If the AM and GM of two positive numbers are $5$ and $4$ respectively, the numbers are

1. Let the two numbers be $a$ and $b$. 2. AM = $(a+b)/2 = 5$, so $a + b = 10$. 3. GM = $\sqrt{ab} = 4$, so $ab = 16$. 4. These two equations describe the SUM and PRODUCT of two numbers — they are roots of the quadratic $x^2 - 10x + 16 = 0$. 5. Use quadratic formula: $x = (10 \pm \sqrt{100 - 64})/2 = (10 \pm 6)/2$, giving $x = 8$ or $x = 2$. 6. So the numbers are $\boxed{2 \text{ and } 8}$. 7. Sanity: AM = $(2+8)/2 = 5$ ✓; GM = $\sqrt{16} = 4$ ✓. 8. Other options don't satisfy both conditions. _Source: NCERT Class 11 Mathematics, Ch 8, §8.4 (AM-GM relationship + Examples), p. 10–11._