REDUCTION OF DECIMAL FRACTIONS.

¶ 75. 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 the decimal of a

To reduce a compound number 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.

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 Ans. 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 dencmination, 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.

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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. 232′6.

7. Reduce 38 gals. 3'52 qts. of beer, to the decimal of a 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.

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

8. What is the value of "72 hhd. of beer?

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?

15. Reduce 3 cwt. 0 qr. 716. What is the value of lbs. 8 oz. to the decimal of a '15334821 of a ton?

ton.

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

7. 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.

T76. 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:

of 20 s. is 2 s.; therefore every 2 s. is, or '1 £. Every shilling is 8, 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 Tobo, or '001 £. But 960 increased by part of itself is 1000; consequently, 24 farthings are exactly 8, 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 11; therefore add 2 to them: then call them so many thousandths, and the result will be correct within less than of robo of a pound. Thus, 17 s. 5 d. is reduced to the

decimal of a pound as follows: 16 s. '8 £. and 1 s. = '05 £. Then, 53 d. 23 farthings, which, increased by 1, (the number being more than 12, but not exceeding 36,) is '024 £., and the whole is '874 £. the Ans.

Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection,-Call every two shillings one tenth of a pound; every odd shilling, five hundredths; and the number of farthings, in the given pence and farthings, so many thousandths, adding one, if the number be more than twelve and not exceeding thirty-six, and two, if the number be more than thirty-six.

T77. Reasoning as above, the result, or the three first figures in any decimal of a pound, may readily be reduced back to shillings, pence and farthings, by inspection. Double the first figure, or tenths, for shillings, and, if the second figure, or hundredths, be five, or more than five, reckon another shilling; then, after the five is deducted, call the figures in the second and third place so many farthings, abating one when they are above twelve, and two when above thirty-six, and the result will be the answer, sufficiently exact for all practical purposes. Thus, to find the value of '876 £. by inspection :

'8

tenths of a pound

'05 hundredths of a pound

'026 thousandths, abating 1, 25 farthings

EXAMPLES FOR PRACTICE.

Ans.

1. Find, by inspection, the decimal expressions of 9 s. 7 d.,

and 12 s. 02 d.

Ans. 479., and ‘603£. 2. Find, by inspection, the value of '523 £., and '694£. Ans. 10 s. 5 d., and 13 s. 104 d. 3. Reduce to decimais, by inspection, the following sums, and find their amount, viz.: 15 s. 3 d.; 8 s. 11 d.; 10 s. 64 d.; 1 s. 81 d. ; † d., and 24 d. Amount, £1'833.

4. Find the value of '47 £.

Note. When the decimal has but two figures, after taking out the shillings, the remainder, to be reduced to thousandths, will require a cipher to be annexed to the right hand, or supposed to be so. Ans. 9 s. 4 d.

5. Value the following decimals, by inspection, and find their amount, viz.: 785 £.; '357 £.; '916 £.; "74 £. ; '5 L.; '25 £.; '09 £.; and '008 £. Ans. 3£. 12 s. 11 d.

SUPPLEMENT TO DECIMAL FRACTIONS.

QUESTIONS.

1. What are decimal fractions? 2. Whence is the term derived? 3. How do decimal differ from common fractions? 4. How are decimal fractions written? 5. How can the proper denominator to a decimal fraction be known, if it be not expressed? 6. How is the value of every figure determined? 7. What does the first figure on the right hand of the decimal point signify? the second figure?

third figure?fourth figure? 8. How do ciphers, placed at the right hand of decimals, affect their value? 9. Placed at the left hand, how do they affect their value? 10. How are decimals read? 11. How are decimal frac tions, having different denominators, reduced to a common denominator? 12. What is a mixed number? 13. How may any whole number be reduced to decimal parts? 14. How can any mixed number be read together, and the whole expressed in the form of a common fraction? 15. What is observed respecting the denominations in federal money? 16. What is the rule for addition and subtraction of decimals, particularly as respects placing the decimal point in the results? multiplication? division? 17. How is a common or vulgar fraction reduced to a decimal? 18. What is the rule for reducing a compound number to a decimal of the highest denomination contained in it? 19. What is the rule for finding the value of any given decimal of a higher denomination in terms of a lower? 20. What is the rule for reducing shillings, pence and farthings to the decimal of a pound, by inspection? 21. What the reasoning in relation to this rule? 22. How may the three first figures of any decimal of a pound be reduced to shillings, pence and farthings, by inspection?

EXERCISES.

1. A merchant had several remnants of cloth, measuring as follows, viz. :

77

6 g

14/

93

81

30

yds.

.....

.....

.....

How many yards in the whole, and what would the whole come to at $3'67 per yard?

Note. Reduce the common fractions to decimals. Do the same wherever they occur in the examples which follow.

Ans. 36'475 yards. $133'863 +, cost.

2. From a piece of cloth, containing 36§ yds., a merchant 'sold, at one time, 7 yds., and, at another time, 12 yds.; how much of the cloth had he left? Ans. 167 yds.

how

3. A farmer bought 7 yards of broadcloth for 8 £., a barrel of flour for 2 £., a cask of lime for 18 £., and 7 lbs. of rice for £.; he paid 1 ton of hay at 37 £., 1 cow at 6 £., and the balance in pork at £. per lb. ; many were the pounds of pork? Note. In reducing the common fractions in this example, it will be sufficiently exact if the decimal be extended to three places. Ans. 1084 lb.

4. At 12 cents per lb., what will 371⁄2 lbs. of butter cost? Ans. $4'7183.

5. At $17'37 per ton for hay, what will 11 tons cost? Ans. $201 92§.

6. The above example reversed. At $201'92 for 111⁄2 tons of hay, what is that per ton? Ans. $1737. 17. If '45 of a ton of hay cost $9, what is that per ton? Consult ¶ 65.

Ans. $20. 8. At 4 of a dollar a gallon, what will '25 of a gallon of molasses cost? Ans. $1. 9. At $9 per cwt., what will 7 cwt. 3 qrs. 16 lbs. of sugar

cost?

Note. Reduce the 3 qrs. 16 lbs. to the decimal of a cwt, extending the decimal in this, and the examples which follow, to four places. Ans. 71'035+.

10. At $69'875 for 5 cwt. 1 qr. 14 lbs. of raisins, what is per cwt.?

that

Ans. $13.

11. What will 2300 lbs. of hay come to at 7 mills per lb. ? Ans. $15 10.

12. What will 765 lbs. of coffee come to, at 18 cents per Ans. $13779

lb.?

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