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REDUCTION OF COMPOUND NUMBERS.
160. The process of changing compound numbers from one denomination into another, without altering their value, is called Reduction.
Ex. 1. Reduce £3 to farthings.
to shillings. This is done by mul£3
tiplying them by 20, because 20s. 20s. in 1£.
make £1. (Art. 147.) That is, 60 shillings.
since there are 20s. in £1, in £3 12d. in ls.
there are 3 times 20s. or 60s. We now reduce the 60s. to pence, by
multiplying them by 12, because 4 lar.in 1d. 12d. make ls. Finally, we re
duce the 720d. to farthings, by Ans. 2880 far.
multiplying them by 4, because 4 far. make ld. The last product, 2880 sar., is the answer, that is, £3=2880 far. 2. Reduce £2, 3s. 6d. and 2 far. to farthings.
Operation. In this example there are shil£.
lings, pence, and farthings.2 3 6 2
Hence, when the pounds are re20s. in £l.
duced to shillings, the given shil
lings (3) must be added mentally 43 shillings. to the product. In like manner, 12d. in ls.
when the shillings are reduced to pence, the given pence (6) must
be added ; and when the pence area 4 far. in ld.
reduced to farthings, the given far2090 far. Ans. things (2) must be added.
Obs. 1. In these examples it is required to reduce higher denominations to lower; as pounds to shillings, shillings to pence, &c., which is done by successive multiplications.
QUEST.-160. What is Reduction? How are pounds reduced to shil. lings? Why multiply by 20? How are shillings reduced to pence ? Why? How, pence to farthings? Why?
2. But it often happens that we wish to reduce lower denominations to bigher, as farthings to pence, pence to shillings, and shillings to pounds. Thus,
3. In 2880 farthings, how many pounds ? Operation. First, we reduce the given farthings
to pence, which is the next higher de
nomination. This is done by dividing 12)720d.
them by 4. For, since 4 far. make ld., 20)60s.
(Art. 147,) in 2880 far, there are as
many pence as 4 is contained times in £3 Ans. 2880. And 4 is contained in 2880, 720
times. We now reduce the pence (720) to shillings, by dividing them by 12, because 12d. make ls. Finally, we reduce the shillings (60) to pounds, by dividing by 20, because 20s. make £i. Thus, 2880 far.=£3, which is the answer required. 4. How many pounds in 2090 farthings ? Operation.
In dividing by 4 there is a 4)2090 far.
remainder of 2 far. ; in dividing 12)522d. 2 far. over. by 12, there is a remainder of
6d. ; in dividing by 20, the quo20)43s. 6d. over.
tient is £2 and 3s. over. The £2, 3s. over.
answer, therefore, is £2, 3s. 6d. Ans. £2, 3s. Od. 2 far.
2 far. That is, 2090 far.=£2, 3s. 6d. 2 far.
Obs. 1. The last two examples are exactly the reverse of the first two; that is, lower denominations are required to be reduced to higher, which is done by successive divisions.
2. Reducing compound numbers to lower denominations is usually called Reduction Descending; reducing them to higher denoininations, Reduction Ascending. The former employs multiplication; the latter division. They mutually prove each other.
Quest.-Obs. Which of the fundamental rules is employed in reducing higher denominations to lower ? Ex. 3. How are farthings reduced to pence? Why divide by 4? How reduce pence to shillings? Why? How, shillinys to pounds? Why? Ohs. Which of the fundamental rules is employed in reducing lower denominations to higher? What is re- . ducing compound numbers to lower denominations usually called ? To higher denominations ? What rule is employed by the former? By the latter?
161. From the preceding illustrations we derive the following
GENERAL RULE FOR REDUCTION.
I. To reduce compound Nos. to lower denominations.
Multiply the highest denomination given, by that number which it tukes of the next louer denomination to make ONE of this higher; to the product, add the number expressed in this lower' denomination in the given example. Proceed in this manner with each successive de nomination, till you come to the one required.
II. To reduce compound Nos. to higher denominations.
Divide the given denomination by that number which it takes of this denomination to make one of the next higher. Proceed in this manner with each successive denomination, till you come to the one required. The last quotient, with the several reznainders, will be the answer sought.
162. Proof.-Reverse the operation ; that is, reduce back the answer to the original denominations, and if the result correspond with the numbers given, the work is right.
Obs. Each remainder is of the same denomination as the dividend from which it arose. (Art. 66. Obs. 2.)
STERLING MONEY. (Art. 147.) 5. In £35, 4s. 6d. how many pence? Operation.
Proof. £ d.
12)8454 pence. 35 4 6
20)704s. 6d. 20
£35, 4s. 6d. 704
12 Ans. 8454d.
Quest.–161. How are compound numbers reduced to lower denominations? How reduced to higher denominations ? Obs. Of what denomination is each remainder? 162. How is Reduction proved ?
6. In 57600 farthings, how many pounds ?
£60 12)14400 d.
20 20)1200 s.
12 £60. Ans.
4 57600 far.
7. In £43, 12s. how many shillings?
TROY WEIGHT. (Art. 148.) 16. In 11 lbs., how many pennyweights ?
Ans. 2640 pwts. 17. In 15 ounces, how many grains ? 18. In 10 lbs. 5 oz. 6 pwts., how many grains ? 19. In 512 pennyweights, how many pounds ? 20. In 2156 grains, how many ounces? 21. In 35210 grains, how many pounds?
AVOIRDUPOIS WEIGHT. (Art. 149.)
22. Reduce 25 pounds to drams. Ans. 6400 drams.
29. Reduce 5672 ounces to tons.
APOTHECARIES' WEIGHT. (Art. 150.)
32. How many drams are there in 70 pounds ?
Ans. 6720 drams, 33. How many scruples in 156 pounds ? 34. How many ounces in 726 scruples ? 35. How many pounds in 1260 drams?
LONG MEASURE. (Art. 151.)
36. In 96 rods, how
5 yds, in 1 r.
3 ft. in 1 yd. Ans. 1584 feet.
We first multiply by 5, then by d, and unite the two results. (Art. 134. a) But to multiply by d, we take half of the mul. tiplicand once. (Art. 131.)
37. In 45 furlongs, how many inches ?
We first reduce the feet to yards,
by dividing by 3; next, reduce the 528
yards to rods, by dividing by 5s. But 2
to divide by 5, we reduce it to 11)1056 halves, and also reduce the dividend Ans. 96 (528 yds.) to halves, then divide 1056
by 11. (Art. 139. I.) 39. In 1728 inches, how many rods? 40. In 26400 feet, how many miles ? 41. In 25 leagues, how many inches? 42. In 40 leagues, 2 inches, 6 furlongs, how many feet? 43. In 750324 inches, how many
miles ? 44. How many barleycorns in the circumference of the earth ?