Find the ratio of two numbers if the ratio of their arithmetic mean to their geometric mean is $25 : 24$
1. Let the two numbers be \(a\) and \(b\). The arithmetic mean (AM) of \(a\) and \(b\) is given by:
   \[
   \text{AM} = \frac{a + b}{2}
   \]
   The geometric mean (GM) of \(a\) and \(b\) is given by:
   \[
   \text{GM} = \sqrt{ab}
   \]

2. According to the problem, the ratio of the arithmetic mean to the geometric mean is \(25 : 24\). Therefore, we can write:
   \[
   \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{25}{24}
   \]

3. Simplify the equation:
   \[
   \frac{a + b}{2\sqrt{ab}} = \frac{25}{24}
   \]

4. Cross-multiplying to eliminate the fraction:
   \[
   24(a + b) = 50\sqrt{ab}
   \]

5. Square both sides to remove the square root:
   \[
   (24(a + b))^2 = (50\sqrt{ab})^2
   \]
   \[
   576(a + b)^2 = 2500ab
   \]

6. Expand and simplify:
   \[
   576(a^2 + 2ab + b^2) = 2500ab
   \]
   \[
   576a^2 + 1152ab + 576b^2 = 2500ab
   \]

7. Rearrange the terms to form a quadratic equation:
   \[
   576a^2 + 1152ab + 576b^2 - 2500ab = 0
   \]
   \[
   576a^2 - 1348ab + 576b^2 = 0
   \]

8. Factorize the quadratic equation:
   \[
   4(144a^2 - 337ab + 144b^2) = 0
   \]
   \[
   4(9a - 16b)(16a - 9b) = 0
   \]

9. Set each factor to zero and solve for \(b\):
   \[
   9a - 16b = 0 \quad \text{or} \quad 16a - 9b = 0
   \]
   \[
   b = \frac{9a}{16} \quad \text{or} \quad b = \frac{16a}{9}
   \]

10. Therefore, the ratio of the two numbers can be:
    \[
    \frac{a}{b} = \frac{16}{9} \quad \text{or} \quad \frac{a}{b} = \frac{9}{16}
    \]

The final answer is \(\boxed{\frac{9}{16}, \frac{16}{9}}\)