The ratio of the number of boys to the number of girls in a school of 640 students, is `5 : 3`. If 30 more girls are admitted in the school, then how many more boys should be admitted so that the ratio of boys to that of the girls, becomes `14 : 9`.

Options

A. 20
B. 15
C. 20
D. 30

Answer

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

### Step 1: Determine the number of boys and girls in the school
Given the total number of students in the school is 640 and the ratio of boys to girls is 5:3.

Let the number of boys be \( 5x \) and the number of girls be \( 3x \).

The total number of students can be expressed as:
\[ 5x + 3x = 640 \]
\[ 8x = 640 \]
\[ x = \frac{640}{8} = 80 \]

Now, we can find the number of boys and girls:
\[ \text{Number of boys} = 5x = 5 \times 80 = 400 \]
\[ \text{Number of girls} = 3x = 3 \times 80 = 240 \]

### Step 2: Calculate the new number of girls after admitting more girls
If 30 more girls are admitted, the new number of girls will be:
\[ \text{New number of girls} = 240 + 30 = 270 \]

### Step 3: Set up the equation for the new ratio of boys to girls
We want the new ratio of boys to girls to be \( 14:9 \). Let \( y \) be the number of boys that need to be admitted. Then the new number of boys will be:
\[ \text{New number of boys} = 400 + y \]

We can set up the ratio:
\[ \frac{400 + y}{270} = \frac{14}{9} \]

### Step 4: Cross-multiply to solve for \( y \)
Cross-multiplying gives us:
\[ 9(400 + y) = 14 \times 270 \]
Calculating \( 14 \times 270 \):
\[ 14 \times 270 = 3780 \]
So, we have:
\[ 9(400 + y) = 3780 \]
Expanding this:
\[ 3600 + 9y = 3780 \]

### Step 5: Solve for \( y \)
Subtract 3600 from both sides:
\[ 9y = 3780 - 3600 \]
\[ 9y = 180 \]
Dividing both sides by 9:
\[ y = \frac{180}{9} = 20 \]

### Conclusion
Thus, the number of boys that should be admitted is \( \boxed{20} \).

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