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is smaller than the proposed integer, 960, you must annex ciphers at the right hand of the given sum. In this manner must any given sum be reduced to the decimal of a higher, or superiour denomination.

2. Reduce 10s. 6d. to the decimal of a pound. Ans.,525. 3. Reduce 109d. 12h. to the decimal of a year. Ans.,3. 4. Reduce 4 calendar months to the decimal of a year. Ans.,375.

5. Reduce 12 drachms to the decimal of a pound avoirdupois. Ans. ,046875.

6. Reduce 2qr. 14lb. to the decimal of a hundred-weight. Ans..,625.

7. Reduce 7oz. 19pwt. to the decimal of a pound troy. Ans. 6625.

8. Reduce 3p. to the decimal of a bushel. Ans.,75.

9. Reduce Iqt. 1pt. to the decimal of a gallon. Ans. ,375. 10. Reduce 5fur. 16po. to the decimal of a mile. Ans.,675. 11. Reduce 4po. to the decimal of an acre. Ans.,025.

12. Reduce 3qr. 2na. to the decimal of a yard. Ans.,875.

EXPLANATIONS.

You have now learned all of Arithmetick, that is, you have learned all of the different operations of WORKING FIGURES. You have learned notation and numeration, both of whole numbers and Decimal Fractions. You have learned to add, substract, multiply, and divide numbers, both simple and compound; and you have also learned to add, substract, multiply, and reduce fractions, or parts of whole numbers. All that you now have to become acquainted with, is the different and various applications of the preceding rules, or operations, in the transactions of the various kinds of mechanical and commercial business. All of these operations are performed either by Addition, Substraction, Multiplication, or Division; in some, you must add, substract, and divide; and in others, you must add, substract, multiply, and divide; and, therefore, as the different fundamental rules are used in the operation, these rules have, for the sake of distinction and convenience in reference, names applicable, or appropriate to the application of them in a particular kind of business, or mechanical or com mercial transaction. Thus, we call the operation of Multi

plication and Division, in a certain manner, Interest, Commission, Ensurance, &c. The operation of Multiplication and Division, in another certain manner, Rule of Three, Discount, Barter, Loss and Gain, Tare and Tret, &c. By a very trifling difference in the operation, we have other rules which we call Square and Cube Roots, Position, Arithmetical Progression, &c. &c. But you must not apprehend any difficulty, or be in the least alarmed at this array of new names or rules, for there is no new principle to be learned: you have merely to observe the different manner of applying the rules, the principles of which you already know, to the various and useful transactions of business, in the different mechanical and com mercial pursuits. Only bear this in mind, and all your anticipated difficulties, with regard to the working of new sums, or rules, will vanish.

REDUCTION.

Q. What is REDUCTION?

A. Reduction teaches to change numbers from one denomination to another, without altering their

value.

Q. How many kinds of Reduction are there? A. Two; Reduction Descending, and Reduction Ascending.

REDUCTION DESCENDING.

Q. What is Reduction Descending

A. Reduction Descending teaches to change, or bring higher denominations into lower; as, pounds into shillings, shillings into pence; pounds into ounces; yards into quarters, &c.

RULE.

Q. How do you reduce high denominations to lower?

A. The number in the highest denomination of the given sum must first be multiplied by that number which it takes of the next lower to make one in that higher, and the figures of the next lower denomination of the given sum must be added in. In this manner must each denomination be multiplied throughout the different denominations; that is, each denomination must be multiplied by that number which it takes of the next lower to make one of that which you are multiplying, always remembering to add in all of the next lower denomination in the given sum when each denominator is multiplied.

REDUCTION ASCENDING.

Q. What is Reduction Ascending?

A. Reduction Ascending teaches to change, or bring lower denominations into higher; as, shillings into pounds, pence into shillings; ounces into pounds; quarters into yards, &c.

RULE.

Q. How do you change low denominations to higher?

A. The lowest denomination given must be divided by that number which it takes of that denomination to make one of the next higher; and in this manner must each denomination be divided up to the denomination required.

Reduction Ascending is precisely the reverse of

Reduction Descending; and, therefore, the different sums in each may be worked reciprocally, as they prove each other.

EXAMPLES

For Theoretical Exercise on a Slate.

FEDERAL MONEY.

1. In $4, how many cents? Ans. 400c.

EXPLANATIONS.

$

100

400c. Ans.

In this example, you multiply the $4 by 100, because 100 cents make a dollar. All that is necessary, however, in reducing federal money, is to add two ciphers to the dollars to reduce them to cents, and three ciphers to reduce them to mills; or, if the sum consist of dollars and cents, add one cipher to reduce them to mills, which is the same as multiplying by 10, 100, 1000, as you remember in the multiplication of whole numbers, by 10, 100, 1000, &c. When the given sum is composed of dollars, cents, and mills, you have only to remove the comma, or separatrix, and the answer will be in mills. To bring mills into cents, you must cut off one figure at the right hand, by the separatrix, and the figures left of the separatrix will be cents; and to bring mills into dollars and cents, you must cut off one figure for mills, and two more for cents, and the figures at the left of the separatrix will be dollars. 2. In 400 cents, how many dollars? Ans. $4. 3. In $8, how many mills? Ans. 8000m.

4. In 8000 mills, how many dollars? Ans. $8.

5. In $800 and 1 mill, how many mills? Ans. 800001m. 6. In 800001 mills, how many dollars? Ans. 800,00,1m. 7. In $1, 11 cents, and 1 miil, how many mills? Ans. 1111m. 8. In 1111 mills, how many dollars? Ans. $1,11,1m.

STERLING OR ENGLISH MONEY.

1. In £31 11s. 10d. Igr., how many farthings? Ans.30329qr.

EXPLANATIONS.

£

S. d. gr.

31 11 10 1
20 shillings in a pound.

631 shillings.
12 pence in a shilling.

7582 pence.

4 farthings in a penny.

In this example, you must multiply the £31 by 20, because twenty shillings make a pound, and add in the 11s., in the given sum, with the product of shillings. You must multiply the 631 shillings by 12, because twelve pence make a shilling, and add in the 10d., in the given sum, with the product of pence. You must multiply the 7582 pence by 4, because four farthings make a penny, and add in the 1gr., in the given sum, with the product of farthings. In this easy manner are all large denominations changed into smaller, by multiplying the given sum by that number which it takes of the next lower to make one of the denomination you are multiplying, always remembering to add in the lower denomination of the given sum, in the product of that of the same denomination; as, in multiplying hours, you multiply by 60, because sixty minutes make an hour, &c.

30329 farthings.

2. In 30329 farthings, how many pounds? Ans. £31 11s. 10d. 1gr.

EXPLANATIONS.

qr.

4)30329

12)7582 1qr.

20(631 10d.

In this example, you must first divide by 4, because four farthings make one penny, the next higher denomination, and the remainder will be farthings. Divide the 7582 pence by 12, because twelve pence make a shilling, the next higher denomination, and the remainder will be pence. Divide 631 shillings by 20, because twenty shillings make a pound, the next higher denomination, and the remainder will be shillings, and you will then have the answer, £31 11s. 10d. 1yr. In this manner are all small denominations changed into larger, by dividing the given sum by that number which it takes of that to make one of the next

£31 11s.

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