Saturday, September 18, 2010

Today's problems are adapted from the NTSE series of books by Tata McGraw Hill for Class 8th (2008).  

1. How many non-overlapping discs of radius 1cms and fit in a disc of radius 2, 4, and 8cms?
2. Given a disc D of radius 2 cm lying on a table, whats the maximum number of discs of the same radius you can place (lying on the table)  so that each of them is touching D and none of the discs overlap.  
3. These are classical packing problems: for example how many balls of radius 1cm can fit in a giant ball of radius 4cms?  How can we estimate this number? How can we estimate the volume of the gaps?
4. The train coming from Toronto to Ottawa, traveling at a speed of 60Km/Hr made me wait for 10 seconds at the signal. How long (in mts) is the train?
5. If Ms. Rich's income is 25% more than Mr. Poor, then by what percentage is Mr. Poor's income less than Ms. Rich?
6. A door-to-door salesman sells two cookie jars, each for $10.00 - one to Ms. Tough-Bargain and the other to Mr. What-ever.  For Ms. Tough-Bargain he make a loss of 10% and for Mr. What-ever he makes a profit of 10%. Did the salesperson made an overall profit or loss?
7. Anant's piggy bank has in all change for $33.00. It consists only of quarters, dimes and nickles, in the ratio of 5:3:2. How many coins of each type Anant has?
8. We know that in an equilateral triangle, each interior angle is 60 degrees; in a square it is 90 degrees; what do you think it will be in a regular pentagon; regular hexagon; and how we will go about finding it.  
9. The Rideau River near my department has pretty fast current. Ottawa police does their rescue drill operations once in a while.  They go down-stream, and then up-stream.  In 10 minutes, the rescuers travel the downstream in their inflatable boats covering the distance of 1 km. Upstream, for the same distance, it takes them twice as long. What is the speed of the current?
10. In the school's Terry Fox Run, I came 19th from the top and 235th from the bottom. How many kids participated in the run? 

September 6. 2010 (Labour Day)

We met last two weeks, and did problems from the book "Learning Maths 6B by Norrin Hasim (Singapore Asian Publications)". I highly recommend this book for middle school kids for a variety of very nice problems - I haven't seen other levels of this book - but 6B has lots of nice geometric problems. Today's Problems 1-3 are based on this.

1. Find the area of the American Football in the square of side 45 cm long, in the above picture.
2. 1/5th of Anant's mm's are same as that of 1/3rd of Aditya's. If Aditya gives Anant 24 mm's (unlikely though!), Anant will have thrice as many mm's as Aditya. How many they had to begin with?
3. A solid iron ball of volume 576 cubic-centimeter is cast into nine identical solid cubes.  What are the dimensions of the cube?
4. What about the other way around? Suppose you have 16 identical cubes, each having a side length  of 2cm long, is cast as a solid ball. What is its radius?
5. A string forms an equilateral triangle, whose sides are 3.14cms long. The same string is now reshaped as a circular ring. What is the radius of the ring? 
6. A circular race track has a inner radius of 56m and outside radius of 63m. What is the total area of the track itself?
7. In the above problem, if the 7mts wide track is divided into 7 equally spaced ones- 1mts each, 7 racers need to run the full length of the track once - how should you place them so that each of them cover the same length? 
8. The difference between the ages of Mr. Young and Ms. Old is 45 years and the ratio of their ages is 3:8. What are the ages of Mr. Young and Ms. Old.
9. Divide 25 into two parts, such that 4 times of one part is the same as 6 times the second. 
10. In a large movie hall, seats are assigned with respect to row and column numbers. In all there are 1296 seats, and the number of rows are the same as columns - how many rows are there?

Sunday June 20, 2010

This is the last week of the school for the kids, and after this week we will take a break for about 7 weeks - as all of us are travelling to different parts of the world.

