Propositiones ad Acuendos Juvenes/Problems/7 - De Disco Pesante Libras XXX

by : Problem $7$

 * De Disco Pesante Libras $\text {XXX}$ : A Dish weighing $30$ Pounds
 * A dish weighing $30$ pounds is made of gold, silver, brass, and lead.
 * It contains $3$ times as much silver as gold;
 * $3$ times as much brass as silver;
 * and $3$ times as much lead as brass.


 * How much is there of each?

Solution

 * Gold: $9$ ounces
 * Silver: $2$ (troy) pounds and $3$ ounces
 * Brass: $6$ (troy) pounds and $9$ ounces
 * Lead: $20$ (troy) pounds and $3$ ounces.

Proof
Let $x$ be the amount in (troy) pounds of gold in the dish.

Then:
 * the amount of silver in the dish is $3 x$
 * the amount of brass in the dish is $3 \times 3 x = 9 x$
 * the amount of lead in the dish is $3 \times 9 x = 27 x$

Thus, knowing the total quantity of metal in the dish:

Given that there are $12$ (troy) ounces in one (troy) pound, we can express this as: