The number of girls in a college was 50 less than the number of boys. Later, some girls joined the college, and the ratio of boys to girls became $2 : 5$. Find the minimum number of girls who joined the college recently, given that the number of boys and girls are positive integers.

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**Solution:**

1. Let the number of girls initially be $x$.

2. Then the number of boys is $x + 50$ (since girls are 50 less than boys).

3. Let $y$ be the number of girls who joined later.

4. After these girls join, the ratio of boys to girls becomes:
   $$\frac{x + 50}{x + y} = \frac{2}{5}$$

5. Cross-multiplying:
   $$5(x + 50) = 2(x + y)$$

6. Expanding:
   $$5x + 250 = 2x + 2y$$

7. Rearranging to isolate $y$:
   $$5x + 250 - 2x = 2y \implies 3x + 250 = 2y$$

8. Therefore,
   $$y = \frac{3x + 250}{2} = \frac{3x}{2} + 125$$

9. Since $y$ must be an integer, $\frac{3x}{2}$ must be an integer, which means $3x$ is even. Since 3 is odd, $x$ must be even.

10. To minimize $y$, choose the smallest positive even integer for $x$, which is $2$.

11. Substitute $x = 2$:
    $$y = \frac{3 	imes 2}{2} + 125 = 3 + 125 = 128$$

**Answer:** $128$

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