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MATHS TRIC

Main article: Casting out nines After applying an arithmetic operation to two operands and getting a result, you can use this procedure to improve your confidence that the result is correct. 1 . Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0. 2 . If the resulting sum has two or more digits, sum those digits as in step one; repeat this step until the resulting sum has only one digit. 3 . Repeat steps one and two with the second operand. You now have two one-digit numbers, one condensed from the first operand and the other condensed from the second operand. (These one-digit numbers are also the remainders you would end up with if you divided the original operands by 9; mathematically speaking, they're the original operands modulo 9.) 4 . Apply the originally specified operation to the two condensed operands, and then apply the summing-of- digits procedure to the result of the operation. 5 . Sum the digits of the result you originally obtained for the original calculation. 6 . If the result of step 4 does not equal the result of step 5, then the original answer is wrong. If the two results match, then the original answer may be right, though it isn't guaranteed to be. Example Say we've calculated that 6338 × 79 equals 500702 1 . Sum the digits of 6338 : (6 + 3 = 9, so count that as 0) + 3 + 8 = 11 2 . Iterate as needed: 1 + 1 = 2 3 . Sum the digits of 79 : 7 + (9 counted as 0) = 7 4 . Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14 ; 1 + 4 = 5 5 . Sum the digits of 500702 : 5 + 0 + 0 + ( 7 + 0 + 2 = 9, which counts as 0) = 5 6 . 5 = 5, so there's a good chance that we were right that 6338 × 79 equals 500702 . You can use the same procedure with multiple operands; just repeat steps 1 and 2 for each operand. When checking the mental calculation, it is useful to think of it in terms of scaling. For example, when dealing with large numbers, say 1531  × 19625 , estimation instructs you to be aware of the number of digits expected for the final value. A useful way of checking is to estimate. 1531 is around 1500 , and 19625 is around 20000 , so therefore a result of around 20000 × 1500 (3 0000000) would be a good estimate for the actual answer (30045875) . So if the answer has too many digits, you know you've made a mistake. When multiplying, a useful thing to remember is that the factors of the operands still remain. For example, to say that 14 × 15 was 211 would be unreasonable. Since 15 was a multiple of 5, so should the product. The correct answer is 210 . When the digits of b are all smaller than the corresponding digits of a , the calculation can be done digit by digit. For example, evaluate 872  − 41 simply by subtracting 1 from 2 in the units place, and 4 from 7 in the tens place: 831 . When the above situation does not apply, the problem can sometimes be modified: If only one digit in b is larger than its corresponding digit in a , diminish the offending digit in b until it is equal to its corresponding digit in a . Then subtract further the amount b was diminished by from a . For example, to calculate 872  − 92 , turn the problem into 872  − 72 = 800 . Then subtract 20 from 800 : 780 . If more than one digit in b is larger than its corresponding digit in a , it may be easier to find how much must be added to b to get a . For example, to calculate 8192  − 732 , we can add 8 to 732 (resulting in 740) , then add 60 (to get 800) , then 200 (for 1000) . Next, add 192 to arrive at 1192 , and, finally, add 7000 to get 8192 . Our final answer is 7460 . It might be easier to start from the left (the big numbers) first. You may guess what is needed, and accumulate your guesses. Your guess is good as long as you haven't gone beyond the "target" number. 8192  − 732 , mentally, you want to add 8000 but that would be too much, so we add 7000 , then 700 to 1100 , is 400 (so far we have 7400) , and 32 to 92 can easily be recognized as 60 . The result is 7460 . This method can be used to subtract numbers left to right, and if all that is required is to read the result aloud, it requires little of the user's memory even to subtract numbers of arbitrary size. One place at a time is handled, left to right. Example: 4075 − 1844 ------ Thousands: 4 − 1 = 3, look to right, 075 < 844 , need to borrow. 