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