. OBS. 2. When there is a remainder, the sign + should be annexed to the quotient, to show that it is not complete. 3. Divide 1.345 by .5 Operation. 9).063 .007 Ans. Ans. 2.69. In this example the dividend has three more places of decimals than the divisor; hence, the quotient must have three places of decimals. We must, therefore, prefix two ciphers to the quotient. 194. From these illustrations we deduce the following RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers, and point off as many figures for decimals in the quotient, as the decimal places in the dividend exceed those in the divisor. If the quotient does not contain figures enough, supply the deficiency by prefixing ciphers. PROOF.--Division of Decimals is proved in the same manner as Simple Division. (Art. 73.) OBS. 1. When the number of decimals in the divisor is the same as that in the dividend, the quotient will be a whole number. 2. When there are more decimals in the divisor than in the dividend, annex as many ciphers to the dividend as are necessary to make its decimal places equal to those in the divisor. The quotient thence arising will be a whole number. (Obs. 1.) 3. After all the figures of the dividend are divided, if there is a remainder, ciphers may be annexed to it and the division continued at pleasure. The ciphers annexed must be regarded as decimal places belonging to the dividend. Note. For ordinary purposes, it will be sufficiently exact to carry the quotient to three or four places of decimals; but when great accuracy is required, it must be carried farther. QUEST.-194. How are decimals divided? How do you point off the quotient? How is division of decimals proved? Obs. When the number of decimal places in the divisor is equal to that in the dividend, what is the quotient? When there are more decimals in the divisor than in the dividend, how proceed? When there is a remainder, what may be done? EXAMPLES. 1. If 1.7 of a yard of cloth will make a coat, how many coats will 10.2 yards make? 2. In 6.75 cords of wood, how many loads are there, allowing .75 of a cord to a load? 3. If a man mows 3.2 acres of grass per day, how long I will it take him to mow 39.36 acres? 4. If 23.25 bushels of barley grow on an acre, how many acres will 556 bushels require? 5. In 74.25 feet, how many rods ? 6. In 99.225 gallons of wine, how many barrels? 7. If a man chops 3.75 cords of wood in a day, how many days will it take him to chop 91.476 cords? 8. If a man travei 35.4 miles per day, how long will it take him to travel 244.26 miles? 9. A dairyman has 187.5 pounds of butter, which he wishes to pack in boxes containing 12.5 pounds apiece: how many boxes will it require ? 10. In 3.575, how many times .25? 195. When the divisor is 10, 100, 1000, &c., the division may be performed by simply removing the decimal point in the dividend as many places towards the left, as there are ciphers in the divisor, and it will be the quotient required. (Arts. 80, 194.) 11. Divide 756.4 by 100. 20. Divide 2071.31 by 65.3. Ans. 7.564. Ans. 1.2682. 14. Divide 1 by 562.5. 16. Divide 2 by .0002. 18. Divide 13.2 by .75. QUEST.-195. When the divisor is 10, 100, 1000, &c. how may the di vision be performed? REDUCTION OF DECIMALS. CASE I. Ex. 1. Change the decimal .25 to a common fraction. Suggestion. Supplying the denominator, .25-25%. (Art. 180.) Now 25 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 12%, or 4. 25 00 196. 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 fraction, 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 common fraction, &c. 9. Reduce .0625 to a common fraction, &c. 10. Reduce .000005 to a common fraction, &c. CASE II. Ex. 1. Change to a decimal. 50 Suggestion. Multiplying both terms by 10 the fraction becomes Again dividing both terms by 5, it becomes (Art. 116.) But.6, (Art. 178,) which is the decimal required. 6 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 .6 A decimal point is prefixed to the quotient, to distinguish it from a whole number. PROOF.-.6 reduced to a common fraction is, and 1% (Art. 120.) 2. Reduce to a decimal. Operation. 8)1.000 .125 Annex ciphers to the numerator and proceed as before. Hence, 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 numerator 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 annexed, 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 too 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 ciphers, what is to be done? 3. Reduce to decimals. Ans. 1.5. 9. Reduce,, and 2 480 to decimals. 3 to decimals. to decimals. 5 1785 to decimals. Ans. 333333+. 12. Reduce to a decimal. 13. Reduce 128 to a decimal. Ans. .128128128+. 999 198. 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 Decimals, Repeating Decimals, or Repetends. CASE III. Ex. 1. Reduce 7s. 6d. to the decimal of a pound. Suggestion. First, reduce 7s. 6d. to pence for the numerator, and £1 to pence for the denominator of a com QUEST.-What are Circulating or Repeating Decimals? |