Prince Braggart of Spoof will be crowned tomorrow as the new king. His 200 year old crown was made from unmixed pure gold and pure silver only. Unfortunately, a fire destroyed the record of how much of the crown in gold and how much is silver. Of course, the main topic of royal conversation among the many kings at the coronation party, is how much gold and how much silver their crowns contain. Prince Braggart wishes to brag about his crown. To do this in a royal manner (no bluffing), he needs to know just how much gold and how much silver his crown contains. As the court mathematician, Prince Braggart has ordered you to find this out before the coronation. Melting down the crown or cutting off a sample for a chemical assay will be harmful to your longevity. You know that the density of gold is close to 20 g/cm³ and the density of silver is close to 10 g/cm³.

Following the lead of the great scientist, Archimedes* (287-121 B.C.) you weigh the crown and find that it has a mass of 3000 grams.  You immerse it in water and measure the volume of the overflow and find that the volume of the crown is 200 cm³. 

A. Write equations in symbols relating the following quantities that will permit you to calculate all the unknowns. The quantities you know are marked with an asterisk (*).  Do not use values but only the symbols shown below.

• the density of gold, ρ_g (*)
• the density of silver, ρ_s (*)
• the mass of the crown, m (*)
• and the volume of the crown, V (*)
• the volume of gold in the crown, V_g
• the volume of silver in the crown, V_s
• the mass of gold in the crown, m_g
• the mass of silver in the crown, m_s 

B. Solve your equations using the numbers given.

From Jerome Dancis, "Supposedly Difficult Arithmetic Word Problems: Keep it Simple for Students."