10. Divide '012 by '005.

Quot. 2'4.

11. Divide three thousandths by four hundredths.

Quot. '075.

12. Divide eighty-six tenths by ninety-four thousandths. 13. How many times is '17 contained in 8?

REDUCTION OF COMMON OR VULGAR FRACTIONS TO DECIMALS.

¶ 74. 1. A man has of a barrel of flour; what is that expressed in decimal parts?

We

As many times as the denominator of a fraction is contained in the numerator, so many whole ones are contained in the fraction. We can obtain no whole ones in , because the denominator is not contained in the numerator. may, however, reduce the numerator to tenths, (¶ 72, ex. 2,) by annexing a cipher to it, (which, in effect, is multiplying it by 10,) making 40 tenths, or 4'0. Then, as many times as the denominator, 5, is contained in 40, so many tenths are contained in the fraction. 5 into 40 goes 8 times, and no remainder. Ans. '8 of a bushel.

2. Express of a dollar in decimal parts.

The numerator, 3, reduced to tenths, is 18, 30, which, divided by the denominator, 4, the quotient is 7 tenths, and a remainder of 2. This remainder must now be reduced to hundredths by annexing another cipher, making 20 hundredths. Then, as many times as the denominator, 4, is contained in 20, so many hundredths also may be obtained. 4 into 20 goes 5 times, and no remainder. of a dollar, therefore, reduced to decimals, is 7 tenths and 5 hundredths, that is, "75 of a dollar.

The operation may be presented in form as follows:

Num.

Denom. 4) 3′0 (75 of a dollar, the Answer.

3. Reduce to a decimal fraction.

The numerator must be reduced to hundredths, by annexing two ciphers, before the division can begin.

66) 4'00 ('0606+, the Answer.

396

400

396

4

As there can be no tenths, a cipher must be placed in the quotient, in tenth's place.

Note. cannot be reduced exactly; for, however long the division be continued, there will still be a remainder.* It is sufficiently exact for most purposes, if the decimal be extended to three or four places.

From the foregoing examples we may deduce the following general RULE:-To reduce a common to a decimal frac

* Decimal figures, which continually repeat, like '06, in this example, are called Repetends, or Circulating Decimals. If only one figure repeats, as '3333 or 7777, &c., it is called a single repetend. If two or more figures circulate alternately, as '060606, ‘234234234, &c., it is called a compound repetend. If other figures arise before those which circulate, as '743333, 143010101, &c., the decimal is called a mixed repetend.

A single repetend is denoted by writing only the circulating figure with a point over it: thus, '3, signifies that the 3 is to be continually repeated, forming an infinite or never-ending series of 3's.

A compound repetend is denoted by a point over the first and last repeating figure thus, "234 signifies that 234 is to be continually repeated.

It may not be amiss, here to show how the value of any repetend may be found, or, in other words, how it may be reduced to its equivalent vulgar fraction.

If we attempt to reduce to a decimal, we obtain a continual repetition of the figure 1: thus, '11111, that is, the repetend 'İ. The value of the repetend 'i, then, is; the value of '222, &c., the repetend 2, will evidently be twice as much, that is, 3. In the same manner, 3: = 3, and 4, and '5 = §, and so on to 9, which 1. What is the value of '8?

= 1.

Ans. &.

2. What is the value of '6? Ans. §. What is the value of "3?

of "? - of '4 ?

=

of '5? - of '9?

If be reduced to a decimal, it produces '010101, or The repetend '02, being 2 times as much, must be and '48, being 48 times as much, must be 18, and 74

· of 'i? the repetend 01.

=

and '03 = 9, - JF, &c.

tion,-Annex one or more ciphers, as may be necessary, to the numerator, and divide it by the denominator If then there ho a remainder, annex another cipher, ana turae wo vojure, and „ continue to do so long as there shall continue to be a remainder, or until the fraction shall be reduced to any necessary degree of exactness. The quotient will be the decimal required, which must consist of as many decimal places as there are ciphers annexed to the numerator; and, if there are not so many figures in the quotient, the deficiency must be supplied by prefixing ciphers.

EXAMPLES FOR PRACTICE. 4. Reduce, 1, 4, and 2 to decimals.

Ans. '5; 25; '025; '00797+.

5. Reduce, Too, 1755, and ob to decimals. Ans. 692; '003; '0028+; '000183 +.

6. Reduce 47, 67, 680 to decimals.

7. Reduce,‰, J8J, 3, 3, 11, 1, 96 to decimals. 8. Reduce §, §, §, 1, £, £, t, zb, z,

1

to decimals.

If 99 be reduced to a decimal, it produces '001; consequently, '002=989, and '037 37 , and 425 435, &c. As this principle will apply to any number of places, we have this general RULE for reducing a circulating decimal to a vulgar fraction,-Make the given repetend the numerator, and the denominator will be as many 9s as there are repeating figures.

