dividend is not divisible by a divisor 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.

14. Divide .13 by 8.

15. Divide 7.2 by .16. 16. Divide 8.7 by .25.

17. Divide 3.6 by 7.5.

NOTE 4. When a decimal is not complete, we sometimes place the sign after it, signifying that there is a remainder.

In Ex. 25, divide by the factors of 300, viz. 100 and 3; i. e. move the point two places to the left and then divide by 3.

26. Divide 3.6412 by 400. 27. Divide 56.427 by 8000. 28. Divide 36.49 by 600.

Ans. .009103.

29. Divide three thousand eight hundred and fifty-three hundred-thousandths by thirty-two millionths. Ans. 1204.0625. 30. Divide eighty-four and eighty-four hundredths by fortyeight thousandths.

PROBLEM 5.

173. To reduce a common fraction to a decimal. Ex. 1. Reduce to a decimal fraction.

3 X 100= 300 75; and 75 100.75, Ans.

If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand

172. For what is the sign + sometimes used?

(Art. 84, c). Now, in the above example, is multiplied by 100 by annexing two ciphers to the numerator; the fraction 300 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,

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.

2. Reduce to a decimal fraction.

X 1000

3. Reduce

3000 =375; and 375 ÷ 1000 =.375, Ans.
to a decimal.

4. Reduce 2 to a decimal.

Ans. .4375. Ans. 1.140625.

9. Reduce,, §, 18, 74, 75, and 17 to decimals.

174. Every decimal fraction is a common 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 common fraction. This proves the rule in Art. 173.

10. Reduce .48 to the form of a common fraction and then to its lowest terms. .48, Ans.

11. Reduce .125 to its lowest terms.

.125=

125=25=4%= }}, Ans. 12. Reduce .17 to the form of a common fraction.

13. Reduce .275, .325, .00025, and .00625.
14. Reduce 2.8.

Ans.

2.828, Ans.

15. Reduce 1.5, 3.75, 8.25, 9.125, and 2.0125.

173. Rule for reducing a common fraction to a decimal? Explanation? 174. Is a decimal also a common fraction? How is this made evident? How may the rule in Art. 173 be proved correct?

PROBLEM 6.

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

Ex. 1. Reduce 2pk. 3qt. to the decimal of a bushel. 1st. 3qt.pk..375pk.; .. 2pk. and 3qt. 2.375pk. 2d. 2.375pk.2-375bush..59375bush., Ans.

=

The principle is the same as in Art. 173. Hence,

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.

2. Reduce 9s. 6d. 3qr. to the decimal of a pound.

NOTE. In dividing by 20 to reduce the decimal of a pound, and in all similar examples, we may point off the O in the divisor, and then divide by 2, but in such a case the point in the dividend must be moved one place toward the left, for by so doing both divisor and dividend are divided by 10, and ... the quotient is unchanged (Art. 84, b).

3. Reduce 2ft. 9in. 1 b. c. to the decimal of a yard.

4. Reduce 3cwt. 2qr. 20lb. 8oz. to the decimal of a ton.

175. Rule for reducing the lower denominations of a compound number to the decimal of a higher denomination? Principle? Mode of dividing when the divisor is 20, 40, etc? When the divisor is a mixed number?

5. Reduce 3oz. 12dwt. 18gr. to the decimal of a pound, Troy Ans. .303125. Weight.

6. Reduce 63 43 19 10gr. to the decimal of a pound.

7. Reduce 5yd. 2ft. 6in. to the decimal of a rod, Long Measure.

Since one of the divisors, in this example, is 5, both divisor and dividend are reduced to halves. The feet and inches are more than a half yard; .. the sum of the given numbers is more than a rod.

8. Reduce 3s. 15° 30′′ to the decimal of a circumference.

Ans. .291689+.

9. Reduce 2d. 6h. 18m. 24sec. to the decimal of a week. 10. Reduce 2qt. 1pt. 1gi. to the decimal of a gallon. 11. Reduce 3fur. 8ch. 2rd. 10 li. to the decimal of a mile. 12. Reduce 8cu. ft. 144c. in. to the decimal of a cubic yard. 13. Reduce 3r. 2rd. 20yd. to the decimal of an acre. 14. Reduce 5fur. 30rd. 5yd. 1ft. 9in. 2 b. c. to the decimal of a mile..

PROBLEM 7.

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

Ex. 1. Reduce .428125£ to shillings, pence, and farthings.

OPERATION.

£.4 28125

20

8.5 6 2 500 s.
12

6.7 5 0 0 d.
4

3.0 0 qr.

Ans. 8s. 6d. 3qr.

This article is the reverse of Art. 175; .. 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.

RULE. Multiply the given decimal by the number it takes of the next lower denomination to make one of this higher, and place the decimal point 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.

2. Reduce .984375 of a bushel to pecks, quarts, and pints. Ans. 3pk. 7qt. 1pt.

3. Reduce .40625 of a gallon to quarts, pints, and gills.
4. Reduce .902288 of a lunar month to weeks, days, hours,
Ans. 3w. 4d. 6h. 20m. 15.1296sec.

minutes, and seconds.

5. Reduce .90625 of a yard to quarters, nails, etc.

6. What is the value of .375°?

7. What is the value of .375 of a ton?

Ans. 22′ 30′′.

8. What is the value of .4658 of a pound, Troy Weight? 9. Reduce .3587 of a mile to furlongs, rods, yards, etc. 10. Reduce .5621b to 3, 3, etc.

MISCELLANEOUS EXAMPLES IN DECIMAL FRACTIONS.

1. What is the cost of 6.25 lb. of beef, at 12 cents per pound? Ans. 75c.

2. Bought 4.5 tons of hay, at $12.50 per ton; what was the cost of the whole? Ans. $56.25. 3. What is the value of 8 acres of land, at $62.50 per acre? 4. Paid $500 for 8 acres of land; what was the price per acre?

5. Paid $500 for a piece of land at $62.50 per acre; how many acres were bought?

6. Bought land at $62.50 per acre, and sold it again at $75 per acre, thereby making $100; how many acres were bought?

7. Bought 8 acres of land at $62.50 per acre, and sold the lot for $600; was there a gain or a loss? How much total? How much per acre?

176. Rule for reducing a decimal of a higher denomination to whole numbers of lower denominations? Explanation?