Our word problem this week is pure genius. Can you solve it in a flash?
24–karat gold is 100% pure gold. How many grams of pure gold are there in a 15 kilogram, 16–karat gold bar?
Our solution waaay down below.
When we think of gold, somehow an ancient Greek flashing the streets springs to mind. Wait! That’s right, his name was Archimedes of Syracuse (287 – 212BC), a famous Greek scholar, mathematician, physicist, engineer, inventor, and astronomer.
This particular story about Archimedes supposedly occurred because the king of Syracuse had a dilemma. 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. He challenged Archimedes to determine if his crown was adulterated gold. Archimedes pondered the problem. He could weigh the crown accurately, and it’s weight matched the quantity of gold provided. However, he could not measure it’s oddly shaped volume. Archimedes knew that the density of gold was nearly twice the density of silver. That means, a given weight of gold occupies half the volume of the equivalent weight of silver.
Thinking deeply, he went to the public baths. 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!”
Taking a mass of pure gold the exact same weight as the crown, he measured the amount of water it displaced. Then he measured the amount of water displaced by the crown. Since the crown displaced more water, it meant that it was less dense, and hence was not pure. We’re told the goldsmith lost his crown over his thievery.
Now for the solution to our problem. 6–karat gold is 16/24 pure gold, or 2/3 g. Since two thirds of the gold bar is pure gold, that means that there are 15 × 2/3 = 10 kg of pure gold. There are 1000 grams per kilogram, so 10 kilograms of pure gold is the same as 10,000 grams pure gold. This is a wonderful example of a wholes-and-parts-multi-step question that we love teaching at Mathnasium. When our student demonstrates understanding, then we'll extend the problem.
Let’s consider Archimedes’ problem. If 15 kg of pure gold were adulterated to 16-karat gold with silver. How much more volume would 15 kg of 16-karat gold occupy? The density of gold is 19.32 grams per cubic centimeter. The density of silver is 10.49 grams per centimeter. 15 kg of pure gold would occupy 15,000 g ÷ 19.32 g/cm3 = 776 cm3. The 16-karat gold has 10 kg of pure gold and 5 kg of pure silver. It’s volume would be 10,000 ÷ 19.32 + 5,000 ÷ 10.49 = 518 + 477 = 995 cm3. Hence, the 16-karat gold occupies 995 – 776 = 219 cm3 more volume.
This eureka moment was the foundation for Archimedes' study in displacement and bouyancy. It's enshired forever as the Archimedes Principle, and is an easy physics experiment done in elementary school.
Inspiration strikes us at odd times, because our minds are an amazing puzzling machine constantly searching to fit puzzle pieces in a way that make sense. It's a processs that usually occurs at rest after repeated viewing the problem. Hence, our method of teaching at Mathnasium via repeated exposure to problems and solutions with restful time off between visits – no homework, yay! As for myself, my problem is when I look at my wedding ring, I think streaking at Syracuse!
Contact:
Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), [email protected]
246 Main St. #A
Fort Lee, NJ 07024
Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, Palisades Park, North Bergen, West New York, and Fairview.
(201) 461-6284
(201) 969-6284
