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44. When there are intermediate denominations between the given denomination, and that to which you would reduce it.

RULE. Reduce the given number step by step in order, through all the intermediate denominations, (by the foregoing rule,) until you have brought it down to the proposed denomination.

Thus to bring pounds into farthings, I first reduce the pounds into shillings, then the shillings into pence, and lastly the pence into farthings.

45. When the given number consists of several denominations. RULE. Begin at the highest denomination, reduce it to the second, and to the result add the second denomination in the given number; reduce this sum to the third denomination, and to the result add the third denomination in the given number; proceed until you have arrived at the denomination required.

Thus to bring pounds, shillings, pence, and farthings, into farthings; I begin with the pounds, reduce them into shillings, and add the given shillings to the result; I then reduce this number into pence, and take in the given pence; and lastly I reduce this last number into farthings, and take in the given farthings.

46. To bring small names into great; that is, to reduce numbers from a lower denomination to a higher.

RULE. Divide by the number denoting how many of the lower denomination make one of the higher.

Thus to bring farthings into pence, I divide the farthings by 4, because 4 farthings make one penny; to bring pence into shillings, I divide the pence by 12, because 12 pence make one shilling; and to bring shillings into pounds, I divide the shillings by 20, because 20 shillings make one pounde.

(since 4 farthings make 1 penny) there will be 20 times 12 times 4, or 960 farthings, in 1 pound; consequently any number of pounds multiplied by 240 will produce the number of pence, and by 960, the number of farthings, in those pounds.

a When there are shillings, pence, or farthings, connected with the given number, it is plain that these must be added in successively, each with its like; viz. shillings with shillings, pence with pence, and farthings with farthings. This being understood, the reason of the rule, as applied to weights and measures, will likewise be evident.

Because there are 4 farthings in 1 penny, 8 farthings in 2 pence, 12 farthings in 3 pence, and in general 4 times the number of farthings in any number of pence; it follows that there will be in any number of farthings one

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.

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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.

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3. How many farthings are there in 231. 14s. 5d.?

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Explanation.

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 471. 8s. 10d. the answer.

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. Divide by 4, 12, and 20.

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., 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 8l. 15s. 4d., how many halfpence? Ans. 4209. 22. In 12627 halfpence, how many pounds? Ans. 261. 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 to 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.

25. Reduce 54120 guineas into pounds.

OPERATION.

guin. 54120

21

54120 108240

20)113652 0

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?

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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.
Multiply by 27, and divide by 21.

30. In 3077. 16s. how many moidores? Ans. 228.
31. In 57015 crowns, how many guineas? Ans. 13575.
32. In 10500 moidores, how many pounds? Ans. 14175.

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,

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

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