Two years ago, Kirti was four times as old as her daughter Sunidhi. Ten years hence, Kirti will be two times as old as her daughter. Find the ratio of the present ages of Kirti and her daughter.

Options

A. `13:4`
B. `12:7`
C. `7:3`
D. `7:4`

Answer

To solve the problem step by step, we will define the present ages of Kirti and her daughter Sunidhi, set up equations based on the information provided, and then solve for their ages.

### Step 1: Define Variables
Let:
- \( K \) = Present age of Kirti
- \( S \) = Present age of Sunidhi

### Step 2: Set Up Equations Based on Given Information
1. **Two Years Ago**:
- Two years ago, Kirti's age was \( K - 2 \).
- Two years ago, Sunidhi's age was \( S - 2 \).
- According to the problem, two years ago, Kirti was four times as old as Sunidhi:
\[ K - 2 = 4(S - 2) \]

2. **Ten Years Hence**:
- Ten years hence, Kirti's age will be \( K + 10 \).
- Ten years hence, Sunidhi's age will be \( S + 10 \).
- According to the problem, ten years hence, Kirti will be two times as old as Sunidhi:
\[ K + 10 = 2(S + 10) \]

### Step 3: Simplify the Equations
1. From the first equation:
\[ K - 2 = 4(S - 2) \implies K - 2 = 4S - 8 \implies K = 4S - 6 \quad \text{(Equation 1)} \]

2. From the second equation:
\[ K + 10 = 2(S + 10) \implies K + 10 = 2S + 20 \implies K = 2S + 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
- \( K = 4S - 6 \) (Equation 1)
- \( K = 2S + 10 \) (Equation 2)

Set the two expressions for \( K \) equal to each other:
\[ 4S - 6 = 2S + 10 \]

### Step 5: Rearrange and Solve for \( S \)
\[ 4S - 2S = 10 + 6 \implies 2S = 16 \implies S = 8 \]

### Step 6: Find \( K \)
Substituting \( S = 8 \) into Equation 2:
\[ K = 2(8) + 10 = 16 + 10 = 26 \]

### Step 7: Calculate the Ratio of Present Ages
Now we have:
- Present age of Kirti \( K = 26 \)
- Present age of Sunidhi \( S = 8 \)

The ratio of their present ages is:
\[ \text{Ratio} = \frac{K}{S} = \frac{26}{8} = \frac{13}{4} \]

### Final Answer
The ratio of the present ages of Kirti and her daughter Sunidhi is \( 13:4 \).

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