A concrete number, whether simple or compound, is often called a Denominate Number.

NOTE 1. All operations in the preceding pages are upon simple num

bers.

NOTE 2. The several parts of a compound number, though of different denominations, are yet of the same general nature; thus, 2 weeks, 3 days, and 6 hours are SIMILAR quantities, and constitute a compound number; but 2 weeks, 3 miles, and 6 quarts are UNLIKE IN THEIR NATURE, and do NOT constitute a compound number.

87. REDUCTION is changing a number of one denomination to one of another denomination, without changing its value.

It is of two kinds, viz. Reduction Descending and Reduction Ascending.

REDUCTION DESCENDING consists in changing a number from a higher to a lower denomination.

REDUCTION ASCENDING is changing a number from a lower to a higher denomination.

ENGLISH MONEY.

88. ENGLISH MONEY is the Currency of Great Britain.

89. REDUCTION DESCENDING is performed by multiplication; thus, to reduce 15£ to shillings, we multiply 15 by 20, because there will be 20 times as many shillings as pounds. So to reduce 15£ and 12s. to shillings, we multiply 15 by 20, and to the product add the 12s.

86. A Concrete Number, what is it called? 87. What is Reduction? How many kinds of Reduction? What are they called? What is Reduction Descending? Reduction Ascending? 88. What is English Money? Repeat the table. 89. How is Reduction Descending performed?

In a similar manner all such examples are reduced.

Hence, 90. To reduce the higher denominations of a com, pound number to a lower denomination:

RULE. Multiply the highest denomination given by the number it takes of the next lower denomination to make one of this higher, and to the product add the number of the lower denomination; multiply this sum by the number it takes of the NEXT lower denomination to make one of THIS; add as before, and so proceed till the number is brought to the denomination required.

Ex. 1. Reduce 11£ 17s. 9d. 3qr. to farthings.

NOTE. Since there are no pence in the 3d example, there is nothing to add to the product obtained by multiplying by 12.

4. Reduce 27£ 15s. 6d. 2qr. to farthings.

5. Reduce 32£ 8d. 3qr. to farthings.

91. REDUCTION ASCENDING is performed by division; thus, to reduce 4299 farthings to pence, we divide the 4299 by 4, because there will be only one fourth as many pence as farthings. Performing the division we obtain 1074d. and a remainder of 3qr. If we wish to reduce the 1074d. to shillings, we divide by 12, because there will be only one twelfth as many shillings as pence, and obtain 89s. and a remainder of 6d. Again,

90. Repeat the rule. Explain the process in Ex. 1. How are the 237 shillings obtained? How the 2853 pence? The 11415 farthings? 91. How is Re duction Ascending performed?

the 89s. may be reduced to pounds, by dividing by 20, giving 4£ and a remainder of 9s. Thus we find that 4299qr. are equal to 4£ 9s. 6d. 3qr.

Like reasoning applies to all similar examples. Hence,

92. To reduce a number of a lower denomination to numbers of higher denominations:

RULE. Divide the given number by the number it takes of that denomination to make one of the next higher; divide the quotient by the number it takes of THAT denomination to make one of the NEXT higher, and so proceed till the number is brought to the denomination required. The last quotient, together with the several remainders (Art. 69, Note), will be the answer.

93. Reduction Ascending and Reduction Descending prove each other.

Ex, 1. Reduce 11415 farthings to pence, shillings, and pounds.

OPERATION.

4) 11 41 5 qr.
12) 28 5 3 d. +3qr.
20) 237 s. 9d.

11 £+17s.

First divide by 4 to reduce the farthings to pence; then divide by 12 to reduce pence to shillings; then by 20 to reduce shillings to pounds, and thus obtain 11£ 17s. 9d. 3qr., Ans.

2. Reduce 17229qr. to pence, shillings, and pounds.

Ans. 17£ 18s. 11d. 1qr.

3. Reduce 6874d. to shillings and pounds.

Ans. 28£ 12s. 10d.

NOTE 1. Since Ex. 3rd is given in pence instead of farthings, the first divisor is 12 rather than 4.

4. Reduce 84697qr. to higher denominations.

5. Reduce 124683qr. to higher denominations.

6. Reduce 347624qr. to pence, shillings, and pounds.

7. Reduce 3746d. to shillings and pounds.

8. Reduce 8793s. to pounds.

'92. Repeat the rule. Explain the process in Ex. 1. How the 9d.? The 178.? The 11£? 93. What is the

How are the 8qr. obtained?
Proof in Reduction?

NOTE 2. The numbers employed in the reduction of a compound num ber are called a Scale. The scale is a descending scale for Reduction Descending and an ascending scale for Reduction Ascending; thus, in English money the descending scale is 20, 12, and 4, and the ascending scale is 4, 12, and 20. The descending scale consists of the numbers at the left hand of the table, taken in order from the bottom to the top of the table, and the ascending scalé consists of the same numbers taken in the reversed order, i. e. from the top to the bottom of the table. In like manner the scale is found in the other tables.

TROY WEIGHT.

94. TROY WEIGHT is used in weighing gold, silver, and precious stones.

7 lb. 11oz. 14dwt. 18gr. 24) 459 5 4 gr.

NOTE 1. In solving Ex. 1, the several numbers of the lower denomina

93. What is a scale? A descending scale? An ascending scale? What are the scales for English money? Where are these scales found? Taken in what order? 94. For what is Troy Weight used? Repeat the table. Descending scale! Ascending?

tions are added mentally, and only the results are written; thus, 12 times 7 are 84, and the 11oz. added give 95oz. Then multiplying the 95oz. by 20, and adding the 14dwt., we have 1914dwt. Finally, in multiplying the 1914 dwt. by 24, first multiply by 4, adding in the 18gr., and then multiplying by 2, and adding the results we have 45954gr. for the answer.

NOTE 2. In reducing Ex. 2, if any divisor is so large that the work is not easily done by Short Division, the numbers may be taken upon the slate and the work done by Long Division, setting down only the results.

3 How many grains in 16lb. 8oz. 19dwt.? 4. Reduce 38695gr. to pounds, etc.

Ans. 96456gr.

Ans. 6lb. 8oz. 12dwt. 7gr.

5. Reduce 87942gr. to pounds, ounces, etc. 6. Reduce 15lb. 8oz. 6dwt. 15gr. to grains. 7. How many spoons, each weighing 2oz. 8dwt. 20gr., can be made from 2lb. 5oz. 6dwt. of silver?

Ans. 12.

8. A jeweller made 8oz. 16dwt. of gold into rings which weighed 3dwt. 16gr. each; how many rings did he make?

APOTHECARIES' WEIGHT.

95. APOTHECARIES' WEIGHT is used in mixing or compounding medicines; but medicines are bought and sold by Avoirdupois Weight.

NOTE 1. The pound, ounce, and grain, in Apothecaries' and Troy Weight are equal, but the ounce is differently subdivided.

94. In solving Ex. 1, what is done with the numbers of the lower denominations? In Ex. 2, how is the work done? 95. For what is Apothecaries' Weight used? Repeat the table. Descending scale? Ascending? What denominations of Apothecaries' Weight are like those of Troy Weight?