Answers to Problems of the Week - July 19 to July 25

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: Babs has a handful of 24 cheese crackers. If she eats half of them, then eats half of what’s left, how many cheese crackers will Babs have left?

Answer:  6 cheese cracker

Solution:  Half of 24 is 12, so Babs will have 12 crackers left after she eats half of them. Half of 12 is 6, so she will have 6 crackers left after she eats half of half the cheese crackers.

Upper Elementary:

Question: Each candle will burn for 30 minutes. How many candles will you need for a power outage that lasts 3 hours and 45 minutes?

Answer:  7¹/₂ candles

Solution:  One candle lasts half an hour, so 2 candles last an hour. For the first 3 hours, you will need 3 × 2 = 6 candles. That leaves another 45 minutes, which is exactly between half an hour and a full hour, so you’re going to need another candle and a half because 1¹/₂ is exactly between 1 and 2 candles. That makes a total of 6 + 1¹/₂ = 7¹/₂ candles.

Middle School: 

Question: What’s the probability of tossing the same face of a coin in a fair coin toss three times in a row?

Answer:  1 out of 4

Solution:  Let’s say we want to toss just heads. Each toss has a one out of 2 chance of landing on heads. So, the chance of tossing 2 heads in a row is ¹/₂ × ¹/₂ = ¹/₄. For the last toss, the probability of tossing heads is one out of 2 again, so we multiply ¹/₄ × ¹/₂ = ¹/₈. So, the probability of tossing 3 heads in a row is 1 out of 8. You have an equal chance of tossing all tails, too, so the probability of tossing all the same face in a fair coin toss is ¹/₈ + ¹/₈ = ¹/₄, or 1 out of 4.

Algebra and Up: 

Question: A bunch of bananas and a cantaloupe together weigh 6 pounds. A papaya and a cantaloupe together weigh 5 pounds. A papaya and a bunch of bananas together weigh 4 pounds. Find the weight of each fruit.

Answer:  The bananas weigh 2.5 pounds, the cantaloupe 3.5 pounds, and the papaya weighs 1.5 pounds.

Solution:  The first two pairs, bananas + cantaloupe and papaya + cantaloupe, have the cantaloupe in common, but bananas + cantaloupe is 1 pound heavier than papaya + cantaloupe, so the bananas must be 1 pound heavier than the papaya: B = P + 1. We also know that the bananas and papaya together weigh 4 pounds: B + P = 4. If we substitute P + 1 in for B in the second equation, we find that 2P + 1 = 4, so 2P = 3, so P = 1.5 pounds. From there, we can find that the bananas weigh 2.5 pounds and the cantaloupe weighs 3.5 pounds.

Challenge problem to take home. 10 stars for the correct answer. 

Pictured below are a regular hexagon and the same hexagon stellated into a star:

If the area of the hexagon is 25 square units, what is the area of the star?

Answer: 50 square units