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EXAMPLES. 1. If yard of cloth that is 2 yards wide, will make a gar. ment, how much of any other cloth that is } yard wide will make the same garment ?
bread. len. bread. 2x3x5
1X4X3 2. If I lend my friend $480 for f of a year, how much ought he to lend me for 1 of a year?
Ans. $864. 3. If í yard of cloth that is 2 yards wide will make a coat, what is the breadth of the cloth whereof it will take 14 yards to make the same co it?
Ans. yd.=3qrs. 2 nails. 4. If 12 men do a piece of work in 102 days, in how
many days will 6 men 'do the same ?
214lays. 5. What length must be cut off a board, that is 7 inches broad, to contain a square foot ?
Ans. 1818 inches. 6. How much in length, of a piece of land that is 1412 poles broad, will make an acre ?
Ans. 1361 poles.
expressed .25, and 125
A Decimal Fraction is such, whose denominator is not expressed, but understood ; to be a unit with as many ciphers annexed as there are places in the numerator : hence 5 To will be expressed .5, 10 100
1000 pressed .125, &c. They have a point or comma prefixed, to distinguish them from integers.
A cipher placed to the left hand of an integer, or to the right of a decimal, neither increaseth nor decreaseth the value : but placed to the right hand of an integer, increaseth the value ten fold; and to the left hand of a decimal, decreaseth the value ten fold :-Two ciphers placed to the right hand of an integer, increaseth the value one hundred fold and to the left hand of a decimal, decreaseth the value one hundred fold, &c. as in the following table of numeration,
o Hundreds of Millions.
Hundreds of Thousands.
is Parts of Ten.
Parts of One Hund. Thou.
Integers, or Whole Numbers. Decimal Parts or Fractions.
In the preceding table it also plainly appears, that as integers or whole numbers increase by a ten fold proportion towards the left hand, so decimal parts decrease towards the right hand by the same proportion. So, 5 Five. And, .5
One Hundred. 500 Five Hund. .005
One Thousand. 5000 Five Thous. .0005
Ten Thousand. A finite decimal is that which ends at a certain number of places; but an infinite, is that which no where ends.
A repeating or circulating decimal, is that wherein one or more figures are continually repeated.
Thus, 3333, &c. or 3, is called a pure repeater. Also, 2666. &c. or .26, is called a mixed repeater.
And.090909, &c. or.181818, &c. are called pure circulates, and consists of the figures of the cite only; .04545, &c. or .32142857142857, &c. are called mixed circulates.
Repeating and circulating numbers are marked by a point over the repeating figures, as in .1 5 or 2.
Note. In all operations, if the result consists of several nines, reject them, and make the next superior place one more. Thus for 17.1999 write 17.2 ; and for 13.999 write 14, &c.
REDUCTION OF DICIMALS. 1. To reduce a vulgar fraction to its equivalent decimal fraction.
Rule-To the numerator of the vulgar fraction affix a point. then annex a competent number of ciphers, and divide by the denominator ; the quotient is the numerator of the dicimal, and the ciphers annexed show the number of decimal places.
EXAMPLES. 1. Reduce }, }, and of a dollar to equivalent decimal fractions, or to cents. Ans. $1=.25c. $i=.5d. $15.75cents. 2. Reduce $} to a dicimal fraction. Ans.
.375. 3. Reduce iz to a decimal.
.04. 4. Reduce to a decimal.
.015625. 5. Reduce to a decimal.
.2916666. 6. Reduce 275 to a decimal.
.0715773. 11 10 7. Reduce Lof to a decimal.
.6043956. 14 13 8. Reduce of } of to a decimal.
.328125. II. To reduce the known parts of money, weight, measure, &c.
into decimals. RULE-Annex ciphers as before, to the lowest denomina. tion, and divide by that number which make one of the high.er denomination given, the quotient will be the decimal. Otherwise bring the parts to a vulgar fraction, and then reduce it to a decimal.
EXAMPLES. 9. Reduce 1s. 24d. to the decimal fraction of a pound.
Ans. £.059375. 10. Reduceus. N. Y. currency to the decimal of a dollar.
$.625. 11. Reduce 9d. to the decimal of a pound. Ans. £.0375 12. Remce 8 oz. 15dwt. 18gr. to the decimal of a lb. troy.
Ans. .73229161b. 13. Keduce 3qrs. 191b. 14oz. to the decimal of a cwt.
Ans. :927455cwt. 14. Reduce 1.5 pint to the decimal of a gallon. Ans. .1875. 15. Reduce .21 pints to the decimal of a peck. -.013125 16. Reduce 24 yards to the decimal of a mile. -.013636. 17. Reduce 3 qrs. 2 nails to the decimal of a yard. - ..875. 18. Reduce 4 poles to the decimal of an acre.
