A student scored `30%` marks and failed by 45 marks. Another student scored `42%` marks and scored 45 marks more than the passing marks. Find the passing marks.

Answer

To find the passing marks based on the information provided, we can follow these steps:

### Step 1: Define Variables
Let the total marks (full marks) be represented by \( x \).

### Step 2: Set Up Equations
From the information given:
- The first student scored \( 30\% \) of the total marks and failed by \( 45 \) marks. This can be expressed as:
\[ \text{Passing Marks} = 30\% \text{ of } x + 45 \]
or
\[ \text{Passing Marks} = \frac{30}{100}x + 45 \]

- The second student scored \( 42\% \) of the total marks and scored \( 45 \) marks more than the passing marks. This can be expressed as:
\[ \text{Passing Marks} = 42\% \text{ of } x - 45 \]
or
\[ \text{Passing Marks} = \frac{42}{100}x - 45 \]

### Step 3: Equate the Two Expressions for Passing Marks
Since both expressions represent the passing marks, we can set them equal to each other:
\[ \frac{30}{100}x + 45 = \frac{42}{100}x - 45 \]

### Step 4: Solve for \( x \)
To solve for \( x \), first eliminate the fractions by multiplying the entire equation by \( 100 \):
\[ 30x + 4500 = 42x - 4500 \]

Now, rearranging the equation:
\[ 4500 + 4500 = 42x - 30x \]
\[ 9000 = 12x \]

Now, divide both sides by \( 12 \):
\[ x = \frac{9000}{12} = 750 \]

### Step 5: Find the Passing Marks
Now that we have the total marks \( x \), we can substitute \( x \) back into one of the equations to find the passing marks. We can use the first equation:
\[ \text{Passing Marks} = \frac{30}{100} \times 750 + 45 \]
Calculating this:
\[ \text{Passing Marks} = 225 + 45 = 270 \]

### Final Answer
The passing marks are \( 270 \).

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