The ratio of boys and girls in a college is `5 : 3`. If 50 boys leave the college and 50 girls join the college, the ratio becomes `9 : 7`. The number of boys in the college is:

Options

A. 300
B. 400
C. 500
D. 600

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Define the Initial Ratio
The initial ratio of boys to girls is given as 5:3. Let's denote:
- Number of boys = 5x
- Number of girls = 3x

### Step 2: Apply the Changes
According to the problem, 50 boys leave the college, and 50 girls join. After these changes, the new number of boys and girls will be:
- New number of boys = 5x - 50
- New number of girls = 3x + 50

### Step 3: Set Up the New Ratio
The new ratio of boys to girls is given as 9:7. We can set up the equation based on this ratio:
\[ \frac{5x - 50}{3x + 50} = \frac{9}{7} \]

### Step 4: Cross-Multiply to Solve for x
Cross-multiplying gives us:
\[ 7(5x - 50) = 9(3x + 50) \]

Expanding both sides:
\[ 35x - 350 = 27x + 450 \]

### Step 5: Rearrange the Equation
Now, we will rearrange the equation to isolate x:
\[ 35x - 27x = 450 + 350 \]
\[ 8x = 800 \]

### Step 6: Solve for x
Now, divide both sides by 8:
\[ x = 100 \]

### Step 7: Calculate the Number of Boys
Now that we have the value of x, we can find the number of boys:
\[ \text{Number of boys} = 5x = 5 \times 100 = 500 \]

### Final Answer
The number of boys in the college is **500**.

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