mals on the right of the quotient as the number of decimals in the dividend exceeds that of the divisor; but if the number of decimals in the quotient and divisor together is not equal to the number in the dividend, supply the defect by prefixing ciphers to the quotient.

2. When the number of decimals in the divisor exceeds that of the dividend, reduce the dividend to the same denomination as the divisor by annexing ciphers, and the quotient will be a whole number. If there is a remainder after the division of the given dividend, ciphers may be annexed to it, and the division continued at pleasure; the ciphers thus annexed being regarded as decimals of the dividend.

NOTE. It is not usually necessary that decimals should be carried to more than six places.

Proof. The proof is the same as in simple division.

EXAMPLES FOR PRACTICE.

3. Divide 183.375 by 489. 4. Divide 67.8632 by 32.8. 5. Divide 67.56785 by .035. 6. Divide .567891 by 8.2. 7. Divide .1728 by 12. 8. Divide 13.50192 by 1.38. 9. Divide 783.5 by 6.25. 10. Divide 983 by 6.6. 11. Divide 172.8 by 1.2. 12. Divide 1728 by .12. 13. Divide .1728 by .12. 14. Divide 1.728 by 12. 15. Divide 17.28 by 1.2. 16. Divide 1728 by .0012. 17. Divide .001728 by 12.

18. Divide one hundred forty-seven and eight hundred twenty-eight thousandths by nine and seven tenths.

19. Divide six hundred seventy-eight thousand seven hundred sixty-seven millionths by three hundred twenty-eight thousandths.

20. Divide seventy-five and sixteen hundredths by five and forty-two thousand eight hundred one hundred thousandths.

21. Divide four and one million twenty thousand three hundred four hundred millionths by thirty one and seventy-six thousandths.

REDUCTION OF DECIMALS.

ART. 187. To reduce a vulgar fraction to a decimal.

Ex. 1. Reduce to a decimal.

Ans. .625.

Since we cannot divide the numerator 5 by 8, we reduce it to tenths by annexing a cipher, and then dividing, we obtain 6 tenths and a remainder of 2 tenths. Reducing this remainder to hundredths by annexing a cipher, and dividing, we obtain 2 hundredths and a remainder of 4 hundredths, which being reduced to thousandths by annexing a cipher, and then dividing again, gives

a quotient of 5 thousandths. The sum of the several quotients, .625,

is the answer.

625

=, Ans.

To prove that .625 is equal to §, we write it in the form of a vulgar fraction and reduce it to its lowest terms. Thus, Hence the following

RULE. Divide the numerator by the denominator, annexing one or more ciphers to the numerator, and the quotient will be the decimal required.

ART. 188. To reduce a compound number to a decimal of a higher denomination.

Ex. 1. Reduce 8s. 6d. 3qr. to the decimal of a pound.

Ans. .428125.

QUESTIONS.-Art. 187. How do you reduce a vulgar fraction to a decimal ? How can you prove the answer correct? What is the rule for reducing a vulgar fraction to a decimal?

OPERATION.

43.00

126.7500

We commence with the 3qr., and first reduce them to hundredths by annexing two ciphers; and then, to reduce these to the decimal of a penny, we divide by 4, since there will be as 208.562500 many hundredths of a penny as of a farthing, and obtain .75d. Annexing this decimal to the since there will be

.428125 6d., we divide by 12,

as many shillings as pence; and then the 8s. as many pounds as shilHence the following

and this quotient by 20, since there will be lings, and obtain .428125 £. for the answer. RULE.-1. Write the given numbers perpendicularly under each other for dividends, proceeding orderly from the least to the greatest; opposite to each dividend on the LEFT hand, place such a number for a divisor as will bring it to the next superior denomination, and draw a line between them.

2. Begin to divide at the lowest denomination, annexing ciphers if necessary, and write the quotient of each division, as decimal parts, on the RIGHT of the dividend next below it, and so on, until they are all divided; and the last quotient will be the decimal required.

NOTE. A compound number may also be reduced to a decimal by first reducing it to a vulgar fraction (Art. 170), and then this fraction to a decimal (Art. 187). Thus, 2s. 6d. 30.125£.

240

EXAMPLES FOR PRACTICE.

