bonusProblemSolutions

Before a plum is dried to make a prune, it is 92% water. A prune is just 20% water. If only water is evaporated in the process, how many pounds of prunes can be made from 100 pounds of plums?

      Plum is 92% water
      Prunes are 20% water

      before: 92 / (92 + 8) = 0.92
      after :   x / (x + 8) = 0.2
      x = 2 lb water
      So 10 lbs prunes (8 pounds water + 2 pounds of fiber)

7) Pet Preferences

Students were surveyed on their preference for pets. Three fourths of the students surveyed liked dogs, three fifths like cats, and one sixth did not like either. What is the fewest number of students surveyed that could like both cats and dogs?

       Draw a Venn Diagram.
      3/4 + 3/5 + 1/6 = (45 + 36 + 10)/60 = 91/60
      which implies that 31/60 like both cats and dogs because the total needs to be 1.
      Since that is an irreducible fraction, the fewest number that could like both is 31.

8) Fluctuating Stock

If a stock that is currently worth $50 per share rises by 160% this year and then falls by 70% the next year. How much will it then be worth?

    first year : $50 + 1.60 * 50 = $130
   second year: 130 - 0.7 * 130 = $39
   You end up with lass than you started with, even though you were probably expecting more!

9) Spiders and Beetles

In a cage full of bugs, there are beetles (6 legs) and spiders (8 legs). There are 162 more legs than bugs, and 18 more beetles than spiders. How many spiders are there?

Use algebra:
Number of legs = 6B + 8S
6B + 8S - (B + S) = 162
B - S = 18
This gives you two equations and two unknowns. Now solve.
B = 18 + S, so now substitute this for B to get
6(18+S) - 2S - 18 = 162
4S = 72
S = 6; B = 24

10) Birthday Paradox

Suppose there are 23 students in Math League. What is the chance that two students share the same birthday?

     The first student you consider can have his b-day on any day of the year, the next on any day but that day and so on. This reasoning gives us
   365/365 * 364/365 * 363/365 * .... * 343 /356
or
    365!
   ------
   365²³ 342!
If you work this out it comes out to a little more than 50%.
   See http://en.wikipedia.org/wiki/Birthday_problem

11) Hour Glasses
How can you measure nine minutes using just 7 minute and 4 minute hour glasses without taking more than nine minutes?

I saw on yahoo news that this is sometimes given as a Google interview question.
1) Turn over both hour glasses
2) When the 4 minute glass runs out of sand, turn it over again. (4)
3) When the remaining 3 minutes runs out from the 7 minute glass, there will be one minute left on the 4 minutes glass. (7)
4) Turn over the 7 minute glass and measure out one minute of sand using the remaining minute on the 4 minute glass. (8)
5) Turn over the 7 minute glass and use that one minute of sand to get to nine minutes (9)

12) Standing in Line
How many different ways can 6 friends stand in line if Alice and Bob refuse to stand next to each others?

   There are 6! ways for six friends (ABCDEF) to stand in line (permutations).
   If we add the restriction that A and B cannot be adjacent, then we must subtract
      10 * 4! for all the different ways that they could be next to each other.
   ( I got 10 because we have AB----, -AB---, --AB--, ---AB-, ----AB, and BA----, ....)
    There for there are 720 - 240 = 480 different ways.

13) Picking Cards

What is the probability that the top 4 cards drawn from a shuffled deck of 52 cards are all the same suit?

14) Selling Flour for a Profit

A miller takes as toll one tenth of the flour that he grinds for a customer. How much did he grind for a customer who had only one kilogram after the toll had been taken?

15) Spiders and Beetles

Aman lived one fourth of his life as a boy, one fifth as a youth, one third as a man in his prime, and thirteen years in old age. How long did he live?



	Bonus Problem Solutions (for 2011 - 2012 year)

