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REDUCTION OF DECIMALS.
Ex. 1. Change the decimal .25 to a common fraction.
Suggestion. Supplying the denominator, .25=266 (Art. 180.) Now 2. is expressed in the form of a common fraction, and as such may be reduced to lower terms, and be treated in the same manner as any other common fraction. Thus 105=mo, or .
196. Hence, to reduce a Decimal to a Common Fraction.
Erase the decimal point ; then write the decimal denominator under the numerator, and it will form a common fruction, which may
be treated in the same manner as all other common fractions.
2. Change .125 to a common fraction, and reduce it to the lowest terms. Ans. §.
3. Reduce .66 to a common fraction, &c. 4. Reduce .75 to a common fraction, &c. 5. Reduce .375 to a common fraction, &c. 6. Reduce .525 to a common fraction, &c. 7. Reduce .025 to a common fraction, &c. 8. Reduce .875 to a coinmon fraction, &c. 9. Reduce .0625 to a conmon fraction, &c. 10. Reduce .000005 to a common fraction, &c.
CA SE II.
Ex. 1. Change to a decimal.
Suggestion. Multiplying both terms by 10 the fraction becomes 30. Again dividing both terms by 5, it becomes 8. (Art. 116.) But 10=.6, (Art. 178,) which is the decimal required.
QUEST.-196. How are Decimals reduced to Common Fractions ?
Now since we make no use of the denominator 10 after it is obtained, we may omit the process of getting it; for if we annex a cipher to the numerator and divide it by 5, we shallobtain the same result. Operation. 5)3.0
A decimal point is prefixed to the quo.6
tient, to distinguish it from a whole number.
Proof.--.6 reduced to a common fraction is 1o, and 16=*. (Art. 120.)
2. Reduce š to a decimal.
Annex ciphers to the numerator and
proceed as before. Hence, .125
197. To reduce a Common Fraction to a Decimal.
Annex ciphers to the numerator and divide it by the denominator." Point off as many decimal figures in the quotient as you have annexed ciphers to the numerator.
OBs. 1. If there are not as many figures in the quotient as you have annexed ciphers to the numerator, supply the deficiency by prefixing ciphers to the quotient.
2. The reason of this process may be illustrated thus. Annexing a cipher to the nuinerator multiplies the fraction by 10. (Arts. 59, 133.) If, therefore, the numerator with a cipher annexed to it, is divided by the denominator, the quotient will obviously be ten times too large. Hence, in order to obtain the true quotient, or a decimal equal to the given fraction, the quotient thus obtained must be divided by 10, which is done by pointing off one figure. (Art. 80.) Annexing 2 ciphers to the numerator multiplies the fraction by 100; annexing 3 ciphers by 1000, &c.; consequently when 2 ciphers are amnexed, the quotient will be 100 times too large, and must therefore be divided by 100; when three ciphers are annexed, the quotient will be 1000 times loo large, and must be divided by 1000 ; &c. (Art. 80.)
Quest.-197. How are Common Fractions reduced to Decimals? Obs. When there are not so many figures in the quotient as you have annexed ciphera, what is to be done?
3. Reduce to decimals. Ans. 1.5.
. 11. Reduce in and out to decimals. 12. Reduce to a decimal. Ans. 333333+. 13. Reduce 12: to a decimal. Ans. .128128128+.
198. It will be seen that the last two examples cannot be exactly reduced to decimals; for there will continue to be a remainder after each division, as long as we continue the operation.
In the 14th, the remainder is always 1 ; in the 15th, after obtaining three figures in the quotient, the remainder is the same as the given numerator, and the next three figures in the quotient are the same as the first three, when the same remainder will recur again.
The same remainders, and consequently the same figures in the quotient, will thus continue to recur, as long as the operation is continued.
199. Decimals which consist of the same figure or set of figures continually repeated, as in the last two examples, are called Circulating Decimuls, Repealing Decimals, or Repetends.
Ex. 1. Reduce 7s. 6d. to the decimal of a pound.
Suggestion. First, reduce 7s. 6d. to pence for the numeraior, and £1 to pence for the denominator of a com
QUEST.What are Circulating or Repeating Decimals ?
mon fraction, and we have £3%. (Art. 161.) Now , reduced to a decimal is £.375. Ans.
200. Hence, to reduce a compound number to the decimal of a higher denomination.
First reduce the given compound number to a common fraction ; (Art. 165;) then reduce the common fraction to a decimal. (Art. 197)
2. Reduce 5s. 4d. to the decimal of £1. Ans. .2666+. 3. Reduce 15s. 6d. to the decimal of £1. 4. Reduce 12s. 6d. to the decimal of £1. 5. Reduce 9d. to the decimal of 1 shilling. 6. Reduce 7d. 2 far. to the decimal of a shilling. 7. Reduce 1 pt. to the decimal of a quart. 8. Reduce 18 hours to the decimal of a day. 9. Reduce 9 in. to the decimal of a yard. 10. Reduce 2 ft. 6 in. to the decimal of a yard. 11. Reduce 6 furlongs to the decimal of a mile. 12. Reduce 13 oz. 8 dr. to the decimal of a pound.
Ex. 1. Reduce £.123 to shillings, pence, and farthings.
Multiply the given decimal of a pound Operation. by 20, as if it were a whole pound, be£.123 cause 20s. make £l, and point off as 20
many figures for decimals as there are shil. 2.460
decimal places in the multiplier and mul12
tiplicand. (Art. 191.) The product is in shillings and a decimal of a shilling.
Then multiply the decimal of a shilling by 4
12, and point off as before, &c. 'The 2.080 numbers on the left of the decimals, viz: Ans. 2s. 5d. 2f. 2s. 5d. 2 far. form the answer required.
Quest.—200. How is a compound number reduced to the decimal of a higher denomination ?
201. To reduce a decimal compound number to whole numbers of lower denominations.
Multiply the given decimal by that number which it takes of the next lower denomination to make one of this higher, as in reduction, (Art. 161, I,) and point off the product, as in multiplication of decimal fractions. (Art. 191.) Proceed in this manner with the decimal figures of each succeeding product, and the numbers on the left of the decimal point in the several products will constituie the whole number required.
2. Reduce L.125 to shillings and pence. Ans. 2s. 6d. 3. Reduce .625s. to pence and farthings. 4. Reduce £.4625 10 shillings, pence, and farthings. 5. Reduce .756 gallons to quarts and pints. 6. Reduce .6254 days to hours, minutes, and seconds. 7. Reduce .856 cwt. to quarters, &c. 8. Reduce .6945 of a ton to hundreds, &c. 9. Reduce .7582 of a bushel to pecks, &c. 10. Reduce .8237 of a mile to furlongs, &c. 11. Reduce .45683 of an acre to roods and rods. 12. Reduce .75631 of a yard to quarters and nails.
FEDERAL MONEY. 202. Federal Money is the currency of the United States. The denominators are, Eagles, dollars, dimes, cents, and mills.
10 mills (m.) make 1 cent, marked ct.
d. 10 diines
1 dollar, "
doll. or $ 10 dollars
E. Obs. Federal money was established by Congress Aug. 8th, 1776. Previous to this, English or Sterling money was the principal currency of the country
Quest.--201. How are decimal compound numbers reduced to whole ones? 202. What is Federal Money? Recite the Table. Obs. When and by whom was it established ?