unless it contains all the factors of the divisor; whereas annexing ciphers to the dividend introduces no prime factor into it, except 2's and 5's.

13. Divide .8746 by .32.

Ans. 2.733125.

14. Divide .45 by .8.

15. Divide .87693 by .64.

NOTE. When a decimal is not complete, we sometimes place the sign+

after it, signifying that there is a remainder.

16. Divide .8742 by .56.

17. Divide .34 by .27. 18. Divide 56.7 by 2.9. 19. Divide 87.69 by 47. 20. Divide 87.69 by .47.

Ans. 1.5610714+.

(d) A decimal fraction may be divided by 10, 100, etc., by moving the separatrix as many places towards the left as there are ciphers in the divisor; for, by moving the point one place to the left, each figure in the dividend is made only before, and consequently the result is only dividend (147); thus, 874.5 ÷ 10 = 87.45; 86.954; 46.87100000.0004687.

21. Divide 7846.987 by 1000. 22. Divide 54.276 by 100000.

23. Divide 46.08 by 1000.

as great as as great as the 8695.4 ÷ 100 =

Ans. 7.846987.

24. Divide .7842 by 1000.

25. Divide 769.428 by 200.

Ans. 3.84714.

In Ex. 25, divide by the factors of 200, viz., 100 and 2; i. e.,

move the point two places to the left and then divide by 2.

26. Divide 48.9632 by 4000.
27. Divide 769.842 by 3200.
28. Divide 3505.6 by 400.
29. Divide 874.69 by 64000.
30. Divide 46.8742 by 16000000.

Ans. .0122408.

CASE 5.

158. X 100 = 300 =75; and 75 ÷ 100 = .75.

If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand (60). Now, in the above example, is multiplied by 100 by annexing two ciphers to the numerator; the fraction 309 is then reduced to the whole number 75, and, finally, 75 is divided by 100 by placing the decimal point before the 75; ...75. Hence,

To reduce a vulgar fraction to a decimal,

RULE.-Annex one or more ciphers to the numerator and divide the result by the denominator, continuing the operation until there is no remainder, or as far as is desirable. Point off as many decimal places in the quotient as there are ciphers annexed to the numerator.

Ex. 1. Reduce to a decimal fraction.

=

5000 =
625; and 625 ÷ 1000 .625, Ans.
to a decimal.
Ans. .1875.

X 1000

2. Reduce

3. Reduce

to a decimal.

4. Reduce to a decimal.

Ans. .20833+.

There being the factor 3 in the divisor, in Ex. 4, and no such element in the dividend, there must necessarily be a remainder, however far the division may be continued (157, c). 5. Reduce to a decimal.

5

6. Reduce,, 74, 75, 15, 1 and 4 to decimals.

159. Every decimal fraction is a vulgar fraction, and, if its denominator be written, it will appear as such. It may then be reduced to lower terms, or modified like any other vulgar fraction.

Ex. 1. Reduce .24 to the form of a vulgar fraction and then to its lowest terms. .2413, Ans.

This process proves the rule in Art. 158.

252

4. Reduce .4765, .87698, .000476 and .0075.

5. Reduce 3.5.

3.518, Ans.

CASE 6.

160. Reduce 6d. and 3qr. to the decimal of a shilling.

To reduce whole numbers of lower denominations to the decimal of a higher denomination,

RULE. Having annexed one or more ciphers to the lowest denomination, divide by the number it takes of that denomination to make one of the next higher, and annex the quotient as a decimal to that next higher; then divide the result by the number it takes of THIS denomination to make one of the NEXT higher, and so continue till it is brought to the denomination required.

Ex. 1. Reduce 8s. 5d. 1qr. to the decimal of a pound.

2. Reduce 1ft. 3in. 2b. c. to the decimal of a yard.

3 2.0 0 0 0 0 0 0 b. c.

12 3.6 6 6 6 6 6 6 in.

3 1.3055555+ ft.

.4351851yd., Ans.

In this example there will be a remainder, however far the operation is carried.

3. Reduce 6oz. 18dwt. 15gr. to the decimal of a pound Troy weight. Ans. .5776041661b.+.

4. Reduce 83 43 29 5gr. to the decimal of a pound.

5. Reduce 12cwt. 3qr. 211b. 8oz. 4dr. to the decimal of a ton. 6. Reduce 5yd. 2ft. 6in. to the decimal of a rod, long meas

7. Reduce 543a. 3r. 36rd. 25yd. 8 ft. 36in. to the decimal of a square mile.

8. Reduce 56ft. 1725in. to the decimal of a cord.

9. Reduce 3qr. 2na. lin. to the decimal of a yard. 10. Reduce 3qt. 1pt. 3gi. to the decimal of a gallon. 11. Reduce 3pk. 7qt. 1pt. to the decimal of a bushel.

12. Reduce 3wk. 6d. 40m. 30sec. to the decimal of a lunar month.

13. Reduce 8s. 24° 50′′ to the decimal of a circumference. 14. Reduce 19cwt. Soz. to the decimal of a ton.

15. Reduce 3fur. 25rd. 2yd. 6in. 2b. c. to the decimal of a mile.

CASE 7.

161. To reduce a decimal of a higher denomination to whole numbers of lower denominations,

RULE.-Multiply the given decimal by the number it takes of the next lower denomination to make one of this higher, and place the separatrix as in multiplication of decimals; multiply the DECIMAL PART of this product by the number it takes of the NEXT lower denomination to make one of THIS, and so proceed as far as necessary. The several numbers at the left of the points

will be the answer.

Ex. 1. Reduce .421875£ to shillings, pence and farthings. Ans. 8s. 5d. 1qr.

£.421875

20

8.4 37 500 s.
12

5.2 5 0 0 d.
4

1.0 0 qr.

This article is the reverse of Art. 160;.. first multiply by 20, because there will be 20 times as many shillings as pounds. For a like reason, multiply the fractional part of a shilling by 12, to reduce it to pence, etc. After having fixed the decimal point in the several products, the ciphers at the RIGHT of the significant figures are disregarded.

2. Reduce .9375 of a gallon to quarts, pints and gills.

Ans. 3qt. 1pt. 2gi.

3. Reduce .84 of a lunar month to weeks, etc.

Ans. 3w. 2d. 12h. 28m. 48sec.

4. Reduce .7694 of an acre to roods, etc.

Ans. 3r. 3rd. 3yd. 1ft. 45.216in.

5. Reduce .6543 of a mile to furlongs, etc.

Ans. 5fur. 9rd. 2yd. Oft. 2in. 1+b.c.

6. Reduce .54324 of a pound Troy to ounces, etc.

7. Reduce .57691b. to 3, 3, etc.

8. Reduce .0876 of a ton to cwt., qr., etc.

9. Reduce .9876 of a mile to fur., ch., etc.

10. Reduce .4698 of a cord to c. ft., cu. ft., etc.
11. Reduce .8694 of a yard to qr., na., etc.
12. Reduce .7564 of a bushel to pk., etc.

162. MISCELLANEOUS EXAMPLES IN DECIMAL FRACTIONS. Ex. 1. Bought 14.75yds. sheeting at 14 cents per yd.; what was the cost of the piece? Ans. $2.065.

NOTE. Decimal fractions are peculiarly adapted to operations in Federa Money, the denominations of which conform to the decimal notation. The dollar is the unit, and dimes, cents and mills are tenths, hundredths and thousandths.

2. Bought 20.5 tons of hay at $12.375 per ton; what was the cost of the whole? Ans. $253.687.