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$ 53. There are other denominate numbers besides those of Federal Money. For example, 6 yards of cloth is a denominate number, the unit, 1 yard of cloth, being denominated or named.
Two numbers are of the same denomination, when they have the same unit, and of different denominations when they have different units.
For example, 8 feet and 10 feet are of the same denomination, the unit being 1 foot; but 30 feet and 60 yards are of different denominations, the unit of the first being 1 foot, and the unit of the second, 1 yard.
Q. What is a denominate number? (see 45.) What is the unit of 6 yards of cloth? When are two numbers of the same denomination | Give an example. When of different denominations? Give an example.
$54. The following Tables show the different kinds of denominate numbers in general use, and also their relative values.
ENGLISH MONEY. The denominations of English Money, are guineas, pounds, shillings, pence, and farthings.
£. 21 shillings
1 guinea £
à far. 1 20 : 240
= 960 12 48
1 4 NOTE.-Farthings are generally expressed in fractions of a penny. Thus, for 1 farthing we write 1, for 2 farthings, 3d, and for 3 farthings, d.
Q. What are the denominations of English Money? Repeat the table. How are farthings generally expressed ?
REDUCTION OF DENOMINATE NUMBERS.
§ 55. Reduction is changing the denomination of a number without altering its value.
For example, 42 dollars and 35 cents are expressed in different denominations. But 42 dollars are equal to 4200 cents,
the sum 4235 cents is equal to 42 dollars and 35 cents. Here we have brought the numbers to the same denomination without altering their value.
Again, if we have 24 shillings, we can reduce them to pounds and shillings: for, since 20 shillings make 1 pound, 24 shillings are equal to £1 4s. Here we have again changed the denomination without altering the value.
We may take, as another example, 3 yards and reduce it to inches. Now, since 3 feet make a yard, and 12 inches a foot, we have
3x3=9 feet; and 9x12=108 inches. If
, on the contrary, it were required to bring inches into yards, we should first divide by 12, to bring them into feet, and then by 3 to bring the feet into yards. Thus,
108 inches -12=9 feet; and 9 feet +3=3 yards.
We therefore see, that reduction of denominate numbers generally, like that of Federal Money, is divided into two parts.
1st. To reduce a number from a higher denomination to a lower.
2d. To reduce a number from a lower denomination to a higher.
Q. What is reduction ? How many pounds and shillings in 24 shillings? How many feet in a yard ? How many inches in a foot ? How many feet in 3 yards? How many inches in 3 yards ? How many feet in 72 inches? How many yards? Into how many parts may reduction of denominate numbers be divided ? Name them.
CASE I. $ 56. To reduce denominate numbers from a higher denomination to a lower.
RULE. J. Consider how many units of the next lower denomination make one unit of the higher.
II. Multiply the higher denomination by that number, and add to the product the number belonging to the lower : we shall then have the equivalent number in the next lower denomination.
III. Proceed in a similar way through all the denominations to the last; the last sum will be the required number.
Q. How do you reduce numbers from a higher to a lower denomination? Repeat the rule.
1. Reduce 9 yards and 6 feet to inches?
We first bring the yards to feet, 27 and then add the 6 feet, after which 6 feet to be added. we reduce the whole to inches.
OPERATION. £27 Os 8d
We first bring the pounds to shillings and then add the 6s; we then bring the shillings to pence and add in the 8d, giving for the answer, 6560 pence.
In reducing, we often add the next OPERATION. lower denomination mentally without £27 6s 8d setting it down. Thus, when we mul
20 tiply by 20, we add the 6s, without
546s writing it down, making in the pro
12 duct 6 in the units place: and when we
6560 multiply by 12 we say, 12 times 6 are 72 and 8d to be added make 80.
3. In £1465 14s 5d how many farthings ? Ans. 1407092. 4. In £45 12s 10d, how many pence?
Ans. 10954. 5. In 87 guineas, how many farthings? Ans. 87696. 6. In £145 16s 11d, how many pence? Ans. 35003.
$ 57. To reduce denominate numbers from a lower denomination to a higher.
RULE, I. Consider how many units of the given denomination make one unit of the next higher; and take this number for a divisor: divide the given number by it and set down the remainder, if there be any.
II. Divide the quotient thus obtained by the number of units in the next higher denomination, and set down the remainder.
IIJ. Proceed in the same way through all the denominations to the highest ; the last quotient with the several remainders annexed, will give the answer sought, and if there be no remainders, the last quotient will be the answer.
Q. In reducing from a lower denomination to a higher what do you first do? What next ? and what next?
EXAMPLES. 1. Reduce 3138 farthings to the denomination of pounds.
In this example we first divide by 4, the number of farthings in 4)3138 a penny; the quotient is 784
12)784 . 2 far. rem. pence, and 2 farthings over. The
2|06|5.. 4d. 784 pence are then divided by 12, the number of pence in a shilling. The quotient is 65 Ans. £3 5s 4d 2 far. shillings and four pence over.
3.. 58. rem.
The 65 shillings are then divided by 20, the number of shillings in a pound, the quotient is £3 and a remainder of 5 shillings. Hence, £3 5s 4d 2 far. is the value of 3138 farthings.
Note.—The same rules apply to all the denominate numbers. 2. Reduce 3658 inches to yards ?
Ans. 101 yards, 1 foot, 10 inch. 3. In 80 guineas, how many pounds ? Ans. £84.
4. In 1549 farthings, how many pounds shillings and pence?
Ans. £i 12s 31d. 5. Reduce 1046 pence to pounds. Ans. £4 7s 2d. 6. Reduce 4704 pence to guineas. Ans. 18 guineas 14s. 7. In 6169 pence, how many £ ? Ans. £25 14s 1d.
PROOF OF REDUCTION. $ 58. After a number has been reduced from a higher denomination to a lower, by the first rule, let it be reduced back by the second; and after a number has been reduced from a lower denomination to a higher, by the second rule, let it be reduced back by the first rule. If the results agree the work is supposed right.
EXAMPLES. 1. Reduce £15 7s 6d to the denomination of pence. OPERATION.
210)307.... 60 Rem., 307
15 ....7s Rem.
Ans. £15 7s 6d. 2. In £31 8s 9d 3 far.: how many farthings ? Also the proof.
3. In £87 14s 8d: how many farthings? Also the proof. 4. In £407 198 112d: how many farthings ? Also the
TROY WEIGHT. § 59. Gold, silver, jewels, and liquors, are weighed by this weight. Its denominations are pounds, ounces, pennyweights, and grains.