Saturday April 16, 2011

This is federal election time in Canada as well as budget talk in the US. Questions are based on those!

• The total US debt  as of March 25, 2011 is  14.26 trillion dollars. How much is 1 trillion? (Just to put this number in perspective - total public parking spaces (metered/road) etc. in San Francisco is about 500,000. Average cost of a car in that area is about $30,000. Compare the debt in terms of some quantity you can imagine.
• Here is an alternate way to understand this figure. The population of US is approx. 311,000,000. Whats the debt per person then? per family? What will a family do if they are in such a debt?
• Canada's debt is approximately 540 billion dollars, and population is 33,739,000. Compare the debt per person and per family v/s US. Growing economy like India has a debt of approximately 750 billion $s. Population is 1.2 billion. 
• What is debt? This is essentially what govt. owns to people (shares/bonds...). Usually it is seen in comparison to GDP, gross domestic production. This is the market value of all the goods and services produced within a country in one year.  Canada's GDP is about 1.6 trillion $, debt around 540 billion $. US GDP is 14.6 trillion and debt 14 trillion. (How can Prime Minister/President reduce debt and increase GDP - whats a healthy economy?) 
• During the federal election in Canada, it is claimed that the cost of election is $300 million dollars. Calculate whats the cost of election per household in Canada? We are having almost one election/year - as opposed to one in 5 years.
• In 2008, in Canadian elections, there were approximately 23,000,000 voters on list and only 14,000,000 voted. The distribution of votes in this election between main parties (Conservatives, Liberal, Bloc, NDP, Green) were  (38%,27%,10%,18%,7%) and the number of seats in the parliament were (143, 77, 49, 37, 0), respectively.  What do these numbers mean? Does parliament reflect the % of votes? Can you think of a better mechanism?
• If you hear the results on May 2, after the polls close, you will see many types of graphs, analysis, trends, ups and downs etc.   Can you think of drawing a line graph whose slopes are 1, 0, -1, 2, 3, -2, -3.  What linear equations these graphs satisfy assuming that they pass through the origin.  
• If I have a line segment, lets say originating at (0,2) and ending at (4,13), then I can measure its length by actually drawing it and then use a ruler. Can I do in some other way as well?  In general if the coordinate of the endpoints are (x1,y1) and (x2,y2), what is its length?

Sunday April 10, 2011

1. On the Boxing Day (26th Dec.), stores need to come up with curious ways to give discounts to attract customer. Here is an interesting one - the store is offering a discount of  an additional 10% every hour  -  what will be the price of an item worth $100, at 9AM, 10AM, 11AM, ..., 5PM. Think of - it the discount is on the original price or the last listed price. Since you know that there are limited quantities of each of the items,  how should you approach?  If all the customers can cooperate what could be your strategy - what if they do not want to cooperate at all?
2. In 10 seconds, the distance that a cheetah, cyclist, and an Olympian  runner can cover are respectively 300m, 160m and 110m. What is their speed? How much they would have covered in 8 seconds or 12 seconds - assuming a constant speed? Whats the best way to find out the distance for any given time duration?
3. A right angled triangle with base 4mts, aligned to x-axis, has an area of 6m^2. What is the slope of its hypotenuse? 
4. Whats is the slope of a Standard Staircase?  (By the way the code for staircases is 7-11, i.e. 7 inch rise for each 11inch.)  Typically the 1st floor ceiling is about 9ft,  how many steps will it require?  Whats the linear distance one needs to reach 9ft, if each step is 11inch deep. Many places steps are made in L, U or semicircular shape - any idea why one builds these shapes?
5.  Last few questions dealt with the slope of a straight line - which is defined to be `rise' over `run'. What can one say about slope of a curve? or a surface? How will one go about computing slope of a curve?
6. Each electron has a charge of  1.6 * 10^-19 Coulomb.  How many electrons you need to make a charge of 1 Coulomb?
7. Current in electrical circuits means how much charge passes through a conductor in 1 second. For example, current of 1 Ampere means a charge of 1 Coulomb has gone through the conductor in 1 second - think of how many electrons have moved through - traffic jams?   If the fuse in the house has a rating of 10 A, then how much charge has gone through the fuse in 1 second? How many electrons? 
8. My house has an electric panel and it is rated as 100 Amp. What does that mean? Lets take any household appliance - for example a computer - and lets look at its ratings? Can we figure out how many computers can we run on a 100 Amp circuit, where we have 110 Volts. What about some more heavy duty appliance like - drier or stove. For example stove are rated for 20Amps.

