Example 4. £183. 158. 48d. = £183•7697916. 41*4= 19 put down 6 for 2nd place and 9 for 3rd place 4x9=36: carry the 3 to 4x6=24 +3=27 put down 7 for 4th place 4x7=28: carry the 2 to 4x9=36+2=38 Put 8 mentally in 5th place, then correction 4x8=32: carry the 3 to 4x7=28+3=31 so 3 should be carried instead of 2- thus for 8) put down 9 for 5th place 4x9=36: carry the 3 to 4x7=28+3=31 put down 1 for 6th place 4x1=4: carry 0 to 4x9=36 put down 6 for 7th place and so on, the 6 repeating. 2°. To convert a decimal into £. 8. d. f. Divide remainder with 3rd place brought down by 4—quotient is pence. Remainder (diminished by 1 if pence are 6d. or over) is farthings. These operations may give results exceeding the actual value by a fraction of a farthing. To decide whether a farthing should be struck off, the fourth place is necessary. (2) If more than three places are known. Proceed as in (1) until the farthings-remainder is reached. Then calculate (mentally) a fourth place from the given 2nd and 3rd by the method of Fours. If this exceed the actual fourth place strike off a farthing from the result obtained by (i), otherwise not. If the two fourth places are equal we must proceed to the 5th absolutely to decide—and so on. (Always using the method of Fours.) Example 1. £73.896= £73. 178. 114d. Mental process 89:5=178. rem. 4 46:4=11d. rem. 2 (above 6d.) 2-1=4. Example 2. £73.89614= £73. 178. 11 d. as in Ex. 1. Mental correction-4th place from •896=8, given place is 1 :. nearest value) below = £73. 178. 11d. to farthings DECIMALS INTO STERLING. 11 Example 3. £73.896916= £73. 178. 11 d. as in Ex. 1. = £73. 178. 11fd. below the actual value Note 1. By striking off one more farthing than the rule requires in (1) all overcharge is avoided without the trouble of finding the 4th place. In doing this there can only be the loss of fd. to the seller. In most Commercial Transactions correctness to pence is all that is required. Note 2. If 3 places are known with allowance for the 4th--this shows that the 4th place is at least 5—hence we may write the 3rd place as one less and consider 4th place as 5. This will in many cases decide absolutely the value correct to farthings. Note 3. Certain decimals and certain decimals only have exact values in £. 8. d. f.; in these cases if sufficient places are given it will be seen that the calculated 4th, 5th etc. places are identical with those given. Note 4. If the decimal to 3 places is exact or is taken to be exact, the rule (1) gives the value above the true value to the nearest farthing and the modification in Note 1 gives the value below the true value to the nearest farthing. To consider three places as exact is to adopt mils as the division of the £. 30. Reduction of Money. An interesting application of Money-Decimalisation gives a very brief way of reducing money to farthings or pence. (1) To reduce £. s. d. f. to farthings. Decimalise the money as usual, to three places. Omit the decimal point. Multiply by 4, commencing two places to the left and carrying from the multiplication of the previous figures what is necessary. Subtract this from the altered Decimal. The answer gives the farthings required. Example 1. £87651. 198. 5 d. 87651'972 3506 078 Ans. 84145 894 Example 2. £896. 178. 11/d. 896'896 35 875 Ans. 861 021 The subtraction can be done as you proceed. Example 3. £6891. 148. 3d. 6891'715 Ans. 6616 047 (2) To reduce £. s. d. to pence. Subtract the Integral part only of too of altered Decimal from the quotient. Answer gives the pence required. Example. £113. 58. 8d. to pence. 4 | 113'283 28 320— 3 rejected. 1 132–83 rejected. Ans. 27 188 Note. The explanation of the accuracy of these approximations is that the Decimal parts are the same. e.g. £87651. 198. 54d. 87651972:916 (Altered) Multiplying by 4 as in the rule 3506078.916 84145894 Also £113. 58, 8d. 4 | 113283.33 28320.83 1132.83 27188 4o. Extension of Method to Eighths, Sixteenths, Thirty-two'ths of a Penny. To decimalise a sum of money including £. 8. d. and lower fractions of a penny than a farthing. Multiply shillings by 5. Put tens-figure of product as first decimal place. Reduce decimal and fraction to farthings and a fraction of a farthing. Add tens-figure to unit-figure of shillings-product for 2nd decimal place. Put unit-figure and fraction (reduced to decimal form) as remaining decimal places. EXTENSION TO SIXTEENTHS. 13 (Increasing by 1 or 0 the unit-figure as the pence are over 6d. or not.) The decimal arrived at is only provisional and has to be corrected to get the exact decimal-in this manner following : The first place is correct absolutely. Consider the correction due to the 4th place in the manner previously shown for £. s. d. f., using 4 to multiply and carrying as before. Add this to 4th place and change the 3rd again if necessary. Now see if this new form will make any difference (when multiplied by 4 in the usual way adopted in this method) to the correction applied to 4th place, make any change arrived at and proceed to correct the 5th place by means of the 3rd and 4th in the same way. Continue the process not only until all the provisional figures are exhausted but until there is an obvious recurrence-remembering that 48 may be written 5 to avoid getting 4999.... Example 1. £25. 138. 878d.= £25.6841= £25.68475 (provisionally). Correction (Mental Process). 4x5=20 4x7=28+2=30 4x4=16+3=19 4x8=32+1=33 .. add 3 to 7 hence decimal now is 25.68505. Trying again with 4 we see that 4 should have been added. So decimal should be 25.68515. Now correct the next place 4x5=20 4x1+2=6 4 x 5 +0=20 .. add 0 to 5 and decimal is now correct to 5 places. Proceeding to the remaining places is 25.6851562 4x2= 8 4x6+0=24 is 25-68515625. Example 2. £36. 98. 5 d.= £36.473}= £36.4735 (provisionally). 36.4735 9 correction for 4th place 36.47447916 method of Fours. Example 3. £74. 8s. 3}}d. = £74.4135 = £74.413625 (provisionally). 74.413625 5 correction for 4th place for 5th for 6th 5o. Converse Extension. To convert any decimal to £. 8. d. f., 8ths, 16ths, 32ths of a penny. Four, Five, Six, Seven places are required respectively to approximate with absolute certainty to farthings, eighths, sixteenths, thirty-two'ths of a penny, but practically sufficiently accurate results can often be obtained with fewer places. Rule. Obtain the product by 4 of the decimal with the 1st place omitted. Subtract this from the original decimal with two zeros added and neglect for the purposes of the subtraction the tens-figure of the 2nd place multiplied by 4. Consider the decimal so obtained. Divide remainder with 3rd place brought down by 4-quotient is pence. If pence are 6d. or over diminish the new remainder by 1. Then this remainder (thus corrected if necessary) is farthings. Divide this remainder with next place brought down by 5—quotient is 8ths. Divide this remainder with next two places brought down by 25—quotient is 16ths. Divide this remainder with next three places brought down by 125—quotient is 32ths. |