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.-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 t, 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 1622.2 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? Divide the following decimals: 10. 46.84: 7.9. 20. 0.000065.003. 11. 1.658;.25. 21. 167342-.002. 12. 67234;.85. 22. 684234.6 ; 2682. 13. 4.00334-6.31. 23. 0.000045-9. 14. 73.8243;.061. 24. 7.231068;.12. 15. 0.00033;.011. 25. 26.3845+.125. 16. 236.041:1.75. 26. 4.00001. 17. 60.0001;1.01. 27. 6:.0000001. 18. 300.402:12.1. 28. 0.8;.0000002. 19. 4.32067:.001. 29. 6541.234567: 21. QUEST.-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 ? CONTRACTIONS IN DIVISION OF DECIMALS. CASE 1. 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. Ans. 46.723. 2. Divide 0.8 by 10000. Ans. 0.00008. 3. Divide 672345.67 by 10. 4. Divide 10342.306 by 100. 5. Divide 42643.621 by 100000. 6. Divide 6723000.45 by 1000000. 7. Divide 1.2300456 by 100000. 8. Divide 2.0076346 by 1000000. CASE II. 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. Common Method. Contraction. 2.4736)3.2682(1.3212 2.4736)3.2682(1.3212 2 47361 2 4736 79460 7946 7420/8 7421 52520 525 494 72 495 30|480 30 24 736 25 5,7440 5 4 9472 5 17968 Explanation.-We perceive the first figure of the quotient will be a whole number; for the number of decimals in the divisor is QUEST.-331. When the divisor is 10, 100, 1000, &o., 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 bringing 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 i 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. Suggestion.-Supplying the denominator, .75=76. (Art. 311.) Now he 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, 178=H, or . 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. ao. 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 ? 13. Change .1814 to a common fraction, &c. 14. Change .0556 to a common fraction, &c. 15. Change .1216 to a common fraction, &c. 16. Change .2005 to a common fraction, &c. 17. Change .0015 to a common fraction, &c. CASE II. 336. Common Fractions reduced to Decimals. Ex. 1. Change to a decimal. Suggestion.-Multiplying both terms by 10, the fraction becomes 0. Again, dividing both terms by 5, it becomes to (Art. 191.) But P=.8, which is the decimal required. (Art. 311.) Note.-Since we make no use of the denominator 10 after it is obtained, we may omit the process of ting it; for, if we annex a cipher to the numerator and divide it by 5, we shall obtain the same result. Operation. 5)4.0 A decimal point is prefixed to the quotient to .8 Ans. distinguish it from a whole number. 2. Reduce to a decimal. Ans. 0.625. 337. Hence, 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 an. nexed ciphers to the numerator, supply the deficiency by prefixing ciphers to the quotient. 2. The reason of this rule may be illustrated thus. Annexing a cipher to the numerator multiplies the fraction by 10. (Arts. 98, 186.) If, therefore, the numerator with u cipher annexed to it, is divided by the denominator, the quotient will obviously be ten times too large. (Art. 141.) 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. 131.) Annexing 2 ciphers to the numerator multiplies the fraction by 100; annexing 3 ciphers by 1000, &c., consequently, when 2 ciphers are an. nexed, 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. 131.) QUEST.-337. How are Common Fractions reduced to Decimals ? |