In a class of 60 students each boy gives as much money as the number of girls and each girl gives as much money as the number of boys. If the total collection of money is 1600. Then find the number of girls in the class?

Options

A. 30, 20
B. 25, 20
C. 50, 60
D. 20, 40

Answer

Let's solve the problem step by step.

### Step 1: Define Variables
Let the number of girls in the class be \( g \) and the number of boys be \( b \). According to the problem, the total number of students is 60. Therefore, we can write:
\[ g + b = 60 \]

### Step 2: Express One Variable in Terms of the Other
From the equation \( g + b = 60 \), we can express \( b \) in terms of \( g \):
\[ b = 60 - g \]

### Step 3: Set Up the Money Collection Equation
According to the problem, each boy gives money equal to the number of girls, and each girl gives money equal to the number of boys. Therefore, the total money collected can be expressed as:
\[ \text{Total Money} = (\text{Number of Boys} \times \text{Number of Girls}) + (\text{Number of Girls} \times \text{Number of Boys}) \]
This can be simplified to:
\[ \text{Total Money} = b \cdot g + g \cdot b = 2bg \]

### Step 4: Substitute for \( b \)
Now, substitute \( b = 60 - g \) into the equation for total money:
\[ \text{Total Money} = 2g(60 - g) \]

### Step 5: Set Up the Equation with Total Money
We know from the problem that the total collection of money is 1600. So we can set up the equation:
\[ 2g(60 - g) = 1600 \]

### Step 6: Simplify the Equation
Now, simplify the equation:
\[ 120g - 2g^2 = 1600 \]
Rearranging gives:
\[ 2g^2 - 120g + 1600 = 0 \]
Dividing the entire equation by 2:
\[ g^2 - 60g + 800 = 0 \]

### Step 7: Solve the Quadratic Equation
Now, we can solve this quadratic equation using the quadratic formula:
\[ g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -60 \), and \( c = 800 \):
\[ g = \frac{60 \pm \sqrt{(-60)^2 - 4 \cdot 1 \cdot 800}}{2 \cdot 1} \]
Calculating the discriminant:
\[ g = \frac{60 \pm \sqrt{3600 - 3200}}{2} \]
\[ g = \frac{60 \pm \sqrt{400}}{2} \]
\[ g = \frac{60 \pm 20}{2} \]

### Step 8: Calculate the Possible Values for \( g \)
Calculating the two possible values:
1. \( g = \frac{80}{2} = 40 \)
2. \( g = \frac{40}{2} = 20 \)

### Step 9: Conclusion
Thus, the number of girls in the class can be either 40 or 20. Since \( g + b = 60 \), if \( g = 40 \), then \( b = 20 \) and vice versa.

### Final Answer
The number of girls in the class is **40**.

---