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REDUCTION.

232. A quantity expressed in different units, such as £3. 11s. 6d. or 15 cwt. 3 qr. 16 lbs., is called a compound quantity; and by the tables, it is evident that any compound quantity can be expressed in several ways. For instance, £4. 9s. 6d. is the same as 89s. 6d., or 1074 pence, or 4296 farthings.

When a quantity is expressed in one or more denominations, reduction shows the method of converting it into one or more others. Since quantities may be either reduced from a high denomination to a lower one, or raised from a lower one to a higher, then there are two kinds of reduction, reduction descending and reduction ascending.

233. First, reduce £24. 8s 74d. into farthings.

Since £1.20s., £24.=24 × 20s. =480s. ; therefore, £24. 8s =480+8=488s. Since 1s. = 12d., 488s.=488 × 12d.=5856d.; therefore, £24. 8s. 7d. 5856d.+7d.=5863d. And since 1d. =4 farthings, 5863 pence=5863 × 4f.=23452f.; and, there fore, £24. 8s. 73d.=23452f.+3f.=23455 farthings.

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234. Secondly, reduce 37849 farthings to pounds, shillings,

and pence

Since 4f. 1d., 1f.d.; therefore, 37849f.=

37849 × 1

4

d. = 94621d.; also, since 12d. 1s., 1d.s.; therefore, 9462d. 9462 × 1 =788s. 6d.; and also, since 20s. £1, 1s. = £20,

12

=

therefore, 788s. = 788 x £= £39. 8s.

==

Hence, 37849f.= 94624d.=788s. 64d. = £39. 8s. 64d., which process is thus

written :

4)37849

12)9462. If.

20)788. 6d.

£39. 8s. 6d.

As a proof of the correctness of these results, operate in a converse way.

235. From what has been said in (§ 233 and 234), the operations in reduction are evident; it is only required to refer to the tables.

236. EXERCISES.

1. Reduce £164. 19s. 74d. to farthings, and prove the operation. 2. In £764. 11s. 114d. how many farthings? Verify the result.

3. What number of pounds, &c., are contained in 3769473 farthings? and prove the correctness of the result.

4. Change 48493 pence to pounds, &c., and conversely.

5. Find the number of farthings in £100. 10s. 103d., and prove it.

6. Required, the number of pence in 564 guineas, and verify the operation.

7. Convert 112546 grains into pounds Troy, and prove the result.

8. Bring 54 lbs. 10 oz. 16 dwts. 19 grs. to grains.

9. In 24 cwt. 2 qrs. 14 lbs., how many pounds?

10. How many drams are there in 3 tons 17 cwt. 2 qrs. 24 lbs. 15 oz. 3 drs. ?

11. Bring 24895 lbs. to cwts., &c.

12. Change 24 lasts 1 wey 4 tods 1 stone 0 cl. 5 lbs. to pounds.

13. Find the number of miles, &c., in 463972 inches.

14. Bring 5 mi. 4 fur. 3 pl. 4 yds. 1 ft. 7 in. to inches.

15. Reduce 14 yds. 3 qrs. 2 nl. 2 in. to inches.

16. How many inches are there in 54 E. e. 4 qrs. 3 nl. 1 in. ?

17. Change 714 inches to Flemish ells.

18. Convert 1476 French ells to inches.

19. Reduce 3 ac. 3 rd. 33 pl. 27 yds. 7 ft. 15 in. to inches

20. In 964893 yds., how many acres?

21. Bring 174 cubic yds. 24 ft. 362 in. to cubic inches.

22. What number of cubic yards, &c., are equivalent to 7945673 cubic inches?

23. Convert 64596 pints to lasts, &c.

24. Reduce 6 lasts 0 wey 3 qrs. 6 bu. 3 pk. 1 gal. 7 pts. to pints.

25. In 37496 pecks how many chaldrons ?

26. In 7 scores 20 ch. 35 bu. 3 pk. how many pecks?

27. Bring 434 hhds. 48 gal. 1 qt. of ale to pints.

28. 76465 pints are equivalent to how many butts, &c.

29. How many tuns, &c., are there in 264054 pints of wine?

30. How many pints are there in 9 hhds. 55 gals. 3 qts. 1 pt. of wine?

31. Convert 365 days 5 hrs. 48 min. 48 sec. to seconds.

32. Convert 4673360 seconds to years, &c.

33. Reduce 24 reams 16 quires 20 sheets of paper to sheets. 34. Convert 1649750 sheets of paper to reams, &c.

35. Find the number of seconds in 164° 45′ 34′′.

36. Find the number of degrees, &c., in 6849".

37. How many barleycorns will reach round the earth, which is 25000 miles in circumference ?

38. The wheel of a railway carriage makes 369600 revolutions in 700 miles. What is the circumference of the wheel?

How

39. The distance between Dover and Calais is 21 miles. many arches, each of 75 feet span, would a bridge between the two places have?

