5 yr ago, age of Sachin was 4 times the age of Vineet and after 10 yr, Sachin will be twice as ld as Vineet. Find the present age of Sachin and Vineet.

Answer

To solve the problem step by step, we will set up equations based on the information provided in the question.

### Step 1: Define Variables
Let the present age of Sachin be \( X \) years and the present age of Vineet be \( Y \) years.

### Step 2: Set Up the First Equation
According to the problem, 5 years ago, Sachin's age was 4 times Vineet's age. Thus, we can write:
\[ X - 5 = 4(Y - 5) \]
Expanding this, we get:
\[ X - 5 = 4Y - 20 \]
Rearranging gives us:
\[ X - 4Y = -15 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that after 10 years, Sachin will be twice as old as Vineet. Therefore, we can write:
\[ X + 10 = 2(Y + 10) \]
Expanding this, we get:
\[ X + 10 = 2Y + 20 \]
Rearranging gives us:
\[ X - 2Y = 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have a system of two equations:
1. \( X - 4Y = -15 \)
2. \( X - 2Y = 10 \)

We can solve these equations simultaneously. Let's subtract Equation 1 from Equation 2:
\[ (X - 2Y) - (X - 4Y) = 10 - (-15) \]
This simplifies to:
\[ 2Y = 25 \]
Dividing both sides by 2 gives:
\[ Y = 12.5 \]

### Step 5: Substitute to Find \( X \)
Now, substitute \( Y = 12.5 \) back into one of the equations to find \( X \). We can use Equation 2:
\[ X - 2(12.5) = 10 \]
This simplifies to:
\[ X - 25 = 10 \]
Adding 25 to both sides gives:
\[ X = 35 \]

### Conclusion
Thus, the present ages are:
- Sachin's present age \( X = 35 \) years
- Vineet's present age \( Y = 12.5 \) years

### Summary of the Solution
- Present age of Sachin: 35 years
- Present age of Vineet: 12.5 years