since the dividend is equal to the product of the divisor and quotient, (Art. 112,) it follows that the dividend must have as many decimals as the divisor an quotient together; consequently, as the dividend has two decimals, and the divisor but one, we must point off one in the quotient. In like manner it may be shown universally, that 329. The quotient must have as many decimal figures, as the decimal places in the dividend exceed those in the divisor; that is, the decimal places in the divisor and quotient together, must bi equal in number to those in the dividend. 2. What is the quotient of 3.775 divided by 2.5? Ans. 1.51 3. What is the quotient of .0072 divided by 2.4. Operation. 2.4).0072(.003 Ans. 72 make up the deficiency. Since the dividend has three decimal more than the divisor, the quotient mus have three decimals. But as it has bu one figure, we prefix two ciphers to it to OBS. It will be noticed that 3, the first figure of the quotient, denotes thousandths; also the product of 2, the units figure of the divisor, into the first quotient figure, is written under the thousandths in the dividend. Hence, The first figure of the quotient is of the same order, as that figure of the dividend under which is placed the product of the units of the divisor into the first quotient figure. 330. From the preceding illustrations we deduce the following general 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. 121.) 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. QUEST.-330. How are decimals divided! How point off the quotient? How is division of decimals proved? 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 arc 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.-1. 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. 2. When there is a remainder, the sign+, should be annexed to the quotient, to show that it is not complete. EXAMPLES. 4. How many boxes will it require to pack 71.5 lbs. of butter, if you put 5.5 lbs. in a box? 5. How many suits of clothes will 29.6 yds. of cloth make, allowing 3.7 yds. to a suit? 6. If a man can walk 30.25 miles per day, how long will it take him to walk 150.75 miles? 7. How many loads will 134642.156 lbs. of hay make, allowing 11622.2 lbs. for a load? 8. If a team can plough 2.3 acres in a day, how long will it take to plough 63.75 acres? 9. How many bales of cotton in 56343.75 lbs., allowing 375 lbs. to a bale! QUEST. Obs. When the number of decimal places in the divisor is equal to that in the ividend, what is the quotient? When there are more decimals in the divisor than in the ividend, how proceed? When there is a remainder, what may be done? CONTRACTIONS IN DIVISION OF DECIMALS. CASE I. 331. 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. 131, 330.) 1. Divide 4672.3 by 100. CASE II. Ans. 46.723. Ans. 0.00008. 332. When the divisor contains a large number of decimal figures, the process of dividing may be very much abridged. 9. It is required to divide 3.2682 by 2.4736, and carry the quotient to four places of decimals. Explanation.We perceive the first figure of the quotient wi!" be a whole number; for the number of decimals in the divisor i QUEST.-331. When the divisor is 10, 100, 1000, &c., how may the division be performed" equal to that of the dividend. (Art. 330. Obs. 1.) Now to obtain the decimals required, instead of annexing a cipher to the several remainders, which multiplies them respectively by 10, (Art. 98,) we may cut off a figure on the right of the divisor at each division, which is the same as dividing it successively by 10. (Art. 130.) When we multiply the divisor by 3, the second quotient figure, we carry 2 to the product of 3 into 3, because the product of 3 into 6, the figure omitted in the divisor, is nearer 20 than 10. Art 327.) We carry on the same principle to the first figure of each product of the divisor into the respective quotient figures. Hence, 333. To divide decimals, carrying the quotient to any required number of decimal places. For the first quotient figure divide as usual; then instead of hringing down the next figure, or annexing a cipher to the remainder, cut off a figure on the right of the divisor at each successive division, and divide by the other figures. In multiplying the divisor by the quotient figure, carry for the nearest number of tens that would arise from the product of the figure last cut off into the figure last placed in the quotient. (Art. 327.) OBS. 1. The reason for this contraction may be seen from the principle, that a tenth of the given divisor is contained in a tenth of the dividend, just as many times as the whole divisor is contained in the whole dividend; (Art. 145;) for, cutting off a figure on the right of the divisor, and omitting to annex a cipher to the dividend or remainder, is dividing each by ten. (Art. 130.) 2. When the divisor has more figures than the quotient is required to have, including the whole number and decimals, we may take as many on the left of the divisor as are required in the quotient, and divide by them as above. 3. If the divisor does not contain so many figures as are required in the quotient, we must divide in the usual way, until we obtain enough figures to make up this deficiency, and then begin the contraction. 10. Divide .4134 by .3243, and carry the quotient to four places of decimals. 11. Divide .079085 by .83497, and carry the quotient to five places of decimals. 12. Divide 2.3748 by 1.4736, and carry the quotient to three places of decimals. 13. Divide .3412 by 8.4736, and carry the quotient to five places of decimals. 14. Divide 1 by 10.473654, and carry the quotient to seven places of decimals. 15. Divide .4312672143 by .2134123406, and carry the quotient to four places of decimals. 16. Divide .879454 by .897, and carry the quotient to six places of decimals. REDUCTION OF DECIMALS. CASE I. 334. Decimals reduced to Common Fractions. Ex. 1. Change the decimal .75 to a common fraction. 100 Suggestion.-Supplying the denominator, .75=5. (Art. 311.) Now 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, 75-15, or 3. 20 335. 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 other common fractions. 2. Change .225 to a common fraction, and reduce it to the lowest terms. Ans. Ans. . 9 3. Reduce .125 to a common fraction, &c. 4. Reduce .95 to a common fraction, &c. 5. Reduce .435 to a common fraction, &c. 6. Reduce .575 to a common fraction, &c. 7. Reduce .656 to a common fraction, &c. 8. Reduce .204 to a common fraction, &c. 9. Reduce .075 to a common fraction, &c. 10. Reduce .012 to a common fraction, &c. 11. Change .0025 to a common fraction, &c. 12. Change .1001 to a common fraction, &c. QUEST.-335. How are Decimals reduced to Common Fractions? |