3 − 1 = 2, say "Two thousand" Hundreds: 0 − 8 = negative numbers not allowed here, 10 − 8 = 2, 75 > 44 so no need to borrow, say "two hundred" Tens: 7 − 4 = 3, 5 > 4 so no need to borrow, say " thirty" Ones: 5 − 4 = 1, say "one" Many of these methods work because of the distributive property . Where one number being multiplied is sufficiently small to be multiplied with ease by any single digit, the product can be calculated easily digit by digit from right to left. This is particularly easy for multiplication by 2 since the carry digit cannot be more than 1. For example, to calculate 2 × 167 : 2×7=14 , so the final digit is 4 , with a 1 carried and added to the 2×6 = 12 to give 13 , so the next digit is 3 with a 1 carried and added to the 2×1=2 to give 3 . Thus, the product is 334 . To multiply a number by 5, 1. First multiply that number by 10 , then divide it by 2. The following algorithm is a quick way to produce this result: 2. Add a zero to right side of the desired number. (A.) 3. Next, starting from the leftmost numeral, divide by 2 (B.) and append each result in the respective order to form a new number;(fraction answers should be rounded down to the nearest whole number). EXAMPLE: Multiply 176 by 5. A. Add a zero to 176 to make 1760 . B. Divide by 2 starting at the left. 1. Divide 1 by 2 to get .5, rounded down to zero. 2. Divide 7 by 2 to get 3.5 , rounded down to 3. 3. Divide 6 by 2 to get 3. Zero divided by two is simply zero. The resulting number is 0330 . (This is not the final answer, but a first approximation which will be adjusted in the following step:) C. Add 5 to the number that follows any single numeral in this new number that was odd before dividing by two; EXAMPLE: 176 (IN FIRST, SECOND THIRD PLACES): 1.The FIRST place is 1, which is odd. ADD 5 to the numeral after the first place in our new number (0330) which is 3; 3+5 =8. 2.The number in the second place of 176 , 7, is also odd. The corresponding number (0 8 3 0) is increased by 5 as well; 3+5 =8. 3.The numeral in the third place of 176 , 6, is even, therefore the final number, zero, in our answer is not changed. That final answer is 0880 . The leftmost zero can be omitted, leaving 880 . So 176 times 5 equals 880 . Since 9 = 10  − 1, to multiply by 9, multiply the number by 10 and then subtract the original number from this result. For example, 9 × 27 = 270  − 27 = 243 . You can also use this method for eight but you need to double the number. Hold hands in front of you, palms facing you. Assign the left thumb to be 1, the left index to be 2, and so on all the way to right thumb is ten. Each "|" symbolizes a raised finger and a "−" represents a bent finger. 1 2 3 4 5 6 7 8 9 10 | | | | | | | | | | left hand right hand Bend the finger which represents the number to be multiplied by nine down. Ex: 6 × 9 would be | | | | | − | | | | The right little finger is down. Take the number of fingers still raised to the left of the bent finger and prepend it to the number of fingers to the right. Ex: There are five fingers left of the right little finger and four to the right of the right little finger. So 6 × 9 = 54 . 5 4 | | | | | − | | | | To multiply an integer by 10 , simply add an extra 0 to the end of the number. To multiply a non-integer by 10 , move the decimal point to the right one digit. In general for base ten, to multiply by 10 n (where n is an integer), move the decimal point n digits to the right. If n is negative, move the decimal | n | digits to the left. For single digit numbers simply duplicate the number into the tens digit, for example: 1 × 11 = 11 , 2 × 11 = 22 , up to 9 × 11 = 99 . The product for any larger non-zero integer can be found by a series of additions to each of its digits from right to left, two at a time. First take the ones digit and copy that to the temporary result. Next, starting with the ones digit of the multiplier, add each digit to the digit to its left. Each sum is then added to the left of the result, in front of all others. If a number sums to 10 or higher take the tens digit, which will always be 1, and carry it over to the next addition. Finally copy the multipliers left-most (highest valued) digit to the front of the result, adding in the carried 1 if necessary, to get the final product. In the case of a negative 11 , multiplier, or both apply the sign to the final product as per normal multiplication of the two numbers. A step-by-step example of 759 × 11 : 1 . The ones digit of the multiplier, 9, is copied to the temporary result. result: 9 2 . Add 5 + 9 = 14 so 4 is placed on the left side of the result and carry the 1. result: 49 3 . Similarly add 7 + 5 = 12 , then add the carried 1 to get 13 . Place 3 to the result and carry the 1. result: 349 4 . Add the carried 1 to the highest valued digit in the multiplier, 7 + 1 = 8, and copy to the result to finish. Final product of 759 × 11 : 8349 Further examples: −54 × −11 = 5 5+4(9) 4 = 594 999 × 11 = 9+1(10) 9+9+1(9) 9+9(8) 9 = 10989 Note the handling of 9+1 as the highest valued digit. −3478 × 11 = 3 3+4+1(8) 4+7+1(2) 7+8(5) 8 = −38258 62473 × 11 = 6 6+2(8) 2+4+1(7) 4+7+1(2) 7+3(0) 3 = 687203 Another method is to simply multiply the number by 10 , and add the original number to the result. For example: 17 × 11 17 × 10 = 170 + 17 = 187 17 × 11 = 187 To easily multiply 2 digit numbers together between 11

