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 girls, becomes 14:9?

Options

A. 25
B. 15
C. 20
D. 30

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the number of boys and girls in the school.
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 \(3x\).

Since the total number of students is 640, we can write the equation:
\[ 5x + 3x = 640 \]
\[ 8x = 640 \]
\[ x = 80 \]

Now, calculate the number of boys and girls:
- Number of boys = \(5x = 5 \times 80 = 400\)
- Number of girls = \(3x = 3 \times 80 = 240\)

### Step 2: Update the number of girls after admitting more girls.
According to the problem, 30 more girls are admitted. Therefore, 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.
Let \(x\) be the number of boys to be admitted. The new number of boys will be:
\[ \text{New number of boys} = 400 + x \]

We want the ratio of boys to girls to be \(14:9\). Thus, we can set up the equation:
\[ \frac{400 + x}{270} = \frac{14}{9} \]

### Step 4: Cross-multiply to solve for \(x\).
Cross-multiplying gives:
\[ 9(400 + x) = 14 \times 270 \]
Calculating the right side:
\[ 14 \times 270 = 3780 \]
So we have:
\[ 9(400 + x) = 3780 \]

### Step 5: Expand and simplify the equation.
Expanding the left side:
\[ 3600 + 9x = 3780 \]

Now, isolate \(9x\):
\[ 9x = 3780 - 3600 \]
\[ 9x = 180 \]

### Step 6: Solve for \(x\).
Dividing both sides by 9:
\[ x = \frac{180}{9} = 20 \]

### Conclusion:
The number of additional boys that should be admitted is **20**.

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