In a school , the ratio of boys and girls is 4:5. When 100 girls leave the school, the ratio becomes 6 :7 How many boy are there in the shool ?

Options

A. 1800
B. 1200
C. 1000
D. 1500

Answer

To solve the problem step by step, we will use the information given in the question about the ratio of boys and girls and the changes that occur when 100 girls leave the school.

### Step 1: Set up the initial ratio
Let the number of boys be \( B \) and the number of girls be \( G \). According to the problem, the ratio of boys to girls is given as:
\[ \frac{B}{G} = \frac{4}{5} \]
This can be expressed as:
\[ 5B = 4G \quad \text{(Equation 1)} \]

### Step 2: Set up the new ratio after girls leave
When 100 girls leave the school, the new number of girls becomes \( G - 100 \). The new ratio of boys to girls is given as:
\[ \frac{B}{G - 100} = \frac{6}{7} \]
This can be expressed as:
\[ 7B = 6(G - 100) \quad \text{(Equation 2)} \]

### Step 3: Expand Equation 2
Expanding Equation 2 gives:
\[ 7B = 6G - 600 \]
Rearranging this, we have:
\[ 7B - 6G = -600 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( 5B - 4G = 0 \) (Equation 1)
2. \( 7B - 6G = -600 \) (Equation 2)

We will solve these equations simultaneously. From Equation 1, we can express \( B \) in terms of \( G \):
\[ B = \frac{4G}{5} \]

### Step 5: Substitute \( B \) in Equation 2
Now, substitute \( B \) in Equation 2:
\[ 7\left(\frac{4G}{5}\right) - 6G = -600 \]
This simplifies to:
\[ \frac{28G}{5} - 6G = -600 \]
To eliminate the fraction, multiply the entire equation by 5:
\[ 28G - 30G = -3000 \]
This simplifies to:
\[ -2G = -3000 \]
Thus:
\[ G = 1500 \]

### Step 6: Find the number of boys
Now that we have \( G \), we can find \( B \) using Equation 1:
\[ 5B = 4G \]
Substituting \( G = 1500 \):
\[ 5B = 4 \times 1500 \]
\[ 5B = 6000 \]
Now divide by 5:
\[ B = \frac{6000}{5} = 1200 \]

### Final Answer
The number of boys in the school is \( \boxed{1200} \).