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REDUCTION OF DECIMALS.
Case I.-To reduce a Vulgar Fraction to its equivalent Decimal.
RULE.-Divide the numerator by the denominator the same as in Division of Decimals, annexing ciphers to the numerator as far as necessary; and the quotient will be the decimal required.
EXAMPLES i. Reduce to a decimal.
Ans. ,5. 2)1,00,5 DEM.—The reason of this operation is plain, be10' capše our numerator arises from a remainder in divi
sion, and our dénominator is the divisor in division; 0
we then only continue the division down below a
unit, by reducing the remainder to tenths, hundredths, thousandths, &c., and finding how often the divisor is contained in the inferiour. denominations, if they may be so called.-And that we preserve the true value of the fraction, may be seen by the quotient, hecause 5 tenths bear the same proportion to a unit, that the numerator does to the denominator. 2, Reduce it and to decimals.
Ans. ,75 and ,25. 3. Reduce š to a décimal.
Ans. ,2. 4. Reduce to a decimal.
Ans. ,625. 5. Reduce $ to a decimal.
Ans. ,125. 6. Reduce to a decimal,
Ans. ,192307. 7. Reduce 375 to a decimal."
Ans. ,0186. Case Il.—To find the value of a Dëcimal in terms of the inferiour denominations.
RULE.- Multiply the given decimal by the number of parts in the next inferiour denomination; and cut off as many places at the right for a remainder, as there are decimal places in the given sum or deeimal. Then multiply the remainder by the parts in the next lower denomination, and cut off as before, and so proceed through all the paris of the integer; then the several denominations standing on the left hand will be the answer.
Note.—The operation here is the same as in Reduction Descending in whole numbers,
EXAMPLES 1. Find the value of ,75 of a pound sterling. Ans. '15s. £
DEM.--As decimals increase in a tenfold propor. ,75 tion, it is plain, all the figures above the given num
20 ber of decimals, must be integers of the next infe. s. 15,00 riour denomination, and so on, because they arise
from the carriage in the place of tenths.
2. What is the value of ,775 of a pound? Ans. 158. 6d. 3. What is the value of ,765 of a pound Troy?
Ans. Joz. 3pwt. 14gr 4. What is the value of ,617 of a cwt. ?
Ans. 2qrs. 13lb. loz. 10dr. 5. What is the value of ,617 of a pound Avoirdupois?
Ans. 902. 13.000 dr. 6. What is, the value of ,981 of a dollar ?
Ans. 98 cents, 1 mill. 7. What is the value.of. ,3 of a year? Ans. 109d. 12h. 8. What is the value of ,367 of a yard?
Ans. lqr. 1,872na. 9. What is the value of :,84 of 5026 Troy?
Ans. 4oz. 4pwt. 10. What is the value of ,8635 of a pound?
Ans., 178.3,24d. 11. What is the value of ,3375 of an acre ?
Ans. I rood, 14 poles. 12. What is the value of ,0375 of a pound? Ans. 9d. 13. What is the value of ,645 of a day?
Ans. 15h. 28m. 48sec. Note.-Addition and subtraetion of Decimals of different denomi. nations, may be performed, after the decimals are redụced to their proper quantities.
14. What is the sum of ,75. of a pound, and ,285 of a shil. ling?
Ans. 15s. 3d. 1;68qrs. 15. What is the sum of ,19 of a pound Troy, and ,85 of an ounce ?
Ans. 3oz. 2,6pwt. 16. What is the difference between ,45 of a pound, and ,45 of a shilling?
Ans. 8s. 6d. 2,4qrs. 17. What is the difference between ,48 of a day, and 35 of an hour
Ans. ith. 10m. ias CASE 111.--To reduce inferiour denominations !o the decimal of any superiour denomination.
RULE.—Divide the given sum reduced to the lowest denomination mentioned, by the proposed integer, reduced to the same denomination, and the quotient. will be the answer or decimal required. Or, set down the given numberš, from the least to the greatest, in a perpendicular column, and divide each denomination-by such a number as will reduce it to the next superioúr denomination; in each place annexing the quotient to the right hand of the next higher denomination, and the last quotient will be the decimal required...
EXAMPLES 1 Reduce 158. 73. 2qrs" to the decimal of a pound.
Ans. ,78125=158. 7d. 2qrs. £
s. d. qr's. 1
15 7 2 20
960)750,00,78125 the decimal 4
0 la thus, which is
DEM.-The reason of the first 412 .he better way.
operation is obvious, because the 127,5
given sum reduced to the lowest
denomination mentioned, stands as 2015,625
the numerator of the integer re,78125 Ans.
duced to the same denomination for 20
a denominator; the decimal or quos. 15,62500 tient must then bear the
portion to the integer, as the given d. 7,50000
sum bears to the integer. In the last 4
operation it is plain, that the quotient, 5 tenths, after the first division,
bears the same proportion to a pen15 7 2
ny that two farthings bear to a pen
ny, and so of the rest, consequently the principle is the same as in the first operation. 2. Reduce 9d. 3qrs. to the decimal of a shiling.
