`55%` of the candidates in a examination were boys `60%` of the boys and `75%` of the girls passed and 315 girls failed . The number of boys who failed were

Options

A. 626
B. 632
C. 616
D. 646

Answer

To solve the problem step by step, we will use the information provided about the percentage of boys and girls, as well as the passing and failing rates.

### Step 1: Define the total number of candidates
Let the total number of candidates in the examination be \( x \).

### Step 2: Calculate the number of boys and girls
Since 55% of the candidates are boys, the number of boys is:
\[ \text{Number of boys} = 0.55x \]
The remaining candidates are girls, which is 45% of the total:
\[ \text{Number of girls} = 0.45x \]

### Step 3: Determine the number of boys and girls who passed
60% of the boys passed the examination, so the number of boys who passed is:
\[ \text{Boys passed} = 0.60 \times 0.55x = 0.33x \]
Similarly, 75% of the girls passed the examination, so the number of girls who passed is:
\[ \text{Girls passed} = 0.75 \times 0.45x = 0.3375x \]

### Step 4: Calculate the number of girls who failed
The number of girls who failed is given as 315. We can express this in terms of \( x \):
\[ \text{Girls failed} = \text{Total girls} - \text{Girls passed} = 0.45x - 0.3375x = 0.1125x \]
Setting this equal to 315 gives us:
\[ 0.1125x = 315 \]

### Step 5: Solve for \( x \)
To find \( x \), we divide both sides by 0.1125:
\[ x = \frac{315}{0.1125} = 2800 \]

### Step 6: Calculate the number of boys
Now that we have the total number of candidates, we can calculate the number of boys:
\[ \text{Number of boys} = 0.55 \times 2800 = 1540 \]

### Step 7: Calculate the number of boys who failed
We already know that 60% of the boys passed, so the number of boys who failed is:
\[ \text{Boys failed} = \text{Total boys} - \text{Boys passed} = 1540 - 0.33 \times 2800 \]
Calculating the number of boys who passed:
\[ \text{Boys passed} = 0.33 \times 2800 = 924 \]
Thus, the number of boys who failed is:
\[ \text{Boys failed} = 1540 - 924 = 616 \]

### Final Answer
The number of boys who failed is **616**.

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