In a class of 135 students, the number of boys is twice that of girls. One-sixth of the boys and one third of the girls faild in the final examination . Find the percentage of students who passed the examination .

Options

A. `75%`
B. `71.45%`
C. `77.78%`
D. `81.23%`

Answer

To solve the problem step by step, we will break down the information given and perform the necessary calculations.

### Step 1: Define Variables
Let:
- \( G \) = number of girls
- \( B \) = number of boys

According to the problem, we know:
1. The total number of students is 135.
2. The number of boys is twice the number of girls.

### Step 2: Set Up Equations
From the information provided, we can set up the following equations:
1. \( B + G = 135 \)
2. \( B = 2G \)

### Step 3: Substitute and Solve for Girls
Substituting the second equation into the first:
\[ 2G + G = 135 \]
\[ 3G = 135 \]
Now, divide both sides by 3:
\[ G = \frac{135}{3} = 45 \]

### Step 4: Calculate the Number of Boys
Now that we have the number of girls, we can find the number of boys:
\[ B = 2G = 2 \times 45 = 90 \]

### Step 5: Calculate Failures
Next, we need to find out how many students failed:
- One-sixth of the boys failed:
\[ \text{Boys failed} = \frac{1}{6} \times 90 = 15 \]
- One-third of the girls failed:
\[ \text{Girls failed} = \frac{1}{3} \times 45 = 15 \]

### Step 6: Calculate Total Failures
Now, we can find the total number of students who failed:
\[ \text{Total failed} = \text{Boys failed} + \text{Girls failed} = 15 + 15 = 30 \]

### Step 7: Calculate Students Who Passed
To find the number of students who passed:
\[ \text{Passed} = \text{Total students} - \text{Total failed} = 135 - 30 = 105 \]

### Step 8: Calculate Percentage of Students Who Passed
Finally, we calculate the percentage of students who passed:
\[ \text{Percentage passed} = \left( \frac{\text{Passed}}{\text{Total students}} \right) \times 100 = \left( \frac{105}{135} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage passed} = 77.78\% \]

### Final Answer
The percentage of students who passed the examination is **77.78%**.