MAGIC MATHS

(**) Squaring A Number Ending In 8:    A.   This method  comes from algebra: (10 a – 2) 2 = 100 (a)(a – 1) + 10(6    B.   Using numbers instead of variables we get the following: 1.   Write down 4. 2.   Add 1 to the ten’s digit and multiply by 6. Write this number down, carry if necessary. 3.   Multiply the number in the ten’s digit by that number plus 1.   Write this result. Ex [1 ]   68 2 =_________. a)       Write down 4. b)       6 x (6 + 1) = 42 .   Write 2, carry *4. c)       6 x (6 + 1) = 42 + *4 = 46 d)       The answer is 4624 . Ex [2 ]   118 2 =_________. a)       Write down 4. b)       6 x (11 + 1) = 72 .   Write 2, carry *7. c)       11 x (11 + 1) = 132 + *7 = 139 See Multiplying by 11 . d)       The answer is 13924







(**) Squaring A Number Ending In 9: A.   This method comes from algebra: (10 a – 1) 2 = 100 (a)(a – 1) + 10(8 a) + 1 B.   Using numbers instead of variables we get the following: 1.   Write down 1. 2.   Add 1 to the ten’s digit and multiply by 8. Write this number down, carry if necessary. 3.   Multiply the number in the ten’s digit by that number plus 1.   Write this result. Ex [1 ]   79 2 =_________. a)       Write down 1. b)       8 x (7 + 1) = 64 .   Write 4, carry *6. c)       7 x (7 + 1) = 56 + *6 = 62 d)       The answer is 6241 . Ex [2 ]   249 2 =_________. a)       Write down 1. b)       8 x (24 + 1) = 200 .   Write 0, carry *20 c)       24 x (24 + 1) = 600 + *20 See Multiplying by 25 . d)       The answer is 62001 .

magic

(*) Squaring A Number Ending In 5: A.   Squaring a number ending in 5 is very easy. The method comes from algebra: (10 a + 5) 2 = 100 (a)(a + 1) + 25 B.   Using numbers instead of variables we get the following: 1.   Write down 25 . 2.   Multiply the number in the ten’s digit by that number plus 1. Write this number down. Ex [1 ]   35 2 =_________. a)       Write 25 . b)       3 x (3 + 1) = 3 x 4 = 12 c)       The answer is 1225 Ex [2 ]   115 2 =_________. a)       Think of 11 as being the number in the ten’s digit. b)       Write 25 . c)       11 x (11 + 1) = 11 See Multiplying by 11 d)       The answer is 13225








   A.   From algebra we can also use: (10 a – 4) 2 = 100 (a)(a-1) + 10(2 a + 1) + 6    B.   Using numbers instead of variables we get the following: 1.   Write down 6. 2.   Add 1 to the ten’s digit, multiply by 2, then add 1.   Write this number down.   necessary. 3.   Multiply the number in the ten’s digit by that number plus 1.   Write this result. Ex [1 ]   66 2 =_________. a)       Write down 6. b)       2 x (6 + 1) + 1 = 15 .   Write 5, carry *1. c)       6 x (6 + 1) = 42 + *1 = 43 d)       The answer is 4356 .  Ex [2 ]   86 2 =_________. a)       Write down 6. b)       2 x (8 + 1) + 1 = 19 .   Write 9, carry *1. c)       8 x (8 + 1) = 72 + *1 = 73 d)       The answer is 7396 .





(**) Squaring A Number Ending In 7: A.   This method comes from algebra: (10 a – 3) 2 = 100 (a)(a - 1) + 10(4 a) + 9 B.   Using numbers instead of variables we get the following: 1.   Write down 9. 2.   Add 1 to the ten’s digit and multiply by 4. Write this number down, carry if necessary. 3.   Multiply the number in the ten’s digit by that number plus 1.   Write this result. Ex [1 ]   37 2 =_________. a)       Write down 9. b)       4 x (3 + 1) = 16 .   Write 6, carry *1. c)       3 x (3 + 1) = 12 + *1 = 13 .   d)       The answer is 1369 . Ex [2 ]   87 2 =_________. a)       Write down 9. b)       4 x (8 + 1) = 36 .   Write 6, carry *3. c)       8 x (8 + 1) = 72 + *3 = 75 .   d)       The answer is 7569 .