.025. 19. Reduce 14 minutes to the decimal of a day. -.009722d.
20. Reduce 52 days to the decimal of a year. -.1424657y. III. To find the value of decimals in known parts of money,
weight, measure, &c. RULE.--Multiply the decimal by the parts in the next inferior denomination, cutting off the decimals from the product ; then multiply the remainder by the next inferior de. nomination ; thus proceeding till you have brought in the least known parts of the integer.
EXAMPLES. 21. What is the value of £:775 ? Ans. 15s. 6d. 22. Required the value of .625 shillings? Ans.
7 d. 23. What is the value of £.8635? Ans. 178. 3.24d. 24. What is the value of .0125 lb. troy?
3dwts. 25. What is the value of .625 cwt ?
2qrs. 1416. 26. What is the value of .009943 miles ?-17yd. 1ft. 5.98848in. 27. What is the value of .875 yd. ?
3qrs. 2na. 28. Quere the value of .025 acre ?
4 poles. 29. What is the value of 3 years?
109dys. 12ho. 30. What is the value of.4765 days? -11ho. 26m. 9s. 36th.
ADDITION OF DECIMALS. RULE-Place the numbers so that the decimal points may stand all exactly under each other, then add as in whole numbers ; and point off as many places for decimals as there are the most in any of the given numbers.
EXAMPLES. 1. Add together 29.0146, 3146.5, 2109, .62417 and 14.16 ?
Ans. 2. What is the sum of 376.25+86.125+637.4725 +6.5+ 41.02+358.865 ? 3. Required the sum of 3.5+47.25+2.007. 1927.01 +1.5?
Ans. 4. Required the sum of £. 9369+£. 101.65+£o. 54.1375 and .625 shillings?
Ans. £165.56875=165. 11s. 4}d.
SUBTRACTION OF DECIMALS. Rule-Place the numbers under each other according to the value of their places, subtract as in whole numbers, and point off the decimal places as in addition.
EXAMPLES. 1. Required the difference between 91.73 & 2.138572 ? Ans. 2. Subtract 1.9185 from 2.73.
Ans. S. Subtract .98765 from 1.
Subtract £.58125 from 158. 3 d. Ans.f.184375=3s. 814
MULTIPLICATION OF DECIMALS. RULE.-Place the factors and multiply as in whole numbers, and from the product point off as many places for decimals as there are in both factors; but if there should not be so many, supply the defect by prefixing ciphers to the left hand.
EXAMPLES. 1. Multiply 2531 by 30.5 7. Multiply 27.35 by 7.70042 2. Multiply 7.346 by .017 8. Multiply 54.32 by .0075 3. Multiply .01538 by .173 9. Multiply .009 by .009 4. Multiply 6 121.3 by.5342 | 10. Multiply 500 by .0002 5. Multiply 72347 by 23.15 | 11. Multiply
.01 6. Multiply 17102 by.3162 | 12. Multiply
.1 13. Multiplyt hirty-one and six tenths by twenty-five.
790. 14. Multiply ninety-five thousandths by six hundredths.
Ans. .0057. 15. Multiply one thousand and eight and four tenths by fiftyfive.
Ans. 55462. QUESTIONS FOR EXERCISE. 16. If 1 bushel of corn .75 cents, what will 64 bushels cest?
Ans. $48. 17. If 1 day's labor be $1 12} cents, what will 27 days amount to ?
Ans. $30 37c. 18. What will 36 lb. of flax amount to at .17 cents per lb ?
Ans. $6 12 c. 19. Multiply 2s. 6d. by 2s. 6d. and let 13. be the integer ?
Ans. 6.25=6s. 3d. 20. Multiply 23. 64. by 2s. 6d. and let £1 be the integer ?
Ans, .015625=01. Os. 31d. 21. What will 13 cwt. 2 qrs. of iron come to at $4 56 c.
Ans. $61 56 c. 22. What will 31496 feet of boards amount to at S8 per M. feet ?
Ans. $251 96 c. 8 m. 23. If 1 yard of cloth cost 78. what will 25 yards cost ?
81. 158. 24. How much will 12 cwt. 3 qrs. 14 lb. of sugar come to at 41. 12s. per cwt ?
Ans. 591. 4s. 6d.
DIVISION OF DECIMALS. RULE-Divide as in whole numbers, and from the right hand of the quotient, point off as many places for decimals as the decimal places in the dividend exceed those in the divisor.
If there should be a remainder, or more decimal places in the divisor than the dividend, ciphers may be annexed to the dividend or remainder, and the quotient extended to any de. gree of accuracy.
If the decimal places of the quotient be not so many as the rule require, prefix ciphers to the left hand.
EXAMPLES. 1. Divide 72117 by 225/3. Divide 71865 by 25.5 2. Divide
759/4. Divide 87 by 317.25