The proportions of gold in three alloys are 40%, 50% and 80% respectively. These alloys are mixed in certain proportions to obtain 30 kg of a new alloy, which has 55% gold. If the amount of the first alloy is 9 kg, what is the amount in kg of the third alloy used?

Given that the total weight of the first alloy = 9 Kg, and the proportion of gold in it is 40%.
So the weight of gold in the first alloy = $$0.4\times\ 9$$ = 3.6 Kg

Let the total weight of the second alloy be x Kg.
It is given that the proportion of gold in the second alloy is 50%. So the weight of gold in the second alloy = $$\ \dfrac{\ x}{2}$$ Kg.

Given that the total weight of the new alloy is 30 Kg.
Total weight of the third alloy = $$\ \ 30\ -\ 9-x$$ = $$21-x$$

Proportion of gold in the third alloy = 80%
Weight of gold in the third alloy = $$0.8*(21-x)$$

Total weight of gold combined = $$0.55*30$$ = 16.5 Kg
We get the following equation:
$$3.6+\dfrac{x}{2}+0.8\left(21-x\right)=16.5$$
$$36 + 5x + 168 - 8x = 165$$
$$3x=39$$
$$x=13$$

Hence the total weight of the third alloy = $$21-x = 8$$ Kg