When our student demonstrates understanding, then we'll extend the problem. This is a wonderful example of a wholes-and-parts-multi-step question that we love teaching at Mathnasium. There are 1000 grams per kilogram, so 10 kilograms of pure gold is the same as 10,000 grams pure gold. Since two thirds of the gold bar is pure gold, that means that there are 15 × 2/ 3 = 10 kg of pure gold. 6–karat gold is 16/ 24 pure gold, or 2/ 3 g. We’re told the goldsmith lost his crown over his thievery. Since the crown displaced more water, it meant that it was less dense, and hence was not pure. Then he measured the amount of water displaced by the crown. Taking a mass of pure gold the exact same weight as the crown, he measured the amount of water it displaced. After he stepped into the bath, he realized that the more his sank in the bath, the more water was displaced – the displaced amount being exactly the volume of his body! He realized he had solved his problem of measuring the volume of the crown and rushed home naked shouting, “Eureka! Eureka!” Or translated: “I’ve found it! I’ve found it!” Thinking deeply, he went to the public baths. That means, a given weight of gold occupies half the volume of the equivalent weight of silver. Archimedes knew that the density of gold was nearly twice the density of silver. However, he could not measure it’s oddly shaped volume. He could weigh the crown accurately, and it’s weight matched the quantity of gold provided. He challenged Archimedes to determine if his crown was adulterated gold. The king had given pure gold to his goldsmith for a crown, but he suspected that the goldsmith had stolen some of the gold and replaced the missing portion with silver. This particular story about Archimedes supposedly occurred because the king of Syracuse had a dilemma. Wait! That’s right, his name was Archimedes of Syracuse (287 – 212BC), a famous Greek scholar, mathematician, physicist, engineer, inventor, and astronomer. When we think of gold, somehow an ancient Greek flashing the streets springs to mind.
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