In a class test , the sum of Arun's marks in Hindi and English is 30 . Had he got 2 marks more in Hindi and 3 marks less in English , the product of the marks would have been 210 . Find his marks in the two subjects .

Answer

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

### Step 1: Define Variables
Let:
- \( H \) = Arun's marks in Hindi
- \( E \) = Arun's marks in English

### Step 2: Set Up the First Equation
According to the problem, the sum of Arun's marks in Hindi and English is 30. Therefore, we can write the first equation as:
\[ H + E = 30 \]

### Step 3: Set Up the Second Equation
The problem states that if Arun had scored 2 marks more in Hindi and 3 marks less in English, the product of his marks would have been 210. This gives us the second equation:
\[ (H + 2)(E - 3) = 210 \]

### Step 4: Solve for E from the First Equation
From the first equation, we can express \( E \) in terms of \( H \):
\[ E = 30 - H \]

### Step 5: Substitute E in the Second Equation
Now, substitute \( E \) in the second equation:
\[ (H + 2)((30 - H) - 3) = 210 \]
This simplifies to:
\[ (H + 2)(27 - H) = 210 \]

### Step 6: Expand the Equation
Now, expand the equation:
\[ H \cdot 27 - H^2 + 2 \cdot 27 - 2H = 210 \]
This simplifies to:
\[ 27H - H^2 + 54 - 2H = 210 \]
Combining like terms:
\[ -H^2 + 25H + 54 = 210 \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ -H^2 + 25H - 156 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ H^2 - 25H + 156 = 0 \]

### Step 8: Factor the Quadratic Equation
We will now factor the quadratic equation:
\[ H^2 - 25H + 156 = (H - 13)(H - 12) = 0 \]

### Step 9: Solve for H
Setting each factor to zero gives us:
1. \( H - 13 = 0 \) → \( H = 13 \)
2. \( H - 12 = 0 \) → \( H = 12 \)

### Step 10: Find Corresponding E Values
Now we find the corresponding values of \( E \) using \( E = 30 - H \):
1. If \( H = 13 \):
\[ E = 30 - 13 = 17 \]
2. If \( H = 12 \):
\[ E = 30 - 12 = 18 \]

### Conclusion
Thus, Arun's marks in Hindi and English can be:
- Hindi: 13, English: 17
- Hindi: 12, English: 18