47. When there are intermediate denominations between the given one, and that to which it is required to be reduced. RULE. Reduce the given number step by step in order from the given denomination upward, through all the intermediate ones, until you have brought it to the proposed denomination. Thus to bring farthings into pounds, I first reduce them to pence, then the pence to shillings, and then the shillings to pounds. When there is a remainder, it is of the same denomination with the dividend from whence it arises'. fourth that number of pence; in like manner, in any number of pence there will be one twelfth that number of shillings; and in any number of shillings one twentieth that number of pounds. This rule is therefore evident, as being the converse of the former rule. f This will be plain, from the consideration that every remainder, being a part of the dividend, is evidently of the same name with it. Here, as well as in each of the weights and measures, there are two tables, the first of which is mostly used in reduction; the second shews what number of every inferior denomination is contained in each superior one. Pounds, shillings, pence, and farthings, are usually denoted by the Latin initials L. s. d. q. L denoting libra, a pound; s, solidus, a shilling; d, denarius, a penny; and q, quadrans, a farthing. 3. How many farthings are there in 237. 14s. 5d.4 ? OPERATION. L. S. d. 5693 pence. Ans. 22773 farthings. Explanation. I multiply the pounds, viz. 23, by 20, and to the product I add the 14 shillings, taking in the 4 in the place of units, and 1 in the place of tens. Next I multiply 474 shillings by 12, and take in the 5 pence to the product. I then multiply 5693 pence by 4, and to the product take in the 1 farthing, and the result is the answer required. 4. Bring 45546 farthings into pounds. OPERATION. Explanation. farthings. 4)45546(2 12)11386(10 20) 948 Ans. 471. 8s. 10d. 471. 8s. 10d. the answer. I divide by 4, 12, and 20, as before, (Ex. 2.) The 2 remainder after the first division are 2 farthings, or d. the 10 remainder after the second are pence, and the 8 cut off in the third division are shillings. These are collected with the last quotient, making together 5. In 1237. how many pence? Ans. 29520. Multiply by 20 and by 12. 6. In 4561. how many farthings? Ans. 437760. Multiply by 20, 12, and 4. 7. In 59040 pence, how many pounds? Divide by 12 and 20. Ans, 246. 8. In 266880 farthings, how many pounds? Ans. 278. 9. In 345 crowns, how many farthings? Ans. 82800. Multiply by 5, 12, and 4. 10. In 165600 farthings, how many crowns? Divide by 4, 12, and 5. Ans. 690. 11. In 487 halfcrowns, how many halfpence? Ans. 29220, Multiply by 30 and 2. 12. In 97440 halfpence, how many halfcrowns? Ans. 1624. Divide by 2 and 30. 13. In 243 guineas, how many farthings? Ans. 244944, Multiply by 21, 12, and 4. 14. In 734832 farthings, how many guineas? Ans. 729. Divide by 4, 12, and 21. 15. In 157 moidores, how many halfpence? Ans. 101736. Multiply by 27, 12, and 2. 16. In 610416 farthings, how many moidores? Ans. 471. Divide by 4, 12, and 27. 17. In 4l. 3s. 2d.4, how many farthings? Ans. 3993. See Example 3. 18. In 11979 farthings, how many pounds? Ans. 121. 9s. 6d. See Example 4. 19. In 5l. 12s. 3d.4, how many farthings? Ans. 5390. 20. In 16170 farthings, how many pounds? Ans. 161. 16s. 10d. 21. In Sl. 15s. 4d., how many halfpence? Ans. 4209. 22. In 12627 halfpence, how many pounds? Ans. 26l. 6s. 1d.✈ 23. In one thousand guineas, how many farthings? Ans. 1008000. 24. In ten thousand farthings, how many crowns? Ans. 41 crowns, and 3s. 4d, over. 48. Sometimes it is necessary to reduce numbers from one denomination to another, such, that there is no number of the one contained exactly in one of the other: operations of this kind require both multiplication and division, and are therefore called Reduction ascending and descending. RULE. Having considered what denomination is given, and what is required, reduce the given one to some inferior denomination common tọ them both, that is, to one which is contained some number of times exactly in each of them, by Art. 43. then reduce it from this into the denomination required", by Art. 46. The truth of this method will be plain from the preceding notes. 21 54120 108240 20)1136520 Ans. 56826 pounds. Explanation. I find that a shilling is the greatest denomination contained some number of times without remainder in both a guinea and a pound; I therefore bring 54120 guineas into shillings by multiplying by 21; (Art. 43.) and lastly, I bring the shillings into pounds by dividing by 20, (Art. 46.) 26. How many half guineas are there in 1234 crowns? 27. In 2142 pounds, how many guineas? Ans. 2040. Multiply by 20, and divide by 21. 28. In 3420 guineas, how many half crowns? Ans. 28728. Multiply by 42, and divide by 5. 29. In 840 moidores, how many guineas? Ans. 1080. 30. In 3071. 16s. how many moidores? Ans. 228. TROY WEIGHT". 49. Troy weight is used for weighing such commodities. as are of a pure nature, and very little subject to waste, as gold, h Troy weight (called in the old books Trone weight) is supposed to have originated in France, and to have taken its name from Troyes, a considerable city in the department of Aube. The origin of all English weights was a corn of sound ripe wheat taken out of the middle of the ear; 32 of these well dried were to make 1 pennyweight, 20 pennyweights an ounce, and 12 ounces a pound, according to 51 Henry III. 31 Edward I. and 12 Henry VII.; but afterwards the penny silver, and jewels; it is likewise employed to determine the comparative strength of liquors. gr. 24 grainsi (gr.) make 1 pennyweight (dwt.) 24 = 1dwt. 33. In 123lb. 4oz. 5dwt. 6gr. how many grains? OPERATION. lb. oz. dwt. gr. 123 4 5 6 12 1480 ounces. 20 Explanation. I begin at the highest denomination (16.), and multiply by 12, taking in the 4; this gives ounces. The ounces I multiply by 20, and take 29605 pennywts. in the 5 for pennyweights. The pennyweights I multiply by 24, and take in the 6 for grains, which gives the answer. 24 118426 I divide grains by 24, and the quotient 1645 is dwts. these I divide by 20, and the quotient is 154 Ans. 6lb. 10oz. 5dwt. 3gr. ounces; these I divide by 12, and 144 108 96 123 120 3 grains. the quotient is pounds. The 3 remaining after the first division are grains, the 5 cut off in the second are dwts. the 10 over in the third are ounces; these placed in order give 6lb. 10oz. 5dwt. 3gr. 35. In 6lb. 10oz. 5dwt. 3gr. how many grains? Ans. 39483. 36. In 710526 grains, how many pounds? Ans. 123lb. 4oz. 5dwt. 6gr. weight was divided into 24 parts, called grains, as at present. This seems to have been the only legal weight used in England from the Norman conquest to the year 1533, when an act was passed, authorizing the use of avoirdupois weight, by which meat was to be bought and sold. Rapin's Hist. of Eng. vol. vi. i The grain troy is thus divided and subdivided by the moneyers; viz. a grain into 20 mites, a mite into 24 droits, a droit into 20 periots, and a periot into 24 blanks. |