There are 100 students in a class, out of which 70% are girls and others are boys. The average score of girls in a test is 20% more than that of boys. If the average score of all the students is 57, then what is the average score of girls?

Options

A. 60
B. 50
C. 45
D. 75

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the number of girls and boys in the class.
Given that there are 100 students in total:
- Number of girls = 70% of 100 = 70 students
- Number of boys = 30% of 100 = 30 students

**Hint:** To find the percentage of a total, multiply the total number by the percentage (in decimal form).

### Step 2: Set up the average score equations.
Let the average score of boys be \( B \). According to the problem, the average score of girls is 20% more than that of boys. Therefore:
- Average score of girls \( G = B + 0.2B = 1.2B \)

**Hint:** To find a value that is a percentage more than another, multiply the original value by (1 + percentage as a decimal).

### Step 3: Calculate the total average score of all students.
The average score of all students is given as 57. The total score of all students can be calculated as:
- Total score of all students = Average score × Total number of students = 57 × 100 = 5700

**Hint:** The total score can be calculated by multiplying the average score by the total number of students.

### Step 4: Set up the equation for total scores.
The total score can also be expressed as the sum of the total scores of girls and boys:
- Total score of girls = Average score of girls × Number of girls = \( G \times 70 \)
- Total score of boys = Average score of boys × Number of boys = \( B \times 30 \)

Thus, we have:
\[ 70G + 30B = 5700 \]

**Hint:** The total score can be represented as the sum of individual scores multiplied by their respective counts.

### Step 5: Substitute \( G \) in terms of \( B \).
Substituting \( G = 1.2B \) into the total score equation:
\[ 70(1.2B) + 30B = 5700 \]
This simplifies to:
\[ 84B + 30B = 5700 \]
\[ 114B = 5700 \]

**Hint:** When substituting, ensure to distribute correctly and combine like terms.

### Step 6: Solve for \( B \).
Now, divide both sides by 114 to find \( B \):
\[ B = \frac{5700}{114} = 50 \]

**Hint:** To isolate a variable, perform the same operation on both sides of the equation.

### Step 7: Calculate the average score of girls \( G \).
Now that we have \( B \), we can find \( G \):
\[ G = 1.2B = 1.2 \times 50 = 60 \]

**Hint:** To find a percentage increase, multiply the base value by the factor representing the increase.

### Conclusion:
The average score of girls is **60**.

**Final Answer:** 60