Tanya's grandfather was 8 times older to her 16 years ago. He would be 3 times of her age 8 years from now. What was ratio of ages of Tanya and her grandfather 8 years ago?

Let Tanya's current age be $x$ $16$ years ago: $y - 16 = 8(x - 16)$ where $y$ is grandfather's current age $8$ years from now: $y + 8 = 3(x + 8)$ From first equation: $y - 16 = 8x - 128$ $y = 8x - 112$ ... $(1)$ From second equation: $y + 8 = 3x + 24$ $y = 3x + 16$ ... $(2)$ Equating $(1)$ and $(2)$: $8x - 112 = 3x + 16$ $5x = 128$ $x = \frac{128}{5}$ Substituting in equation $(1)$: $y = 8(\frac{128}{5}) - 112$ $y = \frac{1024}{5} - 112$ $y = \frac{1024-560}{5}$ $y = \frac{464}{5}$ $8$ years ago: Tanya's age = $\frac{128}{5} - 8 = \frac{128-40}{5} = \frac{88}{5}$ Grandfather's age = $\frac{464}{5} - 8 = \frac{464-40}{5} = \frac{424}{5}$ Ratio = $\frac{88}{5} : \frac{424}{5}$ $= 88 : 424$ $= 11 : 53$ Therefore, the ratio of Tanya's age to her grandfather's age $8$ years ago was $11 : 53$.