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2. Let 375,13758 be multiplied by 16,7324, so that the product may have but four places of decimals.

375,13758 multiplicand.

4237,61 multiplier reversed.

37513758 product with 1
22508255 product with 6 increased with 688
2625963 product witb 7 increased with 7 X 5
112541 product with 3 increased with 3X7

7503 product with 2 increased with 2X3
1500 product with increased with.0

6276,9520 answer.

Let the same example be repeated, and let only one place in decimals be pricked off **

375,13758 multiplicand.
4237,66! multiplier reversed.
37514 product hy 1 with the increase of 1X7
22508 product with 6 increased with 6X3
2626 product with 7 increased with 7X1
113 product with 3 increased with 3X5

7 product with 2-increased with 2X7
1 the increase only of 4 X3

6276,9 answer the same as before.

DIVISION CONTRACTED. Role. Find what is the value of the first figure in the quotient; then, by knowing the first figure's denomination, you may have as many, or as few places of decimals as you please, by taking as many of the left-hand figures of the divisor as you think convenient for the first divisor, and then take as many figures of the dividend as will answer them, and in dividing omit one figure of the divisor at each following operation.

EXAMPLE. 1. Divide 721,17562 by 2,257432, and let there be three places of decimals in the quotient:

Demonstration. In this example, the unit's place of the divisor falls under the hundred's place in the dividend, and it is required that three places of decimals be in the quotient; so there must be six places in all, that is, three places of whole numbers, and three of decimals. Then, because I have the divisor in the first six figures of the dividend, I cut off the 62 as useless, and seek how often the divisor is in the dividend, and the answer is 3 times, which put in the quotient, and multiply and subtract as in common division, and the remainder is 43946. Then prick off the 3 in the divisor, and seek how often the remaining figures may be had in 43946, the remainder, which can be but once; put 1 in the quotient, and multiply and subtract, and the next remainder is 21372. Then prick off the 4 in the divisor, and seek how often the remaining figures may be had in 21372, which will be 9 times, which put in the quotient, and multiply thas, saying, 9 times 4 is 36, for which I carry 4, (in respect of the 4 last prieked off,) and 9 times 7 is 63, and 4 is 67; set down 7, and carry 6, and so proceed till the division be finished, always respecting the increase made from the figure pricked off. See the work.

2,257432)721,175/62(319,467 Ans!

677229

43996
22574

21372
20317

1055
903

152 135

17
15

2

THE COUNOX WAT.

2,25743)721,17562(319,467 Ans.

677229

439466
22574 13

21372 32
20316 87

1055 450
902972

152 4780
1354458

1703220
15 | 80201

Rem. 1123019 NOTE. I have set down the work of the last example at large, according to the common way, that thereby the learner may see the reason of the rule, on the right side of the perpendicular black line, being wholly omitted.

REDUCTION OF DECIMALS. 'To reduce a vulgar fraction to its equivalent decimal,

RULE. Annex ciphers to the numerator, and divide by the denominator, and the quotient will be the decimal required. As many ciphers as you annex to the given numerator, so many places must be pointed in the quotient; and if there be not so many places of figures in the quotient, make up the deficiency by placing ciphers to the left hand of the said quotient.

EXAMPLES.

1. Reduce to a decimal.

8)1,000

,125 Ans.

2. What decimal is equal to ?

Ans. ,25 3. What decimal is equal to 1 ?

,3333-4 4. What decimal is equal to j?

.6666 --5. What decimal is equal to i?

,5 6. What decimal is equal to 7. What decimal is equal to 1?

,875 8. What decimal is equal to To reduce quantities of several denominations to a decimal.

Rule 1. Bring the given denomination first to a vulgar fraction, and reduce said vulgar fraction to its equivalent decimal.

RULE 2. Place the several denominations above each other, letting the highest denomination stand at the bottom, and then divide each denomination, (beginning at the top,) by its value in the next denomination, and the last quotient; will give the decimal required.

EXAMPLES.

1. Reduce 20 cents to the decimal of a dollar,
20

Then 5)1,0
-
100

Ans.

