Appendix VI. — On Decimal Money
Of all the simplifications of commercial arithmetic, none is comparable to that of expressing shillings, pence, and farthings as decimals of a pound. The rules are thereby put almost upon as good a footing as if the country possessed the advantage of a real decimal coinage.
Any fraction of a pound sterling may be decimalised by rules which can be made to give the result at once.
| Two shillings is | £·100 | |
| One shilling is | £·050 | |
| Sixpence is | £·025 | |
| One farthing is | £·001 | 04⅙ |
Thus, every pair of shillings is a unit in the first decimal place; an odd shilling is a 50 in the second and third places; a farthing is so nearly the thousandth part of a pound, that to say one farthing is ·001, two farthings is ·002, &c., is so near the truth that it makes no error in the first three decimals till we arrive at sixpence, and then 24 farthings is exactly ·025 or 25 thousandths. But 25 farthings is ·026, 26 farthings is ·027, &c. Hence the rule for the first three places is One in the first for every pair of shillings; 50 in the second and third for the odd shilling, if any; and 1 for every farthing additional, with 1 extra for sixpence.
| Thus, | 0s. | 3½d. = £·014 |
| 0s. | 7¾d. = £·032 | |
| 1s. | 2½d. = £·060 | |
| 1s. | 11¼d. = £·096 | |
| 2s. | 6d. = £·125 | |
| 2s. | 9½d. = £·139 | |
| 3s. | 2¾d. = £·161 | |
| 13s. | 10¾d. = £·694 |
In the fourth and fifth places, and those which follow, it is obvious that we have no produce from any farthings except those above sixpence. For at every sixpence, ·00004⅙ is converted into ·001, and this has been already accounted for. Consequently, to fill up the fourth and fifth places,
Take 4 for every farthing59 above the last sixpence, and an additional 1 for every six farthings, or three halfpence.
The remaining places arise altogether from ·00000⅙ for every farthing above the last three halfpence; for at every three halfpence complete, ·00000⅙ is converted into ·00001, and has been already accounted for. Consequently, to fill up all the places after the fifth,
Let the number of farthings above the last three halfpence be a numerator, 6 a denominator, and annex the figures of the corresponding decimal fraction.
It may be easily remembered that
| The figures of | ¹/₆ | are | 166666... |
| ″ | ²/₆ | ... | 333333... |
| ″ | ³/₆ | ... | 5 |
| ″ | ⁴/₆ | ... | 666666... |
| ″ | ⁵/₆ | ... | 833333... |
| 0s. | 3½d. = | ·014 | 58 | 3333... |
| 0s. | 7¾d. = | ·032 | 29 | 1666... |
| 1s. | 2½d. = | ·060 | 41 | 6666... |
| 1s. | 11¼d. = | ·096 | 87 | 83333... |
| 2s. | 6d. = | ·125 | 00 | 0000... |
| 2s. | 9½d. = | ·139 | 58 | 3333... |
| 3s. | 2¾d. = | ·161 | 45 | 83333... |
| 13s. | 10¾d. = | 694 | 79 | 1666... |
The following examples will shew the use of this rule, if the student will also work them in the common way.
To turn pounds, &c., into farthings: Multiply the pounds by 960, or by 1000-40, or by 1000(1-⁴/₁₀₀); that is, from 1000 times the pounds subtract 4 per cent of itself. Thus, required the number of farthings in £1663. 11. 9¾.
| 1663·590625 × 1000 | = | 1663590·625 |
| 4 per cent of this, | 66543·625 | |
| No. of farthings required, | 1597047 |
What is 47½ per cent of £166. 13. 10 and ·6148 of £2971. 16. 9?
| 166·691 | |
| 40 p. c. | 66·6764 |
| 5 p. c. | 8·3346 |
| 2½ p. c. | 4·1673 |
| 79·1783 | |
| £79.3.6¾ | |
| 2971·837 | |
| ·6 | 1783·1022 |
| ·01 | 29·7184 |
| ·004 | 11·8873 |
| ·0008 | 2·3775 |
| 1827·0854 | |
| £1827.1.8½ |
The inverse rule for turning the decimal of a pound into shillings, pence, and farthings, is obviously as follows:
A pair of shillings for every unit in the first place; an odd shilling for 50 (if there be 50) in the second and third places; and a farthing for every thousandth left, after abating 1 if the number of thousandths so left exceed 24.
The direct rule (with three places) gives too little, the inverse rule too much, except at the end of a sixpence, when both are accurate. Thus, £·183 is rather less than 3s. 8d., and 6s. 4¾d. is rather greater than £319; or when the two do not exactly agree, the common money is the greatest. But £·125 and £·35 are exactly 2s. 6d. and 7s.
Required the price of 17 cwt. 81 lb. 13½ oz. at £3.11.9¾ per cwt. true to the hundredth of a farthing.
| 3·590625 | ||
| 17 | ||
| 61·040625 | ||
| lb. | 56 ½ | 1·795313 |
| 16 ⅐ | ·512946 | |
| 7 ⅛ | ·224414 | |
| 2 ⅛ | ·064118 | |
| oz. | 8 ¼ | ·016029 |
| 4 ½ | ·008015 | |
| 1 ¼ | ·002004 | |
| ½ ½ | ·001002 | |
| £63·664466 | ||
| £63.13.3½ | ||
Three men, A, B, C, severally invest £191.12.7¾, £61.14.8, and £122.1.9½ in an adventure which yields £511.12.6½. How ought the proceeds to be divided among them?
| A, | 191·63229 | |
| B, | 61·73333 | |
| C, | 122·08958 | Produce of £1. |
| 375·45520)511·62708 | (1·362686 | |
| 136·17188 | ||
| 23·53532 | ||
| 1·00801 | ||
| 25710 | ||
| 3183 | ||
| 180 |
| 1·362686 | 1·362686 | 1·362686 | ||
| 92·236191 | 33·33716 | 85·980221 | ||
| 1·362686 | 8·17612 | 1·362686 | ||
| 1·226417 | 13627 | 272537 | ||
| 13627 | 9538 | 27254 | ||
| 8176 | 409 | 1090 | ||
| 409 | 41 | 122 | ||
| 27 | 4 | 7 | ||
| 3 | 8·41231 | 1 | ||
| 1 | 1·663697 | |||
| 2·611346 |
| 261·1346... | A’s | share | £261.2.8¼ |
| 84·1231... | B’s | ... | 84.2.5¾ |
| 166·3697... | C’s | ... | 166.7.4¾ |
| 511·6274 | £511.2.6¾ | ||
If ever the fraction of a farthing be wanted, remember that the coinage-result is larger than the decimal of a pound, when we use only three places. From 1000 times the decimal take 4 per cent, and we get the exact number of farthings, and we need only look at the decimal then left to set the preceding right. Thus, in
| 134·6 | 123·1 | 369·7 | ||
| 5·38 | 4·92 | 14·79 | ||
| ·22 | ·18 | ·91 |
we see that (if we use four decimals only) the pence of the above results are nearly 8d. ·22 of a farthing, 5½d. ·18, and 4½d. ·91.
A man can pay £2376. 4. 4½, his debts being £3293. 11. 0¾. How much per cent can he pay, and how much in the pound?
3293·553)2376·2180(·7214756
70·7309
4·8598
1·5662
2488
183
18
Answer, £72. 2.11½ per cent.
0.14. 5¼ per pound.