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76. The denominations of English money, are guineas, pounds, shillings, pence, and farthings.
TABLE. 4 farthings marked far. make 1 penny, marked d. 12 pence
1 shilling, 20 shillings
£. 21 shillings
1 guinea. £
d. far. 1 = 20 = 240 = 960
1 4 NOTE 1. This table is read, 4 farthings make 1 penny, 12 pence make 1 shilling, 20 shillings 1 pound.
NOTE 2. Farthings are generally expressed in fractions of a penny. Thus, for 1 farthing, write fd., for 2 farthings Ld., and for 3 farthings, d.
REDUCTION OF DENOMINATE NUMBERS. 77. Reduction is changing the unit or denomination of a number, without altering the value of the number.
For example, 42 dollars and 35 cents are expressed in different denominations.
But 42 dollars are equal to 4200 cents.
76. What are the denominations of English money? Repeat the table. How many farthings in 1 shilling? In 1 pound? How are farthings generally written?
77. 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 ?
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, have
3x3=9 feet ; and 9 x12=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.
78. From the above illustrations we see, that reduction of denominate numbers generally, like that of Federal money, is divided into two parts.
1st. To change the unit of a number from a higher denomination to a lower.
2d. To change the unit of a number from a lower de. nomination to a higher.
79. To reduce denominate numbers from a higher denomination to a lower.
RULE. I. 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.
78. Into how many parts may reduction of denominate numbers be divided ? Name them. Does the term reduction imply a change in value?
79. How do you reduce numbers from a higher to a lower denomination? Repeat the rule.
1. Reduce 9 yards and 6 feet to inches.
3 We first bring the yards to feet, and then add the 6 feet, after
27 which we reduce the whole to
6 feet to be added inches.
2. Reduce £27 6s. ed. to the denomination of pence.
£27 6s. 8d.
We first bring the pounds to shil. lings and then add the 6s.; we then bring the shillings to pence and add in the 8d., giving for the answer,
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 prod
12 uct 6 in the units' place: and when
6560 we 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.
80. To reduce denominate numbers from a lower de. nomination to a higher.
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.
III. 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.
1. Reduce 3138 farthings to the denomination of pounds.
In this example we first divide by 4, the number of far 4)3138 things in a penny; the quotient
12)784 . 2 far. rem. is 784 pence, and 2 farthings over. The 784 pence are then
210)615 . . 4d. rem. divided by 12, the number of
3.. 58. rem. pence in a shilling. The quo
Ans. £3 58. 4d. 2far. tient is 65 shillings, and four pence over. 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 58. 4d. 2far. is the value of 3138 farthings. 2. Reduce 3658 inches to yards.
Ans. 101 yards, 1 foot, 10 inches. 3. In 80 guineas, how many pounds ?
Ans. £84. 80. In reducing from a lower denomination to a higher what do you first do? What next ? and what next? Is this rule applicable to all denominato numbers ?
4. In 1549 farthings, how many pounds, shillings and pence ?
Ans. £1 12s. 344. 5. Reduce 1046 pence to pounds. Ans. £4 7s. 2d. 6. Reduce 4704 pence to guineas.
Ans. 18 guineas 14s. 7. In 6169
£? Ans. £25 14s. 1d.
PROOF OF REDUCTION.
81. 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.
1. Reduce £15 7s. 6d. to the denomination of pence.
6d. rem. 307
78. rem. 12 3690
Ans. £15 78. 6d. 2. In £31 8s. 9d. 3far., how many farthings ? Also the proof.
3. In £87 14s. 8 d., how many farthings ? Also the proof.
4. In £407 198. 11 d., how many farthings? the proof.
82. The standard avoirdupois pound of the United States, as determined by Mr. Hassler, is the weight of 27.7015 cubic inches of distilled water. By this weight are weighed all coarse articles, such as hay, grain, chandlers' wares, and all the metals, except gold and silver.
81. How do you prove reduction?