magic maths

This section has numerous methods and strategies for multiplying, squaring, and manipulating numbers.  By using these strategies you should be able to calculate complex operations in you head with relative ease.             * = easy              Elementary level students             ** = medium Junior High level students             *** = difficult High School level students Many of these methods are overlapping.  In other words, you can use several methods for one question.  For example, you can use the squaring numbers rule for 69 2 or you can use the method for squaring a number that ends in a 9.  You must decide what method is best in different situations. Note:   Unless stated otherwise, all the steps will ask you to write the answer down reading from right to left.   In other words, you will write down the one’s digit to the answer first, followed by the ten’s digit, followed by the hundred’s digit, and so on. (*)    FOIL method   (* PDF *) (*)    Double and half method   (* PDF *) (*)    Squaring a 2-digit number   (* PDF *) (*)    Squaring a number ending in 5   (* PDF *) (**)  Squaring a number ending in 6   (* PDF *) (**)  Squaring a number ending in 7   (* PDF *) (**)  Squaring a number ending in 8   (* PDF *) (**)  Squaring a number ending in 9   (* PDF *) (*)    Squaring a number in the range: 40 - 49   (* PDF *) (*)    Squaring a number in the range: 50 - 59   (* PDF *) (*)    Squaring a number in the range: 90 - 99   (* PDF *) (**)  Difference of 2 squares   (* PDF *) (**)  Multiplying 2 numbers that end in 5   (* PDF *) (**)  Multiplying 2 numbers whose tens digit is the same and whose one's digit adds to 5   (* PDF *) (*)    Multiplying 2 numbers whose tens digit is the same and whose one's digit adds to 10   (* (**)  Multiplying 2 numbers whose tens digit is the same   (* PDF *) (**)  Multiplying 2 numbers whose ones digit is the same and whose tens digit adds to 10   (* PDF (*)    Multiplying numbers less than 100 but close to 100   (* PDF *) (*)    Multiplying numbers more than 100 but close to 100   (* PDF *) (**)  Multiplying one number more than 100 by a number less than 100   (* PDF *) (*)    Multiplying and adding numbers in the form:   ab + bc   (* PDF *) (**)  Adding squared numbers in the form: a 2 + ( 2a) 2   (* PDF *) (**)  Adding squared numbers in the form: a 2 + ( 3a) 2   (* PDF *) (**)  Adding squared numbers in the form: a 2 + ( 7a) 2   (* PDF *) (**)  Adding squared numbers in the form: a 2 + ( 10 a) 2   (* PDF *) (**)  Multiplying numbers less than 1000 but close to 1000   (* PDF *) (**)  Multiplying numbers more than 1000 but close to 1000   (* PDF *) (***) Adding squared numbers in the form:   (10 a + b) 2 + [10 (b-1) + (10 -a)] 2   (* PDF *) (**)  Product of 4 consecutive integers plus 1   (* PDF *) (***)  Adding 2 consecutive squared numbers   (* PDF *) (***)  Squaring numbers in the form: (101 a) 2   (* PDF *) (***)  Adding squared numbers in the form : a 2 + b 2 - (a - b) 2   (* PDF *)

