In an examination, the ratio of passes to failures was 4: 1. Had 30 less appeared and 20 less passed, the ratio of passes to failures would have been 5: 1. Find the number of students who appeared for the examination.

Answer

To solve the problem step by step, we will use the information provided in the question regarding the ratio of passes to failures and the changes in the number of students.

### Step 1: Define Variables
Let:
- \( x \) = number of students who passed
- \( y \) = number of students who failed

### Step 2: Set Up the First Equation
From the problem, we know that the ratio of passes to failures is 4:1. This can be expressed as:
\[ \frac{x}{y} = \frac{4}{1} \]
This implies:
\[ x = 4y \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Scenario
The problem states that if 30 fewer students had appeared and 20 fewer passed, the new ratio of passes to failures would be 5:1.

The total number of students who appeared would then be:
\[ (x + y) - 30 \]
The number of students who passed would be:
\[ x - 20 \]
The number of students who failed would then be:
\[ (y) - (x - (x - 20)) = y - (x - 20) = y - x + 20 \]

### Step 4: Set Up the Second Equation
According to the new scenario, the ratio of passes to failures is now 5:1:
\[ \frac{x - 20}{y - 10} = \frac{5}{1} \]
This implies:
\[ x - 20 = 5(y - 10) \]
Expanding this gives:
\[ x - 20 = 5y - 50 \]
Rearranging this, we have:
\[ x - 5y = -30 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
Now, we can substitute \( x = 4y \) from Equation 1 into Equation 2:
\[ 4y - 5y = -30 \]
This simplifies to:
\[ -y = -30 \]
Thus:
\[ y = 30 \]

### Step 6: Find the Value of \( x \)
Now that we have \( y \), we can find \( x \) using Equation 1:
\[ x = 4y = 4 \times 30 = 120 \]

### Step 7: Calculate the Total Number of Students
The total number of students who appeared for the examination is the sum of those who passed and those who failed:
\[ \text{Total Students} = x + y = 120 + 30 = 150 \]

### Final Answer
The number of students who appeared for the examination is:
\[ \boxed{150} \]