Ratio of age of Ravi to Vicky, 4 years ago was 5:6, while ratio of present age of Rocky to that of Vicky is 5: 4. If 2 years later sum of age of Ravi and Rocky will be 63 years, then find the difference between present age of Ravi and Vicky?

Options

A. A)4 years
B. B)2 years
C. C)8 years
D. D)6 years

Answer

To solve the problem step by step, we will break down the information given and set up equations based on the ratios and conditions provided.

### Step 1: Establish the ages based on the given ratios.
Let Ravi's present age be \( R \) and Vicky's present age be \( V \). According to the problem, the ratio of their ages 4 years ago was 5:6. Therefore, we can express this as:
\[ \frac{R - 4}{V - 4} = \frac{5}{6} \]
Cross-multiplying gives us:
\[ 6(R - 4) = 5(V - 4) \]
Expanding this, we get:
\[ 6R - 24 = 5V - 20 \]
Rearranging gives us our first equation:
\[ 6R - 5V = 4 \quad \text{(Equation 1)} \]

### Step 2: Establish the relationship between Rocky and Vicky's ages.
Let Rocky's present age be \( K \). The problem states that the ratio of Rocky's present age to Vicky's present age is 5:4. Thus, we can express this as:
\[ \frac{K}{V} = \frac{5}{4} \]
Cross-multiplying gives us:
\[ 4K = 5V \quad \text{(Equation 2)} \]

### Step 3: Use the information about their ages in 2 years.
The problem states that in 2 years, the sum of Ravi's and Rocky's ages will be 63 years. Therefore, we can express this as:
\[ (R + 2) + (K + 2) = 63 \]
Simplifying this gives us:
\[ R + K + 4 = 63 \]
Thus, we have:
\[ R + K = 59 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 2 into Equation 3.
From Equation 2, we can express \( K \) in terms of \( V \):
\[ K = \frac{5V}{4} \]
Now, substitute this into Equation 3:
\[ R + \frac{5V}{4} = 59 \]
Multiplying through by 4 to eliminate the fraction gives:
\[ 4R + 5V = 236 \quad \text{(Equation 4)} \]

### Step 5: Solve the system of equations (Equation 1 and Equation 4).
Now we have two equations:
1. \( 6R - 5V = 4 \) (Equation 1)
2. \( 4R + 5V = 236 \) (Equation 4)

Adding these two equations:
\[ (6R - 5V) + (4R + 5V) = 4 + 236 \]
This simplifies to:
\[ 10R = 240 \]
Thus, we find:
\[ R = 24 \]

### Step 6: Find Vicky's age using \( R \).
Now substitute \( R = 24 \) back into Equation 1:
\[ 6(24) - 5V = 4 \]
This simplifies to:
\[ 144 - 5V = 4 \]
Rearranging gives:
\[ 5V = 140 \quad \Rightarrow \quad V = 28 \]

### Step 7: Find the difference between Ravi's and Vicky's ages.
Now we have:
- Ravi's present age \( R = 24 \)
- Vicky's present age \( V = 28 \)

The difference between their ages is:
\[ |R - V| = |24 - 28| = 4 \]

### Final Answer:
The difference between the present ages of Ravi and Vicky is **4 years**.