CONVERSE EXTENSION. 15 Example. £25.6473852 895408 64: 5=12/- 4 rem. 1489: 125=11 thirty-two’ths. Note 1. If a limited number of places are known, the approximations can only be carried to a certain point depending on the number of places correct in the diminished decimal and on what is known of the error in the next place. Note 2. In pricing goods obtain by this method the cost to any approximation and add 1 or more to the corresponding fraction of a penny. This will always ensure against loss by retail. If the retailer wishes a certain profit on the whole sale, he will add this to the total cost and then divide-approximating by the above method and adding 1 to the corresponding fraction of a penny to ensure his full profit. EXAMPLES. 1. Decimalise (a) exactly to three places, (b) to three places with allowance for 4th, (c) exactly: (1) £18. 98. 7 d. (2) £17. 58. 3 d. £931. 178. 54d. (4) £864. 88. O{d. £91. 13s. 5d. (6) £72. 6s. 8]d. £3306. 178. 5 d. (8) £123. 14s. 6 d. £904. 168. 7d. (10) £75. 138. 410. (12) £66. 15s. 9 d. (18) £875. 18s. 10 d. (20) £8. 128. 2d. Any sum written at random is easily decimalised—the answer being verified by actual working. 2. Convert into £. s. d. f.: (1) £81:67154. (2) £793.563 (approx.). (3) £184.4931. (4) £71:336 (exact). £361.1056. (6) £137.004 (approx.). £87.3125. (8) £931.273 (approx.). (9) £710•311 (exact). (10) £876.950 (exact). Any decimal written at random may be turned into money by the method and the answer verified by actual working. 3. Reduce (1) £3751. 198. 1£d. to farthings. £193. 78. 5d. to pence. £815. 38. 9fd. to farthings. (10) £71. 4s. 10d. to farthings. 4. Decimalise (1) £73. 7s. 105d. (2) £18. 68. 4jfd. (3) 78. 3}id. (4) 588d. 13. 97d. (6) 38. 41/d. (7) £3. 10s. 43.d. (8) 3 d. (9) 18. 14.d. (10) 38. 2 d. 5. Convert into £. 8. d. f., 8ths, 16ths, 32ths, consistently with absolute accuracy: (1) £19.3145978. (2) £3:12784. (3) £•3567. £:3786. £7.645918. (6) £13:31267. £27.298314. £139.765. £78.8341. (10) £1.93756. 7. Methods of Approximation. In calculating prices or profits and in all engineering or practical work it is often only necessary to get results correct to a given number of decimal places. This is true for instance in calculating money in decimals of a £—three places ensures correctness to the nearest farthing. Hence all working which brings results to a larger number of places is superfluous and it becomes of importance to use such methods as will give what is necessary with least work. 1°. Addition and Subtraction. In adding and subtracting, the Rule for approximating is simple. ALLOWANCE FOR PLACES. 17 In each line stop at the place one further to the right than the number of places required correct, and make allowance for the place beyond. Then add in the ordinary manner. Example. Add 83.17135 correct to 3 places. 206.34127 1308-12161 1597.6343 Mental Process. Consider 3 as 4 because of the 5 in line 1 2 as 3 7 2 6 as 6 3 Then add in the usual way. The 4th place is not necessarily correct. 2o. On making allowance. (1) In any Decimal Expression to make allowance for the part beyond the place stopped at—if next place is 5 or over, add 1 to the digit stopped at, otherwise add 0. (2) In multiplying any Decimal—to make allowance for the multiplication of the part omitted in the approximation-multiply the figure beyond the place stopped at and add the nearest multiple of 10 to the product of the figure stopped at, counting 5, 15, 25 etc. as being nearer to 10, 20, 30 etc, respectively than to 0, 10, 20 etc. Example 1. 8.634578 x 9 correct to 3 places. 8.634578 9 77.7111 Mental Process. 9x7=63, carry 6, then as usual. Example 2. 13•6235 x 7 correct to 2 places. 7 95.365 Mental Process. 7x5=35, carry 4, then as usual. (3) In dividing any Decimal—to make allowance at any point for further places in the quotient put down as the final figure that exact multiple of the divisor which is nearest (above or below) to the remainder used as the dividend. Example. 84:7312 : 19 correct to 3 places with allowance for 4th. 19 ) 84•7312 ( 4.456 87 113 Mental Process. 6x 19=114, nearest to 113. This is very important in Aliquotation. 3. Multiplication. De Morgan's Method. (1) Absolute Rule for finding a product correct. to a given number of Decimal Places. Reverse the multiplier omitting the decimal point. Arrange this with the original unit-figure under that digit in the multiplicand which is one place further to the right than the number of places required correct. Multiply by the digits of the multiplier, beginning with each at the digit immediately above but adding in the nearest multiple of 10 derived from the digit to the right. Place the products in successive lines with their right-hand digits in a vertical column. Add and mark off one more place to the left than the number of places required correct. Strike off the final figure and the remainder will be the correct answer. (2) Sufficient Rule in many commercial calculations. Reverse the multiplier omitting the decimal point. Arrange this with the original unit-figure under that digit of the multiplicand which is as far to the right as the number of places required correct. Proceed as above with the multiplication. MULTIPLICATION-APPROXIMATION. 19 Add and mark off the number of places required correct. The last figure will never differ by more than 1 from the true result and therefore any answer of 3 places in money will always be correct to pence. (3) Another Form of the Rule valuable in Exchanges. Place the multiplier under the multiplicand so that the points are in a column. Multiply by the digit furthest to the left-commencing with that digit in the multiplicand which is the required number of places further to the right than the multiplying digit is to the left of the decimal point (making allowance in the usual way for the multiplicand-digit to the right of the one taken). Proceed in this way with each digit of the multiplier. Throughout put the decimal point in its place for each line. Add, and the required number of places will lie to the right of the decimal point. N.B. When the decimal parts of the multiplier are reached the starting-point in the multiplicand will get nearer and nearer to the decimal point and ultimately pass to the left of it. The Third Form shows in fact the principle on which the method is founded—the form is however trying to the eye and hence for general purposes the reversing of the multiplier is more convenient. In thus reversing the multiplier the starting-point of the multiplication for each figure of the multiplier is automatically settled. In all cases of direct multiplication of money the second form is sufficient but if any continued operations are involved or there is division as well it is better to |