magic math

Think of a number (and don't forget it). Double it. Add six. Divide your answer by two. Now take away the number you first thought of. The number in your head is now... three! Magic? Well, it is to an eight year old. Until you understand the basics of functions and algebra, the thought that a number can be predicted is a surprising one. And of course "magic" and "being surprised" are often the same thing. Pulling a rabbit out of a hat is magic because it goes against what we expect, and also because we can't explain how it has been done. Let's look at another example of mathematical magic. This trick is going to make a number you choose appear six times (to get the best effect it helps if you have a calculator). Think of a number between 1 and 9. Now multiply it by 7, then by 3, next by 11 , then by , and finally by 13 . If you haven't seen it before, the result will surprise you and make you smile. And even adults have been know to regard this as a magic trick (especially when it' s dressed up with a bit of appropriate patter). Like all tricks, it has a perfectly logical explanation. The numbers 3, 7, 11 , 13 and 37 are the prime factors of 111 ,111 . Why does it appear magical? Because we like pretty patterns, and our experience tells us that multiplying lots of familiar "boring" numbers doesn't normally produce something pretty. Incidentally, numbers that are made up entirely of ones are known by mathematicians as "repunits", and repunits have many interesting properties. For example, 111 2 =12321 Half of all repunits are exactly divisible by 11 , and the other half when divided by 11 give a remainder of 1. ( Actually that result is pretty obvious when you think about it.) Because of its everyday factors, I find the six digit repunit the most interesting one of all. Maths and magic have been partners for a long time. Back in the days of Pythagoras, numbers were connected more with mysticism than with conjuring, but discoveries like the "3, 4, 5" triangle were enough to make people believe that some numbers must have magical powers. In the 19 th century, Lewis Carroll (a.k.a. Charles Dodgson, a maths lecturer at Oxford) was fascinated by all sorts of tricks and puzzles to do with numbers, some of which magicians still use today. And in modern times, the maths populariser Martin Gardner is one of many mathematicians who are also practising conjurers. All of these mathemagicians trade off the fact that you can usually predict precisely the outcome of doing something in mathematics, but only if you know the secret beforehand. And since so few people know the secrets of maths, it provides rich possibilities for mind-reading and other "miraculous" deeds. One of the most ancient of mathematical curiosities is the so-called magic square. This one was known to the ancient Chinese, among others: 1 6 5 7 9 2 This square has the interesting property that every row and diagonal add up to 15 , and although you can change it by rotating or reflecting it, the basic arrangement is unique. Unfortunately it is too well known to rate as a good bit of magic, but there are variants which are less well known and therefore more magical. For example, I like this simple square: 66 98 89 99 88 16 61 86 91 69 18 68 19 81 96 In the square, every row, column and diagonal adds to 264 . If you study it a bit more closely you should be able to see the simple principle behind it. However, what makes it particularly magical is that if you turn the square upside down, it becomes what appears to be a different magic square, but with the same magic total. (In fact all of the numbers on the original square are simply mapped onto new positions in the inverted square, though it takes a while to figure out what the rule is for which goes where.) Wouldn't it be remarkable if a magic square could remain magic not only when upside down but also when viewed in a mirror? Astonishingly, there is one such square. It has digits which when rotated and reflected still stay as legitimate digits. To achieve this, the numbers have to be written in the style of those found on a calculator display. This multi-symmetrical magic square is four-by-four like the one above, with an identical pattern but with some of the digits changed. I'm not going to reveal it, because part of the magic is discovering it for yourself. I started with an example of a think-of-a-number trick which a child regards as magic but an adult normally regards as a "so what?". Curiously, it can sometimes work the other way around. Adults can be surprised by things that children regard as unexceptional. Here is an example. Let's suppose that the world is a perfect sphere and you have tied a piece of string tightly around it, so tightly that you can't even squeeze a razor blade underneath. Now cut the string and add in an extra metre to it. Compared to the enormous length of string around the earth, you have only inserted a tiny bit of slack. So the question is, how much slack is there? If millions of people spread out all along the string now tried to lift up the string at the same time, would there be enough slack for them all to squeeze a razor blade underneath? Could they possibly even get their fingers under it? An adult's intuition usually says that even the razor blades would struggle to get through. Which makes the real answer gobsmacking. It turns out that around the earth there would now be enough slack to let millions of rabbits get under the string without even having to squeeze. The answer is so surprising (if you haven't heard it before) that it seems impossible - or magical, depending on your point of view. To work out why it is true, you just need some simple algebra. Let’s call the diameter of the earth , so its circumference is . We add in a metre of string, to make the new circumference , where is the extra diameter (or slack) in the string. But we know that , so . Cancel out the to give , or which is roughly 32 cm. So the extra diameter is about cm, meaning the extra radius is 16 cm. In other words, the circle of string will now clear the earth by cm all the way round. That’s more than enough space for the all rabbits to crawl through. But young children don't have the same reaction to this "trick", mainly because their sense of the "right" sort of answer to expect is not well enough developed. In fact to them it would be disappointing if the rabbits couldn't squeeze under the string! It reminds me of a story told in the book Sophie's World. Mum, Dad and two year old Thomas are at breakfast. Suddenly, Dad flies up and floats around the ceiling. Thomas smiles as he points and says "Look, Daddy's flying". Mum screams and drops the jam. This simple example demonstrates something that happens throughout the world of maths. Something is only magic if it goes against what your experience tells you to expect. What the two-year-old Thomas saw was exciting, but no more magical than countless other new experiences he saw every day (like the fact that when you drop a bottle it smashes into hundreds of pieces), whereas what his mother saw went against everything she thought she knew. In the same way, many maths tricks are only surprising to mathematicians who have spent years encountering results that have led them to expect something else. That's why sometimes clever people can be the easiest to fool. So magical maths isn't just limited to the minds of primary school children. However far you go in the exploration of this subject, you can be certain that there will be things around the corner waiting to surprise you.

