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REDUCTION ASCENDING.

To reduce a quantity from a lower to a higher denomi

nation.

49. RULE. Divide the number expressing the quantity by the number which shows how many of the denomination in which it is make one of the higher denomination the quotient will be of the higher denomination, and if there be any remainder, it will be of the lower.

When there are intermediate denominations between that in which the quantity is given and that to which it is to be reduced, it is generally better to reduce it by successive steps; first to one or more of the intermediate denominations, and then to the required denomination.

Method of Proof.

To the answer found by this rule, apply the rule for Reduction Descending, and if the result be the same as the given number, the work is correct. Hence Reduction Descending may be proved in a similar manner, by applying the rule for Reduction Ascending.

Ex. 1. In 350600 cents how many eagles and dollars?

100)3506|00

10)3506 dollars.

350 eagles and 6 dollars.

other measures are derived, is the ten-millionth part of an arc of the meridian extending from the equator to the pole, and is called a metre -ten metres constitute a measure called the decametre-ten decametres a hectometre, &c.: and the tenth part of a metre is called a decimetre-the tenth part of a decimetre a centimetre, &c. The adaptation of this system to the decimal notation, gives it the advantage of facilitating in an extreme degree almost all the mercantile operations in arithmetic.

2. Reduce 83913 farthings to pounds.

:

4)83918 farthings.

12)209793

20)1748 32

£87 8 32d.

In this example, the farthings are divided by 4, because in the same sum there are 4 times as many farthings as there are pence for a similar reason the pence are divided by 12 to reduce them to shillings, and the shillings thus found by 20 to reduce them to pounds. Hence it appears that 83918 farthings are equivalent to 20979 pence, with two farthings or a halfpenny; to 1748 shillings and 3 pence halfpenny, or finally, to £87 8 32. The operation is proved by reducing the operation to farthings, thus: £ S. d.

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Proof, 591241 lbs.

In this example the pounds are divided by 28 (or, which is equivalent, by 4, and the quotient by 7) to reduce them to quarters-the quarters by 4, to reduce them to hundreds and the hundreds by 20 to reduce them to tons. From the operation it appears that 591241lbs. are equivalent to 21115qrs. 21lbs.; or to 5278cwt, 3qrs. 21lb.; or to 263 tons, 18cwt. 3qrs. 21lb.

4. In 37030 mills how many dollars and cents?

Ans. $37.03.

5. In 2200 pence, English money, how many pounds? Ans. £9 11 6d.

6. Reduce 1785dwts. to pounds. Ans. 7lb. 5oz. 5dwts. 7. In 12960 grains of Guinea gold how many ounces ?

Ans. 27.

8. In 14769 ounces of Swedish iron, how many hundredweight? Ans. 8cwt. 27lb. 1oz.

9. In 21444 drams of Persian silk, how

many pounds? Ans. 83lb. 12oz. 4dr.

10. The battering-ram which was employed by Titus to demolish the walls of Jerusalem weighed, according to Josephus, 100,000lb.; how many tons, &c. do they contain? Ans. 44 tons, 12cwt. 3qrs. 12lb. 11. How many tons does the large bell of Moscow weigh, its weight being 432000/6.?

Ans. 192 tons, 17cwt. Oqrs. 16lb. 12. How many tons, &c. does the enormous pedestal of reddish granite, a single stone, on which stands the equestrian statue of Peter the Great, at Petersburg; its weight being 3,200,000lb.? Ans. 1428 tons, 11cwt. 1qr. 20lb. 13. In 159022 grains, how many pounds, ounces, drams, scruples, and grains? Ans. 271b 73 23 19 2gr. 14. In 3 dozen napkins, each containing 20 nails, and 1 dozen tablecloths, each containing 38 nails, how many yards? Ans. 73yds. 2qrs.

15. Hannibal, the Carthaginian general, set out from New Carthage (now Carthagena) in Spain, about 217 years, B. C. to invade Italy ;-and the distance from New Carthage to the plains of Italy to computed at 63,360,000 inches, how many miles are in this space? Ans. 1000.

16. The ruins of the city of London, in the great fire in the year 1666, are computed to have covered 69760 square poles of ground, from the Tower, by the Thames side, to the Temple Church, and from the Northeast gate, along the

city-wall, to Holborn-bridge. How many acres are in that quantity? Ans. 436. 17. The base of the great pyramid of Egypt covers 1760 square poles, how many acres are in this quantity?

Ans. 11.

18. In 96,000,000 pints of beans, how many bushels? Ans. 1,500,000. 19. In 607024 cubic inches how many cubic yards? Ans. 30 cubic yds. 23 cubic feet, and 600 cubic inches. 20. In 126 pint-bottles of wine, how many hogsheads? Ans. 3. 21. It is said that 20,160,000 pints of Port wine are annually exported from Oporto in Portugal; how many pipes are in that quantity ? Ans. 20,000.

22. The island of Madeira, which belongs to Portugal, produces 8064000 bottles of wine annually, each containing 3 pints and a half; how many pipes might be filled with that quantity? Ans. 28000. 23. How much time, in the course of 40 years, does a person who rises at 5 o'clock in the morning gain over another who continues in bed till 7, supposing them both to go to rest at the same hour at night?"

Ans. 3 years, 121 days, and 16 hours. 24. The fortunate war which America maintained with England continued nearly 252460800 minutes, how many years did it last?

Ans. 8.

Questions.

What is reduction ?

How many kinds, and what are they called?

What is reduction descending?

What is reduction ascending?

How is the operation of reduction descending performed ? Repeat the rule.

What are the denominations of Federal money?

Repeat the table, and also the particular rule for reducing this money, from a higher to a lower denomination. What are the denominations of English money?

Repeat the rule, &c. as in Federal money.

What are the denominations of Troy weight?
Repeat the table, &c.

For what purpose is Troy weight used?

What are the denominations of Avoirdupois weight?

Repeat the table, &c.

What are the denominations of Apothecaries weight?
Repeat the table, &c.

What are the denominations of Cloth measure?
Repeat the table, &c.

What are the denominations of Long measure?
Repeat the table.

What are the denominations of Square measure? and repeat the table.

What are the denominations of Land measure? and repeat the table.

What are the denominations of Cubic measure? and repeat the table.

What are the denominations of Liquid measure? and repeat the table.

What are the denominations of Dry measure? and repeat the table.

What are the denominations of Time? and repeat the table.

Repeat the table for the division of the circle.

By whom is this table used?

Repeat the rule for performing the operation of reduction ascending.

How is reduction descending proved?

How is reduction ascending proved?

Exercises in Reduction ascending and descending.

Ex. 1. During the year 1826, there were made at the mint in the United States, 18069 half and 7802 quarter eagles, and 4004180 half dollars. The gold, silver, and copper coinage, made from the commencement of the institution to the 31st of December, 1825, was $25390966. How much is the grand total, allowing there was $17161 in cents, made in 1826 ? Ans. $27502462.

2. In the mint in London there are eight coining presses, which, with a child to supply each, strike 19000 coins in an hour. Now, if they were employed 12 hours each day, for 313 days, in coining halfpence, what would be the number and the value of all the pieces coined during that time? Ans. number, 71364000; value, £148675.

3. The quantity of linen imported into the United States from Ireland, in 1806, was two millions, six hundred and

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