Ans. ,8125, 3. Reduce 10s. 6d. to the decimal of a pound. Ans. ,525.
4. Reduce 2qrs. 3na. to the decimal of a yard.
Ans. ,6875. .5. Reduce 10cwt. 3qrs. 14lb. to the decimal of a tủņ.
Ans. ,54375. 6. Reduce £14 12s. to a decima! expression Ans. £14,6. 7. Reduce £47 155. 60. to a decimal expression.
Ans. £47,775. 8. Reduce 3qrs. and 2na. to the decimal of an Ell English.
Ans. ,7 E. E. 9. Reduce 109 days, 12 hours, to the decimal of a year.
Ans. ,3 of a year. 10. Reduce 3qrs, 121b. 5oz. 1,92dr. to the decimal of a cwt.
Ans. 86. 11. Reduce 5fur. 16pol. to the decimal of a mile.
Ans. ,675 of a mile. 12. Reduce 7 calendar months, to the decimal of a year,
Ans. ,625 of a year. QUESTIONS ON DECIMAL FRACTIONS. What is a decimal fraction ? A. It is a fraction that has for its denominator a unit, 1, with as many ciphers annexed as the numerator has figures, and is usually expressed by setting down the numerator only, with a point before it. What is a mixed number} A. It is made up of a whole number with some decimal fraction, one being separated from the other by a point. What is a pure decimal ? A. Something less than a unit, 1, with a point at the left hand. Would a pure decimal become a unit by annexing significant figures at the right? A. It would not, a pure decimal, like certain mathematical lines which always approach each other, but never meet, may continually approach towards a unit, yet never become a unit, thus 9 tenths want one tenth of a unit, 99 want one hundredth of being a unit, and ,999 want one thousandth of becoming a unit; so we might carry the decimal to any number of figures whatever: it would want something of being a unit, 1, and yet continually approaching towards a unit. How do you set down decimals for adding? A. According to the value of their places; tenths under tenths, hundredths under hundredths, &c. How do you point off, in addition of decimals ? A. Directly below the given decimal points. How are numbers set down in subtraction of decimals? A. The same as in addition, and point off from the difference the same for decimals. How do you point off in multiplication of decimals? A. As many figures from the right hand of the product as there are decimal places in both the fac
When you multiply by a pure decimal, is the product greater or less than the multiplicand ? A. Less, because it is repeating the multiplicand by something less than a unit. What proportion does the product bear to the multiplicand ? A. The same proportion that the multiplier bears to a unit. How do you pcint off in division of deci.
mals ? A. As maxy figures from the right hand of the quotient, as the decimal places in the dividend exceed those in the divisor. When you divide by a pare decimal, is the quotient greater or less than the dividend? A. Greater. Why should it be greater?. A. Because the divisor is less than a unit, and it is evident, that the dividend will contàin a part oftener than a whole. What proportion does the dividend bear to the quotient? A. The same proportion as the divisor bears to a’unit. If you divide 10 by one tenth, what will be the quotient? A 100. If you multiply 10 by one tenth, what will be the product ? A One. How do you reduce a vulgar fraction to a decimal ? A. By dividing the numerator by the denominator. How do you reduce inferiour denominations to the decimal of a superiour denomination? A. I reduce the given sum to the lowest denomination mentioned for a dividend, and then reduce an integer to the same denomination for a divisor, and the quotient will be the decimal required. How do you find the value of a decimal in the inferiour denominations of the integer? A. By multiplying the decimal by the inferiour denominations of the integer, the same as in reduction descending, and pointing off from the products in each place as many figures for decimals as will equal the deciinals in the given number; the figures at the left of the decimal points will be the value of the fraction in the inferiçur de. nominations of the integer.
REDUCTION OF CURRENCIES. Reduction of Currencies is finding the value of the coin or currency of one state or country in that of another. Although the same denominations and coin are generally used in the different countries and states, yet the standard value frequently differs in each.
Thus, a dollar is reckoned in
8s. called Ncu-York currency.
6s. called New England currency.
4s. 8d. called Georgia currency.
7s. Od. called Pennsylvania currency.
4.6.. called English or Slerling money.
5s. called Canada or Halifax currency. To reduce the Currency of each Slale to Federal Money.
RULE.—Divide the given sum, reduced to shillings, to sixpences or to pence, by the number of shillings, sixpences, or pence in a dollar as it is reckoned in each state.