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,36

2. Reduce 30 cts. to the decimal of a dollar. Ans. ,3 3. Reduce 36 cts. to the decimal of a dollar. 4. Reduce 2 dollars to the decimal of an eagle, 5. Reduce 2 cents to the decimal of a dime. 6. Reduce 6 dimes to the decimal of an eagte.

,2 ,

7. Reduce 5 mills to the decimal of a dime.

Ans. ,5 8. Reduce 38 cents to the decimal of an eagle.

Ans. ,038 9. Reduce 4 dollars 25 cents to the decimal of an eagle.

Ans. ,425 10. Reduce 3 qrs. 2 na. to the decimal of a yard.

Ans. ,875 11. Reduce a gallon to the decimal of a hogshead.

Ans. ,015873+ 12. Reduce 21 feet to the decimal of a yard.

Ans. ,83333+: 13. Reduce 3 qrs. 21 lbs. to the decimal of an cwt.

Ans. ,9375 14. Reduce 4 cwt. 2 qrs. to the decimal of a ton.

Ans. ,225 15. Reduce 14 drams to the decimal of a pound Avoirdupois.

Ans. ,0546875 16. Reduce 14 cwt. to the decimal of a ton.

Ans. ,7 17. Reduce 7 minutes to the decimal of a day.

Ans. ,0048611+ 18. Reduce 2 days to the decimal of a week.

Ans. ,2857142+ 19. Reduce a pint to the decimal of a gallon.

Ans. ,125 20. Reduce 76 yards to the decimal of a mile.

Ans. ,04318181+ 21. Reduce 4 perches to the decimal of an acre.

Ans. ,025 22. Reduce 4 inches to the decimal of a yard.

Ans. ,1111111111+ 23. Reduce 72 days to the decimal of a year.

Ans. ,1972602+

To find the value of a decimal, in the known parts of an

integer. RULE. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for a remainder, to the right hand, as there are places in the given decimal. Multiply the remainder by the next inferior denomination, and cut off a remainder as before, and so on through all the parts of the integer, and the several denominations standing on the left hand make the answer.

EXAMPLES.
R? What is the value of ,875 of a yard ?

Ans. 3 qrs. 2 na.

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2. What is the value of ,617 of an cwt. ?

Ans. 2qrs. 13 lbs. 1 oz. 10 drs. 3. What is the value of ,875 of a hogshead of wine?

Ans. 55 gallons 1 pint. 4. What is the value of ,712 of a furlong

Ans. 23 poles 2 yds. 1 ft. 11,04 inc. 5. What is the value of ;17 of a tun of wine ?

Ans. 42 gallons 3,36 quarts. 6. What is the value of ,7 lb, of silver ?

Ans. 8 oz. 8 dwt. 7. What is the proper quantity of ,761 of a day?

Ans. 18 hrs. 15 min. 53,4 sec. 8. What is the proper quantity of ,71 of 4 oz. of gold?

Ans. 2 oz. 16 dwt. 19,2 grs. 9. What is the proper quantity of ,4712 of an ell English?

Ans. 2 qrs. 1,424 na. 10. What is the value of ,67 of a league?

Ans. 2 miles 3 poles 1 yd. 3+ inc. 11. What is the value of ,12785 of a year?

Ang. 46 days 15 hrs. 57 min. 57+ sec. 12. What is the proper quantity of ,3 of a year?

Ans. 109 days 12 hours. 13. What is the proper quantity of ,5 of an hour?

Ans. 30 minutes. Practical Questions. 1. What is the sum of ,17 ton ,19 cwt. ,17 qr. ,7 lb.?

Ans. 3 cwt. 2 qrs. 15,54 lbs. 2. What is the sum of ,17 lb. Troy, and ,84 oz.

Ans. 2 oz. 17 dwts. 14,4 grs. 3. What is the difference between 941 of a day, and ,16 of an hour?

Ans. 9 hrs. 40 min, 48 seca

EXTRACTION OF TAE SQUARE ROOT. To find the root thereof, that is, to find out such a pumber, as being multiplied into itself, the product shall be equal

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