \]
This simplifies to:
\[ 2y = 90 \]
Dividing both sides by 2 gives:
\[ y = 45 \]

### Step 5: Find the Present Age of Kanika
Thus, the present age of Kanika is:
\[ \text{Present age of Kanika} = y = 45 \]

### Step 6: Find the Present Age of Kanika's Father
Now, we can find the present age of Kanika's father by substituting \( y \) back into either equation. Using Equation 1:
\[ x - 5(45) = -60 \]
This simplifies to:
\[ x - 225 = -60 \]
Adding 225 to both sides gives:
\[ x = 165 \]
So, the present age of Kanika's father is 165.

### Final Answer
The present age of Kanika is **45 years**.

---