1. 9 zebras weight same as 4 yaks, 8 yaks weigh same as 15 xantus, 10 xantus  weigh same as 27 wolves. How many wolves weigh the same as 4 zebras?
2. On a 200m circular race track in your school yard, Bruce and Anant are competing in a race, a long 10KM race. Bruce runs at the speed of 5m/sec and Anant runs at the speed of 4m/sec. How many meters has Anant run when they meet for the first time? Second time?  (Both of them are running in the same direction and start at the same time.)  
3. Aditya and Mark smuggled 72 candies for their trip to the Arrowhead camp. Aditya being scared of getting caught, gives half of his candies to Mark, and since he is still very very scared, he gives in addition  12 more. Mark is thrilled - he has now become the Candy King, he has three times more than what Aditya is left with. How many candies each of Aditya and Mark had to begin with? 
4. 3/4 of 30 is same as 9/10th of what number?
5. Radhika buys her favourite sandal for a discount of 20%. Raju,  while strolling in Bayshore, spots the same sandal in two different stores. The one is selling it at  an additional discount of 20%  on what Radhika paid, the other store is selling it at a discount of 37% at the original price. Which deal is better for Raju?
6. Two twins, Dharmesh and Prince, have a birthday on Feb 29th. Geeta, his mentor, decided to give Dharmesh $1 on Feb 1, $2 on Feb 2, $3 on Feb 3, ..., $29 on Feb 29th as the B'day gift.  Remo, Prince's mentor, decided to give him $20 everyday from Feb 1 to Feb 29th as his B'day gift. Who had a better present?
7. Today's geometric construction - we will draw a parallelogram whose sides are  6cm long, angle is 45-degrees.  What is its area?       
 

 

Sunday, June 6, 2010

1. Suppose you have many carpet runners. Their dimension is typically 2ft by 6ft. Your room measure 8ft by 12ft. What kind of interesting patterns you can make to cover your room wall to wall.
2. I have tossed a coin 10 times in a row, and so far I haven't got the Heads.  (Wow! I am so unlucky!). Does this increases my chance of getting a head in the next toss?
3. What's  common between golf, notebook and doughnut (or medhu vada).
4. Take a loonie (Canadian 1$ coin) and roll it along its edge on a table. Observe the eye of the loon (the bird/ or the Queen), and trace the curve it follows. What shape is this curve? (Learn More: Cycloid)  
5.  Alice and Bob are in two different rooms, and they have no way to communicate with each other as well as they do not trust each other. Think both from Alice and Bobs viewpoint and reason what you will do in the following scenario:  Alice gets a $10 bill with the following promise - either she can keep the money or send it to Bob, and if Bob returns the favor then her amount will be doubled (Bob may just walk off).  Same is told to Bob, that either he can keep the money which came from Alice or send it back to Alice, and if she returns the favor then the amount which he had will be doubled. They can keep sending back and forth the money and keep doubling!   What will you do if you are Alice? or Bob?
6.  In the news we have heard a lot about the oil spill in the Gulf of Mexico. Here is a simple Math problem related to this. They claim that 26,500 barrels of oil is spilled in a day (1barrel= approx. 160 l).  The diameter of the pipe from which the crude oil is oozing out is 20 inches.  How many barrels/per day you think they will be spilling out in case they made the diameter of that pipe 24 inches instead of 20 inches.
7. An Olympic size swimming pool measures L=50m x W=25m x D=2m. What is the total volume of water it contains?  1 cubic meter requires 1000 liters of water. In terms of the Gulf oil spill, it started 48 days ago - how many pools of oil it is? If the pipe was 24 inches wide then how many swimming pools it would have filled?
8. Can you arrange 9 dots on a piece of paper so that there are 10 straight lines, each passing through at least 3 dots.