2. Reduce 15s. 6d. to the fraction of a pound.

3. Reduce 5cwt. 2qr. 14lb. to the decimal of a ton.

4. Reduce 3qr. 21lb. to the decimal of a cwt.

5. Reduce 6fur. 8rd. to the decimal of a mile.

6. Reduce 3R. 19p. 167ft. 72in. to the decimal of an acre.

ART. 189. To find the value of a decimal in whole numbers of a lower denomination.

Ex. 1. What is the value of .9875 of a pound. Ans. 19s. 9d.

OPERATION.

.9875

20

19.7500

12

9.00 0 0

There will be 20 times as many ten thousandths of a shilling as of a pound; therefore, we multiply the decimal .9875 by 20, and reduce the improper fraction to a mixed number by pointing off four figures on the right, which is dividing by its denominator 10000. The figures on the left of the point are shillings, and those on the right decimals of a shilling. This decimal of a

QUESTIONS. Art. 188. Will you explain the operation for reducing a compound number to a decimal of a higher denomination? Repeat the rule. By what other method can this be done? - Art. 189. Explain the operation for finding the value of a decimal in whole numbers of lower denominations.

shilling we multiply by 12, and, pointing off as before, obtain 9d., which, added to the 19s., gives 19s. 9d. for the answer.

RULE.Multiply the given decimal by the number required of the next lower denomination to make ONE of the given denomination, and point off on the RIGHT, for a REMAINDER, as many places as there are places in the given decimal. Multiply this remainder by the number that will reduce it to the next lower denomination, pointing off for a remainder as before, and thus proceed, until the reduction is carried to the denomination required. The several numbers standing at the left hand of the point will be the answer, in whole numbers, of the different lower denominations.

EXAMPLES FOR PRACTICE.

1. What is the value of .628125 of a pound?

2. What is the value of .778125 of a ton ?

3. What is the value of .75 of an ell English?

4. What is the value of .965625 of a mile ?

5. What is the value of .94375 of an acre?

6. What is the value of .815625 of a pound Troy?

7. What is the value of .5555 of a pound apothecaries' weight?

MISCELLANEOUS EXERCISES IN DECIMALS.

1. What is the value of 15cwt. 3qr. 14lb. of coffee at $9.50 per cwt.?

2. What cost 17T. 18cwt. 1qr. 7lb. of potash at $53.80 per ton?

3. What cost 37A. 3R. 16p. of land at $75.16 per acre?

4. What cost 15yd. 3qr. 2na. of cloth at $3.75 per yard?

5. What cost 15g cords of wood at $4.62 per cord?

6. What cost the construction of 17m. 6fur. 36rd. of railroad at $3765.60 per mile?

7. What cost 27hhd. 21gal. of temperance wine at $15.371 per hogshead?

8. What are the contents of a pile of wood, 18ft. 9in. long, 4ft. 6in. wide, and 7ft. 3in. high?

9. What are the contents of a board 12ft. 6in. long, and 2ft. 9in. wide?

10. Bought a cask of vinegar containing 25gal. 3qt. 1pt. at $0.37 per gallon; what was the amount?

11. Bought a farm containing 144A. 3R. 30p. at $ 97.621 per acre; what was the cost of the farm?

12. Sold Joseph Pearson 3T. 18cwt. 21lb. of salt hay, at $9.37 per ton. He having paid me $20.25, what remains

due?

13. If of a cord of wood cost $ 5.50, what cost one cord? What cost 72 cords?

14. If 4 yards of cloth cost $ 12ğ, what cost 17% yards?

§ XXI. REDUCTION OF CURRENCIES.

ART. 190. REDUCTION OF CURRENCIES is finding the value of the denominations of one currency in the denominations of another.

The nominal value of the dollar, expressed in shillings and pence, differs in the different States of the Union and in different countries, as may be seen by the following

TABLE.

In New England, Indiana, Illinois, Missouri, Virginia, Kentucky, Tennessee, Mississippi, Texas, Alabama, and Florida, the dollar is valued at 6 shillings; $1 = £. = £. fo

In New York, Ohio, and Michigan, the dollar is valued at 8 shillings; $1 = &£. = ££.

90

In New Jersey, Pennsylvania, Delaware, and Maryland, the dollar is considered 7 shillings and 6 pence; $1 = ££. Z£.

240

What is the

QUESTIONS. Art. 190. What is reduction of currencies ? value of a dollar, in the different States, expressed in shillings and pence?