Sunday April 3, 2011

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•  The above graph  (taken from a Forex Blog) shows the trend of CAD/USD loonie over last  5 years. Any conclusions can you draw?
• This is year of India in Canada. Indian Government decided to take up a survey to see how are Indian's performing outside India. They needed to compare Indians living in `similar' type of countries - for example US, Canada, UK and Australia. What should be a good hypothesis to test? What kind of primary/secondary data can be used? What kind of conclusions can be drawn? Is there any point of doing this kind of study?
• Mr. Ad. needs to form a team of 30 players who will participate in several of summer sports (sort of mini summer olympics) where the sports include running, field hockey, tennis, soccer, jumping, swimming, etc. What selection criteria should he use (name at least 5).  Design some hypothesis to test whether the selection criteria leads to medals. What kind of data primary or secondary can be used to verify his hypothesis.
• Here is a chart showing  (taken from here) Speed v/s Safe stopping distance in icy v/s normal conditions. Why do you think its not a straight line curve?

• This web-site lists some of the super cars, their price and the time (in seconds) it takes them to reach the speed of 100Km/hr.  For example, 2005 Ferrari FXX reaches that speed in 2.5 secs, and it costs $1.5 million.  What kind of plot do you expect in terms of price v/s time it takes to attain 100Km/hr. (For example my car takes 7.4 secs - though I never tried that).
• What kind of distance-time graph you will expect when you hit a Home run in Baseball - the launch of a satellite - a train entering a station to stop - tiger chasing its kill - in general a typical commute from home to office  (In distance time graph, you will plot the distance of the ball (or an object) from its original position as time increases).
• How many odd 1-digit, 2-digit, 3-digit, 7-digit numbers can be formed using the digits 1,2,3,4,5,6,7, if each number consists of distinct digits (e.g. 223 is not valid!).
• This one is based on Zero Knowledge Proofs - its an interesting  concept.  See wikipedia entry on this (this picture is from there) Idea is pretty simple. Both of these persons don't trust each other. Person standing outside the cave (call him Bob), needs to know the number-key of  the door which is at the far end of the cave. Person standing in the cave (call her Alice) claims that she knows the number key, and is willing to give that key for $100. Bob, can gamble, and pay Alice $100, and hope that she is telling the truth. But she may not!  How can Alice convince Bob that she knows the key, without revealing the key, especially before Bob pays her $100! This kind of technique is used, for example the password you have on your bank card.

 

Sunday, March 27 2011

• Why does long division works? This is an exercise in number representation. Divide 123456 by 11 - explain why your method works? Divide 10000250 by 10 and then explain why your method works? Did your reasoning hold with consecutive zeros?
• Now try dividing x^3+x^2-3x+1 by x-1. Follow the same steps, and try to get rid of the highest powers of x in each step. Cross check your solution by multiplying your result with (x-1).  Did the same reasoning hold for long division?
• Mr. A has 30% sens card, 25% pens card, 15% ducks card, and rest of them are equally distributed among 27 different teams. He has been buying 4 cards per week, costing him 50c/week. He does this whenever he plays his hockey game. The season in all consists of 32 games - how much money he spent - how many sens, pens, and ducks card he has?  How many HABS card he has? What are chances that he has a PK Subban card? How much money he should invest to be more or less certain that he has a Subban's card?
• Consider a square whose each side is of length one (unit square). How long is its diagonal? How will we do this for a unit cube?
• Lets do the above problem for a cylinder, whose base has a radius of 1.5m, and its height is 4m. How long is the diagonal.
  

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, 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?

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?