In a class, there are 54 students. `33(1)/(3)%` of the number of students are boys and rest are girls. The average score in mathematics of boys is 50% more than that of the girls. If the average score of all the students is 70, then what is the average score of the boys?

Options

A. 81
B. 84
C. 87
D. 90

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Determine the number of boys and girls
We know that there are 54 students in total and that \(33 \frac{1}{3}\%\) of them are boys.

First, convert \(33 \frac{1}{3}\%\) to a fraction:
\[ 33 \frac{1}{3}\% = \frac{100}{3}\% = \frac{100}{3 \times 100} = \frac{1}{3} \]

Now, calculate the number of boys:
\[ \text{Number of boys} = \frac{1}{3} \times 54 = 18 \]

Next, calculate the number of girls:
\[ \text{Number of girls} = 54 - 18 = 36 \]

### Step 2: Set up the average scores
Let the average score of girls be \(x\). According to the problem, the average score of boys is 50% more than that of girls:
\[ \text{Average score of boys} = x + 0.5x = 1.5x \]

### Step 3: Use the overall average score
The average score of all students is given as 70. We can set up the equation using the average scores and the number of students:
\[ \frac{(36 \cdot x) + (18 \cdot 1.5x)}{54} = 70 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ \frac{36x + 27x}{54} = 70 \]
\[ \frac{63x}{54} = 70 \]

### Step 5: Solve for \(x\)
Multiply both sides by 54:
\[ 63x = 70 \times 54 \]
\[ 63x = 3780 \]
Now, divide by 63:
\[ x = \frac{3780}{63} = 60 \]

### Step 6: Calculate the average score of boys
Now that we have \(x\), we can find the average score of boys:
\[ \text{Average score of boys} = 1.5x = 1.5 \times 60 = 90 \]

### Final Answer
Thus, the average score of the boys is **90**.

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