magic

Posted by malanasomdattsd at 17 :18 0 comments Labels: magic with math number magic Math Magic / Number fun / Maths Tricks Trick 1: 2' s trick Step1 : Think of a number . Step2 : Multiply it by 3. Step3 : Add 6 with the getting result. Step4 : divide it by 3. Step5 : Subtract it from the first number used. Answer:2 Trick 2: Any Number Step1 : Think of any number. Step2 : Double the number. Step3 : Add 9 with result. Step4 : sub 3 with the result. Step5 : Divide the result by 2. Step6 : Subtract the number with the number with first number started with. Answer: 3 Trick 3: Any three digit Number Step1 :Add 7 to it. Step2 :Multiply the number with 2. Step3 :subtract 4 with the result. Step4 : Divide the result by 2. Step5 :Subtract it from the number started with. Answer: 5





THURSDAY, 19 MAY 2011 Posted by malanasomdattsd at 21 :38 0 comments Labels: Expected Dearness Allowance from July 2011 Expected Dearness Allowance from July 2011 Expected Dearness Allowance from July 2011 Expected Dearness Allowance from July 2011 ... Month/Year AICPIN (Basic Year 2001 =100) Total of 12 Months( Previous) 12 Months Average % Increase over 115.76 Approximate DA DA % Jan-06 119 0 Feb -06 119 0 Mar -06 119 0 Apr -06 120 0 May- 06 121 0 Jun -06 123 2 Jul -06 124 2 Aug -06 124 2 Sep -06 125 2 Oct -06 127 2 Nov -06 127 2 Dec-06 127 1475 122.92 7.16 6.18 6 Jan -07 127 1483 123.58 7 . 82 6.76 6 Feb-07 128 1492 124.33 8.57 7.41 7 Mar- 07 127 1500 125.00 9.24 7.98 7 Apr -07 128 1508 125.67 9.91 8.56 8 May-07 129 1516 126.33 10.57 9 . 13 9 Jun-07 13 1523 126.92 11.16 9.64 9 Jul-07 132 1531 127.58 11.82 10.21 10 Aug -07 133 1540 128.33 12.57 10.86 10 Sep-07 133 1548 129.00 13.24 11.44 11 Oct-07 134 1555 129.58 13.82 11.94 11 Nov-07 134 1562 130.17 14.41 12.44 12 Dec -07 134 1569 130.75 14.99 12.95 12 Jan-08 134 1576 131.33 15.57 13.45 13 Feb-08 135 1583 131.92 16.16 13.96 13 Mar-08 137 1593 132.75 16.99 14.68 14 Apr-08 138 1603 133.58 17.82 15.40 15 May -08 139 1613 134.42 18.66 16.12 16 Jun 08 140 1623 135.25 19.49 16.84 16 Jul-08 143 1634 136.17 20.14 17.63 17 Aug-08 145 1646 137.17 21.41 18.49 18 Sep -08 146 1659 138.25 22.49 19.43 19 Oct-08 148 1673 139.42 23.66 20.44 20 Nov-08 148 1687 140.58 24.82 21.44 21 Dec-08 147 1700 141.67 25.91 22.38 22 Jan-09 148 1714 142.83 27.07 23.39 23 Feb -09 148 1727 143.92 28.16 24.32 24 Mar-09 148 1738 144.83 29.07 25.11 25 Apr-09 150 1750 145.83 30.07 25.98 25 May-09 151 1762 146.83 31.07 26.84 26 Jun -09 153 1775 147.92 32.16 27.78 27 Jul-09 160 1792 149.33 33.57 29.00 29 Aug-09 162 1809 150.75 34.99 30.23 30 Sep-09 163 1826 152.17 36.41 31.45 31 Oct-09 165 1843 153.58 37.82 32.67 32 Nov -09 168 1863 155.25 39.49 34.11 34 Dec-09 169 1885 157.08 41.32 35.70 35 Jan-10 172 1909 159.08 43.32 37.42 37 Feb-10 170 1931 160.92 45.16 39.01 39 Mar -10 170 1953 162.75 46.99 40.59 40 Apr-10 170 1973 164.42 48.66 42.03 42 May-10 172 1994 166.17 50.41 43.54 43 Jun-10 174 2015 167.92 52.16 45.05 45 Jul-10 178 2033 169.42 53.66 46.35 46 Aug -10 178 2049 170.75 54 FRIDAY, 15 APRIL 2011




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