SECTION I. ARITHMETICAL OPERATIONS, 1. Mental or Very Brief Rules in Multiplication and Division, (1) To multiply by a number entirely of nines. Add to multiplicand as many O's as there are 9's. Subtract the multiplicand. (2) To multiply by a number entirely of nines except in the unit-place. Add to multiplicand as many O's as there are digits. Subtract the multiplicand x (10 – unit-figure). e.g. 999997—add six O's and subtract 3 x multiplicand. (3) To multiply by any power of 5. Add as many O's to multiplicand as the given power. Divide by same power of 2. e.g. 625 = 5% -add four O's and divide by 16 (24). (4) To divide by any power of 5. Multiply dividend by same power of 2. Mark off as many places as the power given. e.g. 3125 = 55. Multiply by 32 (29), mark off 5 places. (5) To multiply by 11 in one line. Add each digit of multiplicand in succession to its immediate lefthand adjacent digit, carrying when necessary and putting the unitdigit to the right and the final digit (plus remainder if any) to the left. e.g. 65178 x 11 = 716958. J. 1 Qe EXAMPLES 739645 by 625, 3125, 250. 2. To multiply in one line in the case of a two-figure multiplier. Example. 83561 x 37. Ans. = 3091757. Mental Process 7x1= 17 7x6=42+3x1 =45 4+7x5=39+3 x 6 = 57 5+7 x 3=26+3 x 5 =41 4+7x8=60+3x3 =69 6+3x8 3 3 This method may be extended to three-figure multipliers, but it is specially useful for those of two figures. =30 EXAMPLES. Multiply in one line: 1. 65372489 by 73, 45, 84. 2. 8371656 by 85, 94, 19. 3. 5030512 by 47, 39, 26. 4. 3125671 by 24, 37, 71. 5. 4123874 by 85, 123, 304. 3. Involution. In squaring and cubing numbers the following Algebraic principles are very useful : (1) The square of the sum of two numbers is equal to the sum of the squares of the numbers + twice the product. (2) The square of the difference of two numbers is equal to the sum of the squares of the numbers – twice the product. (3) The cube of the sum of two numbers is equal to the sum of the cubes of the numbers + 3 x product x sum of the numbers. MULTIPLICATION. 3 (4) The cube of the difference of two numbers is equal to the difference of the cubes – 3 x product x difference of the numbers. (5) The difference of the squares of two numbers is equal to the product of the sum and difference of the mbers. Examples. 1. 3052 = (300+5)2 = 3002 + 25 + 10 x 300 = 90000 + 25 + 3000= 93025. 2. 4932 =(500 – 7)2 = 5002 + 72 - 2.500.7 = 250000 + 49 – 7000=243049. 3. 263 = (20+6)3 = 203 +63 + 3. 120.26 8000 +216 +9360= 17576. 4. 393 = (40 - 1)3= 403 – 13 – 3.40.39 = 63999 - 4680 = 59319. 1. Square 631, 8007, 9013, 15012, 848. 2. 739, 891, 747, 9995, 761. 3. Cube 39, 410, 937, 97, 105. 4. 95, 38, 729, 409, 521. 5. Find value of 812 – 294, 3752 – 3253, 962 – 842, 7202 – 7112, 3122 - 3052. 4. Abbreviation in Multiplication. This arises when certain digits or sets of digits are multiples of other digits or sets of digits following or preceding them. Much labour is also thrown away in writing down lines twice where by a judicious arrangement of the work the repetition might be avoided. The two principles following are of great use with easily rememberable multipliers. 1°. When 1 occurs among the digits, write down the multiplicand adding O's according to the place of the 1. Then multiply by the remaining digits using the original multiplicand as it stands in the line written down. 2. Factors among the digits should be constantly noticed and made use of to shorten the number of lines or lessen the multipliers. Example 1. 9673 x 315=300+5 x 3. 2901900 145095 3046995 Example 2. 83561 x 369 =9+40 x 9. 752049 3008196 80834009 Example 3. 5612 x 852=840+12=12+70 x 12. 67344 471408 4781424 The cases in which one or other or both of these principles may be used are literally without number. A decimal point does not hinder their application. EXAMPLES. 1. 8345712 x 3106. 3. 936412 x 78013. 5. 3912451 x 96412. 7. 5614273 x 13758. 9. 378964 x 65131. 2. 78934 x 1768. 4. 1973256 x 832. 6. 9178 x 2136. 8. 8176541 x 42731. 10. 129875 X 8124. |