A student got `20%` marks and failed by 30 marks. Another student got `32%` marks and scored 42 marks more than passing mark. Find the passing mark in an examination.

Answer

To find the passing marks in the examination, we can follow these steps:

### Step 1: Define the variables
Let the full marks of the examination be \( x \).

### Step 2: Set up the equations based on the information given
1. The first student got 20% of the full marks and failed by 30 marks. This can be expressed as:
\[ \text{Marks obtained by first student} + 30 = \text{Passing marks} \]
Therefore, we can write:
\[ 0.2x + 30 = P \quad \text{(Equation 1)} \]

2. The second student got 32% of the full marks and scored 42 marks more than the passing marks. This can be expressed as:
\[ \text{Marks obtained by second student} - 42 = \text{Passing marks} \]
Therefore, we can write:
\[ 0.32x - 42 = P \quad \text{(Equation 2)} \]

### Step 3: Set the two equations equal to each other
Since both equations represent the passing marks \( P \), we can set them equal to each other:
\[ 0.2x + 30 = 0.32x - 42 \]

### Step 4: Solve for \( x \)
1. Rearranging the equation gives:
\[ 30 + 42 = 0.32x - 0.2x \]
\[ 72 = 0.12x \]

2. Now, divide both sides by 0.12 to find \( x \):
\[ x = \frac{72}{0.12} = 600 \]

### Step 5: Find the passing marks \( P \)
Now that we have the full marks \( x = 600 \), we can substitute this value back into either Equation 1 or Equation 2 to find the passing marks.

Using Equation 1:
\[ P = 0.2 \times 600 + 30 \]
\[ P = 120 + 30 = 150 \]

### Final Answer
The passing marks in the examination are \( \boxed{150} \).

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