3. What is the vulgar fraction equivalent to "704? 4. What is the value of '003?

014?

Ans. 18

324?
Ans. to last,

01021 ?

33333

5. What is the value of '43? In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is to, and the repetend, 3, is of To, that is, ; these two parts must be added together. + 3% = 38 18, Ans. Hence, to find the value of a mixed repetend,—Find the value of the two parts, separately, and add them together.

=

6. What is the value of 153?

7. What is the value of '0047?

8. What is the value of '138?

70%+880 = 188 = 1%, Ans.

It is plain, that circulates may be added, subtracted, multiplied, and divided, by first reducing them to their equivalent vulgar fractions

REDUCTION OF DECIMAL FRACTIONS.

T75. Fractions, we have seen, (T 63,) like integers, are reduced from low to higher denominations by division, and from high to lower denominations by multiplication.

To reduce a compound num- To reduce the decimal of a ber to a decimal of the highest higher denomination to integers denomination. of lower denominations.

1. Reduce 7 s. 6 d. to the decimal of a pound.

Ans.

2. Reduce 375 £. to integers of lower denominations. 375 £. reduced to shillings,

6 d. reduced to the decimal of a shiling, that is, divided that is, multiplied by 20, is by 12, is '5 s., which annexed 750 s.; then the fractional to the 7 s. making 7'5 s., and part, '50 s., reduced to pence, divided by 20, is 375 £. the that is, multiplied by 12, is 6 d. Ans. 7 s. 6 d. The process may be pre- That is,-Multiply the given sented in form of a rule, thus:- decimal by that number which Divide the lowest denomina- it takes of the next lower detion given, annexing to it one nomination to make one of this or more ciphers, as may be higher, and from the right necessary, by that number hand of the product point off which it takes of the same to as many figures for decimals make one of the next higher as there are figures in the denomination, and annex the given decimal, and so conquotient, as a decimal to that tinue to do through all the dehigher denomination; so con- nominations; the several numtinue to do, until the whole bers at the left hand of the shall be reduced to the deci- decimal points will be the mal required.

EXAMPLES FOR PRACTICE.

3. Reduce 1 oz. 10 pwt. to the fraction of a pound.

OPERATION.

20)10'0 pwt.

12)1'5 oz.

$125 lb. Ans.

N

value of the fraction in the proper denominations. EXAMPLES FOR PRACTICE. 4. Reduce '125 lbs. Troy to integers of lower denominations.

OPERATION.

lb. '125

12

oz. 1'500

20

pwt 10'000. Ans. 1oz.10pwt.

5. Reduce 4 cwt. 23 qrs. to the decimal of a ton.

Note. 22'6.

7. Reduce 38 gals. 3'52 qts.

6. What is the value of 2325 of a ton?

8. What is the value of "72

of beer, to the decimal of a hhd. of beer? hhd.

9. Reduce 1 qr. 2 n. to the decimal of a yard.

11. Reduce 17 h. 6 m. 43 sec. to the decimal of a day.

13. Reduce 21 s. 10 d. to the decimal of a guinea.

15. Reduce 3 cwt. 0 qr. 7 lbs. 8 oz. to the decimal of a ton.

10. What is the value of 375 of a yard?

12. What is the value of 713 of a day?

14. What is the value of 78125 of a guinea?

16. What is the value of '15334821 of a ton?

Let the pupil be required to reverse and prove the following examples:

17. Reduce 4 rods to the decimal of an acre.

18. What is the value of "7 of a lb. of silver?

19. Reduce 18 hours, 15 m. 50'4 sec. to the decimal of a day.

20. What is the value of '67 of a league?

21. Reduce 10 s. 94 d. to the fraction of a pound.

¶ 76. There is a method of reducing shillings, pence and farthings to the decimal of a pound, by inspection, more simple and concise than the foregoing. The reasoning in relation to it is as follows

T of 20 s. is 2 s.; therefore every 2 s. is, or '1 £. Every shilling is 180, or '05 £. Pence are readily reduced to farthings. Every farthing is £. Had it so happened, that 1000 farthings, instead of 960, had made a pound, then every farthing would have been, or '001 £. But 960 increased by part of itself is 1000; consequently, 24 farthings are exactly 125, or '025 £., and 48 farthings are exactly 88, or '050 £. Wherefore, if the number of farthings, in the given pence and farthings, be more than 12,4 part will be more than ; therefore add 1 to them if they be more than 36, 4 part will be more than 14; therefore add 2 to them: then call them so many thousandths, and the result will be correct within less than of Tobo of a pound. Thus, 17 s. 5 d. is reduced to the