Sunday, February 27, 2011

Saturday February 26, 2011

  • Look at the figure on the left. Came from ancient Chinese Math (Zhou bi, 1045 BC onwards) )! It is drawn on a 7x7 square. Each triangle (yellow or green) is of dimension 3x4. The black square is of dimension 1x1.  Each triangle is right angled. Whats the area of the square made up of green triangles and black square. What's the side of this square - do you see the Pythogorean theorem! 
  • You can try to do the same with outer square being of size 14x14, each triangle of dimension 6x8, and the black square of dimension 2x2.
  • In general, you can prove the Pythogorean theorem as follows: assume that the sides of the right-angled triangle are a and b, and we need to show the hypotenuse is sqrt(a^2+b^2). Assume a is greater than or equal to b. Draw the above picture by taking the sides of the outer square to be a+b. The dimension of the inner square are a-b times a-b.   Now it should be straightforward to see that the area of the green square (inclusive of the black one) is  4*area of green triangles + area of the black square = 4*1/2*ab+(a-b)(a-b)= a^2+b^2, and hence the side of this square will be sqrt(a^2+b^2).
  • This is not really a math problem - sort of related to do with string manipulation -You need to change WIDE to RISE, where the rules of the game is to change only one character at a time and each intermediate word is meaningful. Whats the smallest number of transformations you need to do? Try doing this from LOVE to RIFT.
  • An outdoor swimming pool is 25ft by 50ft and is 8 ft deep. In the morning it is full of water, and by the end of the hot summer day, water drops down by 1.5 feet, due to evaporation. How much water is lost? How many buckets it is? What is the rate of evaporation - lets say we have 16 hours of sunlight in Ottawa in summer - but the peak is from 11AM till  7PM. How can we minimize the evaporation?
  • The ratio of the number of goals between  Alfie and Alex is 3:4 and between Alex and Sid is 5:6.  Whats the ratio of goals between Alfie and Sid.
  • Anant in his grade 5/6  class found the following stat when he conducted the chocolate poll.  In all 80% liked the chocolate. The ratio of Grade 5 to Grade 6 kids in his class is 2:3. What are the chances that when you pick a `random' kid from Anant's class - that this one really likes chocolate and is in grade 5?
  • Four identical cubes are placed next to each other to make a rectangular prism. The surface area of this prism is 360 sq cms less than the sum total of the surface area of the  four cubes.  Can you determine the dimension of the cube?
  • Next year my age and Mr. A's age  will be prime numbers, and the product of our ages will be 611. How old are we now? Of course, there is exactly one way to non-trivially factor 611, since its a product of primes. How will we do it, if in place of 611, its a very large number - for example a number made up of 500 digits! The computationally difficulty of finding factors of such large numbers lies at the heart of most of the secure transactions over the internet!
  • Whats the last digit in the product of five consecutive numbers, where one of those numbers has 7 as its last digit.
  • What is the smallest possible number that can be multiplied to 120, so that the product is a cubic number?

    Sunday, February 13, 2011

    Sunday, Febrauary 13, 2011

    Problems are inspired by the Grade 9 text book today!
    • Typically, in bikes (cycles), there is a plastic reflecting light which is attached to a wheel, so that the bike is visible to a car coming across. What  will be the shape of the path traversed by this light, when the bike moves along?
    • Mr. Forget started  to walk back to his home from the grocery store, located 650 mts from his home. He started to walk back in the direction of his house, but went 800 mts, then suddenly remembers that he has gone past, and switches direction, and then walks back half the distance this time (400 mts), OOPS, he again remembers and then switches direction and walks back, and this continues .. Will he ever reach his home?
    • What is the total distance a puck travels in a hockey game? How will go about estimating this distance.  Ice rink is 200ft by 85 ft?  Assume that the hockey game is 60 minutes long, an average pass is approximately 15ft.
    • You  take a very long string, and double it up, then redouble it, then reredouble it, and so on. Lets say you did it 50 times. Now cut the string from the middle. How many pieces will you get?
    • How will you show that the sum of three consecutive numbers is divisible by 3. Can one make the same statement for 4 consecutive numbers being divisble by 4 and so on. It of course doesn't work for two.
    • Lets do some fun experiments with the Mobius strip. Take a ribbon, twist and then tape it. Mark the center line - and cut along, see what happens?  What if you have two lines - say 1/3rd and 2/3rd away and then cut along them, and see what happens?
    • Half of a fraction, decreased by 0.75 results in 7/12. What was the fraction?
    • How many different bracelets can be made consisting of 2 red and 3 green beads?
    • What is the smallest and largest possible value of a/b + c/d, where a,b,c,d need to take values from {1,2,3,4}?
    • What will be the last digit in 2^1234?

    Sunday, February 6, 2011

    February 6, 2011

    • Dr. A. for the sleepover party teased his friends. While they were sleeping, he put a pink sticker on the forehead of all of his friends. When they woke up, they all started laughing looking at each other! Then suddenly Mr. B stopped laughing - Why?
    • How can we find the surface area of a tetrahedron whose sides are of length 1. What about an octahedron? What about their volumes?
    • Mr A. has two types of cards in his pocket. One card is red on both the sides and other one has red on one side and green on the other side. He takes a card out - and sees that one side is red - what are the chances that the other side is red as well? (50% is not the right answer BTW)
    • How heavy is the water mattress? Recall that Queen mattress dimensions are 152x200x20 cms and 1 cubic meter of water weighs 1000Kilos.
    • How to measure volume of a stone? Suppose you have a beaker (which is a cylinder). Lets say you fill the beaker of radius 6cms with water, and the height of the water in the beaker reads 14.2 cms.  Now gently drop the stone in the beaker and water rises to 18.7 cms. What is the volume of the stone? 
    • Next set of questions are from Grade 9 EQAO testing
    1. If x=1/3, then what is 6x^2.
    2. Typically car sales (and many other sales)  are paid a fixed amount per week and certain percentage of their sales. Lets say that this person earns $500/week and 2.5% of the sales. If  the total payment for the week was $700, then how much were the sales in that week?
    3. What is the sum of the interior angles of a regular pentagon; hexagon; 12 sided figures?
    4. You ordered CDs from your favourite store online. Each CD costs $11.44 + Tax. Total you paid is $90.49 which includes HST.  How many CDs did you buy?