1. The problem states that green and yellow dyes are mixed in a ratio of 5:6 to make 44 litres of lime-coloured dye. 2. We want to find how much green and yellow dye there are in these 44 litres. 3. The total parts in the initial mix are $5 + 6 = 11$ parts. 4. Each part therefore represents $\frac{44}{11} = 4$ litres. 5. The amount of green dye is $5 \times 4 = 20$ litres. 6. The amount of yellow dye is $6 \times 4 = 24$ litres. 7. Next, 8 litres of yellow dye is added, so the new quantity of yellow dye is $24 + 8 = 32$ litres. 8. The quantity of green dye remains $20$ litres. 9. The new ratio of green to yellow dye is $20 : 32$. 10. To simplify, find the greatest common divisor (GCD) of 20 and 32, which is 4. 11. Divide both terms by 4: $\frac{20}{4} : \frac{32}{4} = 5 : 8$. 12. Therefore, the new ratio of green to yellow dye in simplest form is $\boxed{5 : 8}$.