40. The hind wheel of a carriage is 15 ft. 8 in. in circumference, and the fore wheel 12 ft. 3 in. How many revolutions

will the latter make more than the former between London and Edinburgh, the distance being 389 mi. 6 fur. 20 p. ? 41. An equal number of moidores, guineas, pounds, shillings, and pence make £389. 8s. 4d. How many are there of each?

42. A sack of flour weighs 22 stones 12 lbs. of 4 lbs., 3 lbs., 3 lbs. 2 lbs., and 1 from it?

How many loaves lb. can be made

43. How many seconds have elapsed from the Christian era to the beginning of 1853?

44. In what time would sound travel from the earth to the moon, the distance being 240000 miles, and sound moving about 1143 feet per second?

45. How long would it take a cannon ball, at the velocity of 1960 feet per second, to travel from the earth to the sun, the distance being 95 millions of miles ?

46. The piston of a steam engine moves at the rate of 240 feet per minute. What is the rate in miles, per day of 16 hours? 47. If 7 yds. 3 qrs. of cloth make a suit of clothes, how many suits can be made from 24 pieces, each of 45 yards?

48. How many times does a clock, which strikes the hours and quarters, strike, between noon on the 20th of June, and midnight on the 31st of December, of the same year?

49. An indolent youth loses 8 minutes per hour. How much time will he lose from August 12th to December 17th, reckoning 10 hours work per day?

50. A wheel is 5 yards in circumference. How long is that part of the circumference which measures 48° 24'?

51. If a person counts 80 sovereigns per minute for 15 hours each day, in how many days will he count a million?

52. How many canisters of tea can be filled out of 8 cwt. 2 qrs., the canisters holding respectively 2 lbs., lb., lb., and there being the same number of each?

ADDITION OF COMPOUND QUANTITIES.

237. Little need be said upon the principles of performing the fundamental operations of compound quantities; references to the corresponding operations of simple quantities will suffice. The former differ from the latter in this, that the subdivisions of the unit are not uniform, as they do not follow the decimal system.

238. A owes to B £564. 2s. 73d.; to C, £324. 16s. 11d.;

to D, £216. 14s. 104d.; and to E, £949. 15s. 6d. How much does he owe in all?

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Having written the numbers under each other, taking care that the units of the same kind are under each other, begin by adding together the farthings, which amount to 7f., or lad., set down the under the farthings; the sum of the pence with the 1 penny resulting from the addition of the farthings, is 35 pence, or 2s. 11d., set down 11d. under the pence; the sum of the shillings, with the 2s. resulting from the addition of pence, is 49 shillings, or £2. 9s., the 9s. are set down under the shillings, and the £2 carried to pounds, the sum of which is £2055.

The same method of proof may be employed as for simple `addition.

When the quantities are of any other sort, the same method is followed. The tables render the process easy.

239. EXERCISES.

1. Find the sum of £3489. 11s. 63d.; £267. 7s. 74d; £5671. 14s. 10d.; £367. 19s. 114d.; and £1304. 19s. 11 d. 2. The expenses of building a house were: surveyor, £340; bricklayer, £5696 17s. 4d.; mason, £2740. 16s. 7d.; carpenter, £4169. 17s. Od.; plumber, £1565. 15s. 3d.; glazier, £473. 10s. 6d.; painter, £375. 18s. 6d.; paper-hanger, £124. 1s. 4d.; and locksmith, £275. 19s. 8d. What did the house cost?

3. The distance from A to B is 24 mi. 7 fur. 20 p.; from B to C, 35 mi. 4 fur. 28 po. 3 yds.; from C to D, 47 mi. 6 fur. 31 po. 4 yds.; from D to E, 15 mi. 3 fur. 16 po. 2 yds. What is the distance between A and E ?

4. Of five pieces of timber, the first contains 3 cub. yds. 24 ft 416 in.; the second, 5 cub. yds. 16 ft. 94 in.; the third, 4 cub. yds. 18 ft. 1014 in.; the fourth, 2 cub. yds. 12 ft. 96 in.; and the fifth, 3 cub. yds. 10 ft. 184 in. How many cubic yards, &c., are there in all ?

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