Sunday May 30, 2010

1. Ethiopian Tune ran 10K in 32 minutes and 11.5 seconds. What was her speed in Km/Hr? 
2. In comparison, in Vancouver 2010 olympics, 10k speed skating record for men was just under 13 minutes. What the speed in km/Hr.
3. Usain Bolt ran 100m in 9.58 seconds. What is his speed?
4. 100m swimming free style record is 46.91 seconds by Cesar Cielo. Whats the speed?
5. Niemi save percentage in the goal is .916. If Philly wants to score 5 goals in a game, how many shots, on average, needs to be directed at the Chicago's net?
6. In a class, there are 15 young men and 15 young women. When polled, it turns out that each  men dates 4 women, and each women dates 3 men. Is it possible? 
7. You have invited 10 guests for a party.   A custard can serves three guests. How many cans you need?
8. On a windless Sunday morning, you biked on the Colonel By Drive (which is essentially a straight North-South road)  for its entire 10 km length, starting at one end, reaching the other end, and then coming back to the start point. Following Sunday you did exactly the same, but there is a 25 KM/Hr wind blowing from North.   Suppose the amount of effort you put in is identical on both the days -  will the total time of trip on both the Sundays will be same?
9. What music to play when climbing the Eiffel Tower. There are in all 1665 steps to reach the top of the tower (actually we can't since this is an Emergency Exit!).  Suppose you can climb at a constant pace, determine how many steps you can climb in a minute - whats the right beat - which is perfect music - and how long will it take?
10. According to the Canadian Tire Paint Calculator, a room of dimensions 4m times 4m times 3m requires 5.58 l of paint for the walls and 1.86 lit for the ceilings. Given this information how will you determine how much paint you will require for your room?
11. Geometric Construction: Draw a triangle with sides 9cm, 7cm, and 5cm and then determine its area.   

Monday, May 24, 2010 (Victoria Weekend)

The first two problems are motivated from "Our days are numbered" by Brown.

1. Here is a simple game about mind reading:  Bob tells Alice to guess two numbers between 0 and 9.  Bob asks Alice to perform the following four steps (a) multiply the first number by 5 (b)  add 9 to the product (c) double the sum (d) lastly add the second number to the result. Then he asks Alice to tell him the result - and then he subtracts 18 from her answer, and now he knows both the numbers which Alice guessed! Why it works? (For example if Alice guessed 3 and 5, then the steps results in (a) 3x5=15 (b) 15+9=24 (c) 2x24=48 (d) 48+5=53. Bob subtracts 18 from 53 and it equals 53-18= 35 - the two digits which
2. Show that the remainder, when a number is divided by 9, is the same as the sum of its digits (if the sum exceeds 9, then keep taking the sum of the digits, till it is below 9). For example, we know that 345/9 results in a remainder of  3. The sum of the digits of 345= 3+4=5=12, which in turn is 1+2= 3, and it is the same as the remainder.
3. Continuing on the theme of the angle bisector from the last week, how will you draw two lines which are perpendicular to each other (only compass and ruler).
4. Give a line L and point p (p is not on L), how will you find the closest point to p on L geometric construction..
5. Given a triangle ABC, how will you draw an incircle of ABC. An incircle is the largest circle completely contained in the triangle. By the way the center of this circle is the meeting point of the angle bisectors.
6. What about circumcircle of ABC - that is the smallest circle that completely contains ABC. Its center is on the perpendicular bisectors of the  sides of triangle.

May 16, 2010

1. The four circles of equal radius (say 1cm each)  are arranged in such a way that their centers make a perfect square, where each side of the square is 2cm long. What will be the dimension of the little circle that can be placed between all these four circles, such that the little circle just barely touches the four big circles. What is its area compared to the big circles?   (Lean More: Pythagoras Theorem).
2. Actually it may be best that you try to draw the above figure using your geometry set. First draw the square, then the four circles, and if your drawing is perfect then the red circle will touch all the other four circles!
3. How many numbers are there between 1 and 30, which are not prime, not a multiple of 4 or 5, and are  divisble by 2 and 6.
4. Consider two points A and B which are at distance 5cm apart on the plane. Rotate A around B at each possible angle (300,270, 180, 90, 45, 30, ...).  Consider all the images of A. What shape they form?  What happens when A and B are in space - what does the angle of rotation means?
5. In the previous problem - say A is a disc and B is a point. Then what shape the union of the images of B form?
6. How many of the following statements are true? (a) The number of False statements are 1  (b) The number of False statements are 2  (c) The number of False statements are 3 (d) The number of False statements are 4.
7. The sum of the ages of a couple is 77 years. Mr. He is now twice as old as Mrs. She was when he was as old as she is now. What are the current  ages of Mr. He and Mrs. She?
8. How many pennies are there if in your pocket you have 50 coins making a change for a dollar?
9. If 5 is added to 1/3rd of a number, it result in  1/2 of that number. Whats that number?
10. How will you bisect an angle between two lines, without actually measuting the angle (you are allowed to use the compass).