Answers to Problems of the Week - March 21 to March 27

Rewards for Problems of the week:

5 stars for the correct answer at your level; 10 stars for the next level; 20 stars for 2+ levels up.

Lower Elementary:

Question: A woolly mammoth weighs 6 tons. A ton is 2,000 pounds. If a caveman weighs 200 pounds, then how many pounds heavier is the woolly mammoth than the caveman?

Answer: 11,800 pounds

Solution: The woolly mammoth weighs 2,000 + 2,000 + 2,000 + 2,000 + 2,000 + 2,000 = 12,000 pounds, which is 12,000 - 200 = 11,800 pounds heavier than the 200-pound caveman.

Upper Elementary:

Question: The probability that a saber-toothed tiger catches its prey is 3 out of 4. If a saber-toothed tiger has caught the past 3 rhinos it chased after, what is the probability that it will catch the next rhino it chases?

Answer: 3 out of 4

Solution: The probability that the saber-toothed tiger catches the next rhino is still 3 out of 4. The first three catches don’t affect whether or not the saber-toothed tiger catches the next rhino, just like the probability of a flipped penny landing on heads isn’t affected by the penny landing on tails three times in a row.

Middle School:

Question: A Neanderthal hunting party caught between 20 and 30 animals. If a sixth of the animals were reindeer, a fourth of them were goats, a third of them were sheep, and the rest were rabbits, then how many rabbits did the Neanderthals catch?

Answer: 6 rabbits

Solution: Since the animals can be divided into sixths, fourths, and thirds, we know that the total must be a multiple of 6, 4, and 3. The only number between 20 and 30 that is a multiple of 6, 4, and 3 is 24. If we subtract the reindeer (24 ÷ 6 = 4), the goats (24 ÷ 4 = 6), and the sheep (24 ÷ 3 = 8) sheep from the total, we find that the Neanderthals caught 24 – 18 = 6 rabbits.

Algebra and Up:

Question: The Siberian unicorn, a ginormous prehistoric rhinoceros-like animal, survived until around 37,000 BCE. Assume the population of Siberian unicorns was exactly 262,144 in the year 39,000 BCE and followed the model f(t) = 262,144(2)^-t, a function of t time in centuries. According to the model, what year did the population of Siberian unicorns decline to exactly 1? You may use your calculator to solve this problem.

Answer: 37,200 BCE

Solution: One way to solve this problem is to notice that since 262,144 is being multiplied by 2 raised to a negative power, the population will halve itself each century. We can find out how many centuries by finding what power we raise 2 to in order to get 262,144. Since 2¹⁸ = 262,144, it takes 18 centuries for the population to fall from 262,144 to 1, making the year 39,000 – 1,800 = 37,200 BCE.

Challenge problem to take home. 5 stars for the correct answer. 

The number 10 can be written as the sum of four consecutive integers: 1 + 2 + 3 + 4 = 10. The number 100 can be written as the sum of five consecutive integers: 18 + 19 + 20 + 21 + 22 = 100. Is there a way to write 100 as the sum of fewer than five consecutive integers? If so, show which integers you can use. If not, why not?

Answer: No, five is the fewest number of consectuive numbers that add up to 100.