The total number of students in a school was 660. The ratio of number of boys and girls was 13 : 9. After some days, 30 girls joined the school and some boys left. The new ratio of number of boys and girls became 6:5. Find the number of boys who left the school.

Options

A. 50
B. 40
C. 60
D. 30

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Determine the initial number of boys and girls
Given:
- Total number of students = 660
- Ratio of boys to girls = 13:9

Let the number of boys be 13x and the number of girls be 9x.

The total number of students can be expressed as:
\[ 13x + 9x = 660 \]
\[ 22x = 660 \]

Now, solve for x:
\[ x = \frac{660}{22} = 30 \]

Now, we can find the number of boys and girls:
- Number of boys = \( 13x = 13 \times 30 = 390 \)
- Number of girls = \( 9x = 9 \times 30 = 270 \)

### Step 2: Calculate the new number of girls after some join
After some days, 30 girls joined the school:
- New number of girls = \( 270 + 30 = 300 \)

### Step 3: Set up the equation for the new ratio of boys to girls
Let \( y \) be the number of boys who left the school. The new number of boys will then be:
- New number of boys = \( 390 - y \)

We are given that the new ratio of boys to girls is 6:5. Therefore, we can set up the equation:
\[ \frac{390 - y}{300} = \frac{6}{5} \]

### Step 4: Cross-multiply to solve for y
Cross-multiplying gives:
\[ 5(390 - y) = 6 \times 300 \]
\[ 1950 - 5y = 1800 \]

Now, isolate \( y \):
\[ 1950 - 1800 = 5y \]
\[ 150 = 5y \]
\[ y = \frac{150}{5} = 30 \]

### Step 5: Conclusion
The number of boys who left the school is:
\[ \boxed{30} \]

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