## Task 2 - 170732

Uli has four different types of steel balls, labeled $A, B, C$, and $D$. Balls of the same type always have the same weight. Using a balance scale, he found that two balls of type $B$ weigh exactly as much as one ball of type $A$. He also found that three balls of type $C$ weigh as much as one ball of type $B$, and that five balls of type $D$ weigh the same as one ball of type $C$.

a) How many balls of type $D$ must Uli place in one (empty) scale pan if they are to balance one ball of type $A$ in the other scale pan?

b) In one scale pan, there are 20 balls of type $D$ and 5 balls of type $C$. How many balls of type $B$ must Uli place in the other (empty) scale pan to achieve balance?
a) The weight of a ball of type $A$ is equal to twice the weight of a ball of type $B$. A ball of type $B$ has three times the weight of a ball of type $C$, so a ball of type $A$ has the same weight as 6 balls of type $C$.

A ball of type $C$ has five times the weight of a ball of type $D$; therefore, the weight of 6 balls of type $C$ is equal to the weight of 30 balls of type $D$. It follows that Uli must place 30 balls of type $D$ on one side of the scale to achieve balance.

b) The weight of 5 balls of type $D$ is equal to the weight of one ball of type $C$, so 20 balls of type $D$ have the same weight as 4 balls of type $C$. The weight of 20 balls of type $D$ and 5 balls of type $C$ is therefore equal to the weight of 9 balls of type $C$.

Since three balls of type $C$ weigh as much as one ball of type $B$, 9 balls of type $C$ therefore weigh as much as 3 balls of type $B$. Uli must therefore place 3 balls of type $B$ on the other side of the scale to balance 20 balls of type $D$ and 5 balls of type $C$.