	Here are the solutions for the bonus problems that I have given so far this year. 1) Chimps Bert and Alice Bert is twice as old as Alice was when Bert was as old as Alice is now. There combined age is 105. How old are they? Two ways to solve. First I will try non-algebra a) No matter how you approach this, you must first realize that Alice's when Bert was as old as Alice is now is going to be Alice's current age minus the difference between their ages. In other words, A = A - (B - A). Here is a chart to illustrate: Bert's age now : ******************************************* Alice's age now : **************************[some difference] Bert's age then : ************************** Alice's age then : ************[same difference] Now using guess and check, we try some possible ages. We know that Bert is older than Alice, and they must sum to 105, so lets try 70 and 35 as a random guess. Assuming those candidate ages we see that Alice was 0 when Bert was as old as Alice is now, so that can't be right. We need to adjust our guess so they are close together. Next try 65 and 40. For this we get that Alice was 15 when Bert was as old as Alice is now. So twice that would give 30, but that is not 60, so lets shrink the gap some more. So finally we try 60 years for Bert and 45 years for Alice, and find that Alice was 30 when Bert was as old as Alice is now. So twice is 60, and that matches our assumption and we are done! b) Now the algebraic approach. It may be a little harder to setup the algebraic equations initially, but it should be more straightforward to solve. No matter what approach you pick, there is no way around understanding the problem first, however. Give the insights from above, our equations are: A + B = 105 B = 2 ( A - (B - A)) Now it is a simple matter to solve B = 2 (2A - B) B = 4A - 2B 3B = 4A B = 4/3 A A + 4/3 A = 105 7/3 A = 105 A = 3/7 105 = 45 years, and B = 60 years. 2) Collaborative Jigsaw Puzzle It takes Bert 4 hours to complete a jigsaw puzzle, but Alice can do it in 3 hours. How long would it take them if they worked together? If Bert worked for 12 hours he could complete 3 puzzles. If Alice worked for 12 hours she could complete 4 puzzles. Working together, they solve puzzles at a rate of 7 / 12 puzzles per hour. Working together on a single puzzle they could complete it in 12 / 7 hours per puzzle. Hence the answer is 1 and 5/7 hours. 3) Crossing a Bridge Four people (Alice, Bert, Charlie, Diane) need to cross a bridge in the dark with only one flashlight. Only two can cross at a time. Alice can cross in 1 minute, Bert in 2, Charlie in 5, and Diane in 7 minutes. When a pair goes across they can only go as fast as the slower of the two. They have to catch a bus in 15 minutes. Can they make it? If so, how? Try to have them cross such that people with similar costs cross at the same time. This reduces minimizes overall cost. Is a bit like load balancing for maximum throughput in a concurrent program. One set of steps that will work is: 1) Have Alice and Bert cross first because they can then be used to shuttle the flashlight back quickly (2 min) 2) Alice carries back the flashlight (1) 3) Next have Charlie and Diane cross because they are both slow (7 min). 4) Bert carries back the flashlight (2) 5) Bert and Alice cross again (2) Total is 14 minutes, so they make the bus. 4) Roots What is the least integer N greater than 1 for which the square root of N, the cube root of N, and the fourth root of N are all integers? a² = N b³ = N c⁴ = N where a, b, c, are integers. c will be the smallest. a must be c². Find LCM of 2, 3, 4 = 12. Find x such that x¹² = N, x cannot be 1, so use 2. N = 2¹² = 4096 ( =8^4 = 16^3 = 64^2) So a=8, b=16, c=64 5) Investing Suppose you invest $1,000 in a volatile stock. Each year, with equal probability, it either rises 40% or falls by 30%. What would be the expected (in other words the mean) value of the stock after 50 years? What would be the most likely (in other words the median) value after that time? (hint: they are not the same). See explanation here and applet for demonstration. 6) Making Prunes Before a plum is dried to make a prune, it is 92% water. A prune is just 20% water. If only water is evaporated in the process, how many pounds of prunes can be made from 100 pounds of plums? Plum is 92% water Prunes are 20% water before: 92 / (92 + 8) = 0.92 after : x / (x + 8) = 0.2 x = 2 lb water So 10 lbs prunes (8 pounds water + 2 pounds of fiber) 7) Pet Preferences Students were surveyed on their preference for pets. Three fourths of the students surveyed liked dogs, three fifths like cats, and one sixth did not like either. What is the fewest number of students surveyed that could like both cats and dogs? Draw a Venn Diagram. 3/4 + 3/5 + 1/6 = (45 + 36 + 10)/60 = 91/60 which implies that 31/60 like both cats and dogs because the total needs to be 1. Since that is an irreducible fraction, the fewest number that could like both is 31. 8) Fluctuating Stock If a stock that is currently worth $50 per share rises by 160% this year and then falls by 70% the next year. How much will it then be worth? first year : $50 + 1.60 50 = $130 second year: 130 - 0.7 * 130 = $39 You end up with lass than you started with, even though you were probably expecting more! 9) Spiders and Beetles In a cage full of bugs, there are beetles (6 legs) and spiders (8 legs). There are 162 more legs than bugs, and 18 more beetles than spiders. How many spiders are there? Use algebra: Number of legs = 6B + 8S 6B + 8S - (B + S) = 162 B - S = 18 This gives you two equations and two unknowns. Now solve. B = 18 + S, so now substitute this for B to get 6(18+S) - 2S - 18 = 162 4S = 72 S = 6; B = 24 10) Birthday Paradox Suppose there are 23 students in Math League. What is the chance that two students share the same birthday? The first student you consider can have his b-day on any day of the year, the next on any day but that day and so on. This reasoning gives us 365/365 * 364/365 * 363/365 * .... * 343 /356 or 365! ------ 365²³ 342! If you work this out it comes out to a little more than 50%. See http://en.wikipedia.org/wiki/Birthday_problem 11) Hour Glasses How can you measure nine minutes using just 7 minute and 4 minute hour glasses without taking more than nine minutes? I saw on yahoo news that this is sometimes given as a Google interview question. 1) Turn over both hour glasses 2) When the 4 minute glass runs out of sand, turn it over again. (4) 3) When the remaining 3 minutes runs out from the 7 minute glass, there will be one minute left on the 4 minutes glass. (7) 4) Turn over the 7 minute glass and measure out one minute of sand using the remaining minute on the 4 minute glass. (8) 5) Turn over the 7 minute glass and use that one minute of sand to get to nine minutes (9) 12) Standing in Line How many different ways can 6 friends stand in line if Alice and Bob refuse to stand next to each others? There are 6! ways for six friends (ABCDEF) to stand in line (permutations). If we add the restriction that A and B cannot be adjacent, then we must subtract 10 * 4! for all the different ways that they could be next to each other. ( I got 10 because we have AB----, -AB---, --AB--, ---AB-, ----AB, and BA----, ....) There for there are 720 - 240 = 480 different ways. 13) Picking Cards What is the probability that the top 4 cards drawn from a shuffled deck of 52 cards are all the same suit? 14) Selling Flour for a Profit A miller takes as toll one tenth of the flour that he grinds for a customer. How much did he grind for a customer who had only one kilogram after the toll had been taken? 15) Spiders and Beetles Aman lived one fourth of his life as a boy, one fifth as a youth, one third as a man in his prime, and thirteen years in old age. How long did he live? [ Solutions will appear hear after the day they are due ]