In an examination, the ratio of the number of candidates who passed, to those who failed was 5:2. If the mumber of failed candidates had been 14 more, then the ratio of the number of passed candidates to those who failed would have been 4 : 3. How many candidates passed the examination?

Options

A. 126
B. 70
C. 100
D. 98

Answer

To solve the problem step by step, we will use algebraic expressions based on the information provided in the question.

### Step 1: Define Variables
Let the number of candidates who passed the examination be \( P \) and the number of candidates who failed be \( F \). According to the problem, the ratio of passed to failed candidates is given as:
\[ \frac{P}{F} = \frac{5}{2} \]
This can be expressed in terms of a variable \( x \):
\[ P = 5x \quad \text{and} \quad F = 2x \]

**Hint:** Use variables to represent the quantities based on the given ratio.

### Step 2: Adjust the Number of Failed Candidates
The problem states that if the number of failed candidates had been 14 more, the new ratio of passed to failed candidates would be:
\[ \frac{P}{F + 14} = \frac{4}{3} \]
Substituting the expressions for \( P \) and \( F \):
\[ \frac{5x}{2x + 14} = \frac{4}{3} \]

**Hint:** Set up the new ratio based on the change in the number of failed candidates.

### Step 3: Cross Multiply to Solve for \( x \)
Cross-multiplying gives us:
\[ 5x \cdot 3 = 4 \cdot (2x + 14) \]
This simplifies to:
\[ 15x = 8x + 56 \]

**Hint:** Use cross-multiplication to eliminate the fractions.

### Step 4: Isolate \( x \)
Now, isolate \( x \) by subtracting \( 8x \) from both sides:
\[ 15x - 8x = 56 \]
This simplifies to:
\[ 7x = 56 \]
Now, divide both sides by 7:
\[ x = 8 \]

**Hint:** Rearranging the equation helps to isolate the variable.

### Step 5: Calculate the Number of Passed Candidates
Now that we have \( x \), we can find the number of candidates who passed:
\[ P = 5x = 5 \cdot 8 = 40 \]

**Hint:** Substitute back the value of \( x \) to find the required quantity.

### Conclusion
The number of candidates who passed the examination is \( \boxed{40} \).