In a certain examination, the number of those who passed was 4 times the number of those who failed. If there had been 35 fewer appeared candidates and 9 more had failed, the ratio of passed and failed candidates would have been 2:1, then the total number of candidates was

Options

A. 135
B. 155
C. 145
D. 150

Answer

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

### Step 1: Define Variables
Let:
- \( x \) = number of candidates who failed
- \( 4x \) = number of candidates who passed (since the number of those who passed was 4 times the number of those who failed)

### Step 2: Total Candidates
The total number of candidates can be expressed as:
\[ \text{Total candidates} = \text{Passed candidates} + \text{Failed candidates} = 4x + x = 5x \]

### Step 3: Adjust for New Conditions
According to the problem, if there had been 35 fewer candidates and 9 more had failed, the new scenario would be:
- New total candidates = \( 5x - 35 \)
- New number of failed candidates = \( x + 9 \)

### Step 4: New Number of Passed Candidates
The number of passed candidates in this new scenario would be:
\[ \text{New passed candidates} = (5x - 35) - (x + 9) = 5x - 35 - x - 9 = 4x - 44 \]

### Step 5: Set Up the Ratio
According to the new conditions, the ratio of passed to failed candidates is given as 2:1. Thus, we can set up the equation:
\[ \frac{\text{New passed}}{\text{New failed}} = \frac{2}{1} \]
Substituting the expressions we found:
\[ \frac{4x - 44}{x + 9} = 2 \]

### Step 6: Cross-Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ 4x - 44 = 2(x + 9) \]
Expanding the right side:
\[ 4x - 44 = 2x + 18 \]

### Step 7: Rearranging the Equation
Now, rearranging the equation to isolate \( x \):
\[ 4x - 2x = 18 + 44 \]
\[ 2x = 62 \]
\[ x = 31 \]

### Step 8: Calculate Total Candidates
Now that we have \( x \), we can find the total number of candidates:
\[ \text{Total candidates} = 5x = 5 \times 31 = 155 \]

### Final Answer
The total number of candidates is **155**.

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