Two numbers are in the ratio 5 : 4. If the difference of their cubes is 61; the numbers are :

1. 5 and 4

2. 25 and 16

3. 10 and 18

4. none of these

Ratio of two numbers are 5:4.

Lets the numbers be $5x$ and $4x$. Hence,

$(5x)^3 - (4x)^3 = 61\\[1em] ⇒ 125x^3 - 64x^3 = 61\\[1em] ⇒ (125 - 64)x^3 = 61\\[1em] ⇒ 61x^3 = 61\\[1em] ⇒ x^3 = \dfrac{61}{61}\\[1em] ⇒ x^3 = 1\\[1em] ⇒ x = \sqrt[3]{1}\\[1em] ⇒ x = 1$ So the numbers are 5 and 4.

Hence, option 1 is the correct option.