The ages of Vinay, Varsha, Veera and Vikram are in arithmetic progression, but not in order. The ratio fo ages of Vinay and Varsha is 6:5 and Veera is to Vikram is 7:8. Two years later the age of Varsha and Vikram will be 2:3. Find the ratio of ages of Vinay and Veera:

Options

A. `7:6`
B. `5:8`
C. `6:7`
D. `8:9`

Answer

To solve the problem step by step, we will follow the information given in the question regarding the ages of Vinay, Varsha, Veera, and Vikram.

### Step 1: Assign Variables Based on Ratios
Let:
- Varsha's age = \( 5x \)
- Vinay's age = \( 6x \)

Since the ratio of Vinay's age to Varsha's age is \( 6:5 \).

Next, for Veera and Vikram:
- Veera's age = \( 7y \)
- Vikram's age = \( 8y \)

Since the ratio of Veera's age to Vikram's age is \( 7:8 \).

### Step 2: Establish the Arithmetic Progression Condition
Since the ages are in arithmetic progression (AP), we know that:
\[ \text{Middle term} = \frac{\text{First term} + \text{Last term}}{2} \]

This gives us two equations based on the arrangement of ages:
1. \( 6x - 5x = 8y - 7y \)
2. Simplifying gives us: \( x = y \)

### Step 3: Use Future Age Ratios
According to the problem, two years later, the ratio of Varsha's age to Vikram's age will be \( 2:3 \). Therefore:
- Varsha's age in 2 years = \( 5x + 2 \)
- Vikram's age in 2 years = \( 8y + 2 \)

Setting up the equation based on the ratio:
\[ \frac{5x + 2}{8y + 2} = \frac{2}{3} \]

### Step 4: Cross Multiply and Simplify
Cross multiplying gives us:
\[ 3(5x + 2) = 2(8y + 2) \]
Expanding both sides:
\[ 15x + 6 = 16y + 4 \]

### Step 5: Substitute \( y \) with \( x \)
Since we established earlier that \( y = x \), we substitute \( y \) with \( x \):
\[ 15x + 6 = 16x + 4 \]

### Step 6: Solve for \( x \)
Rearranging the equation:
\[ 15x + 6 - 4 = 16x \]
\[ 2 = 16x - 15x \]
\[ x = 2 \]

### Step 7: Calculate the Ages
Now that we have \( x \):
- Vinay's age = \( 6x = 6 \times 2 = 12 \) years
- Varsha's age = \( 5x = 5 \times 2 = 10 \) years
- Veera's age = \( 7y = 7 \times 2 = 14 \) years
- Vikram's age = \( 8y = 8 \times 2 = 16 \) years

### Step 8: Find the Ratio of Ages of Vinay and Veera
Now, we need to find the ratio of Vinay's age to Veera's age:
\[ \text{Ratio of Vinay to Veera} = \frac{12}{14} = \frac{6}{7} \]

### Final Answer
The ratio of ages of Vinay and Veera is \( 6:7 \).

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