MULTIPLICATION OF DECIMAL FRACTIONS. Ex. 1. Multiply.48 by .5. Suggestion. Multiplying by a fraction, is taking a part of the multiplicand as many times as there are like parts of a unit in the multiplier. (Art . 132.) Hence, multiplying by .5, which is equal to for }, is taking half of the multiplicand once. Now .48, or ** -2=206. (Art. 138.) But Yo=.24. (Art. 179.) Operation. Multiplying as in whole numbers, and .48 pointing off as many decimals in the pro.5 duct as there are decimal figures in both .240 Ans. factors, we have .2 10. But since ciphers placed on the right of decimals do not affect their value, the O may be omitted. (Art. 183.) Thus .24=100 which is the same result as before. 4. Multiply 8.45 96.071 456.03 4.5 228015 182412 191. From the preceding illustrations we deduce the following RULE FOR MULTIPLICATION OF FRACTIONS. Multiply as in whole numbers, and point off as many figures from the right of the product for decimals, as there are decimal places both in the multiplier and multiplicand. If the product does not contain so many figures as there are decimals in both factors, supply the deficiency by prefixing ciphers. Quest.–191. How are decimals multiplied together? How do you point off the product? When the product does not contain so many fig. ures, what is to be done ? Proof.--Multiplication of Decimals is proved in the same manner as Simple Multiplication. (Arts. 53, 74.) 25 100 Obs. The reason for pointing off as many decimal places in the product as there are decimals in botlı factors, may be illustrated thus : Suppose it is required to multiply .25 by .5. Supplying the denomi 125 nators .25=106, and .5=10 (Art. 180.) Now Xió= -1007 (Art. 135.) "Thus the new denominator is 1 with as many ciphers annexed to it, as there are ciphers in both denominators; (Art. 59 ;) or as there are decimal places in both the given fractions. Now the common fraction 1o do expressed decimally, is 125 ; (Art. 179 ;) consequently the product of the given factors .25X.5, contains just as many decimals as the factors themselves. EXAMPLES. feet are Ex. 1. In 1 piece of cloth there are 31.7 yards : how many yards are there in 7.3 pieces ? 2. In 1 barrel there are 31.5 gallons : how many gallons are there in 8.25 ? 3. In one rod there are 16.5 feet: how many there in 35.75 rods ? 4. How many cords of wood are there in 45 loads, allowing 8.25 of a cord to a load ? 5. How many rods are there in a piece of land 25.35 rods long, and 20.5 rods wide ? 6. If a man travel 38.75 miles per day, how far can he travel in 12.25 days ? 7. How many pounds of coffee are there in 68 sacks, allowing 961.25 pounds to a sack? 8. If a family consume .85 of a barrel of flour in a week, how much will they consume in 52.23 weeks? 9. What is the product of 10.001 into .05 ? 192. When the multiplier is 10, 100, 1000, &c., the multiplication may be performed by simply removing the decimal point as many places towards the right, as there are ciphers in the multiplier. (Arts. 59, 191.) Quest.—How is multiplication of decimals proved? 192. How proceed when the multiplier is 10, 100, 1000, &c. 11. Multiply 4.6051 by 100. DIVISION OF DECIMAL FRACTIONS. Ex. 1. Divide .75 by .5. Operation. .5).75 1.5 Ans. We divide as in whole numbers, and point off 1 decimal figure in the quotient. Obs. 1. We have seen in the multiplication of decimals, that the product has as many decimal figures, as the multiplier and multiplicand. (Art. 191.) Now since the dividend is equal to the product of the divisor and quotient, (Art. 65,) it follows that the dividend must have as many decimals as the divisor and quotient together; consequently, as the dividend has two decimals, and the divisor but one, we must point off one in the quotient ; that is, we must point off as many decimals in the quotient, as the decimal places in the dividend exceed those in the divisor. 2. Divide .289 by 2.4. Operation. 24 Substitute long division for short, and point off the quotient as before. 1 rem. 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 Ans. 2.69. 4. Divide .063 by 9. Operation. In this example the dividend has three more places of decimals than the divisor; 9).063 hence, the quotient niust have three places .007 Ans. 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 fig. ures 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 divi. dend, 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 rust 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 m:n mows 3.2 acres of grass per day, how long 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 ? how 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. Ans. 7.564. 12. Divide 1268.2 by 1000. Ans. 1.2682. 13. Divide 1 by 1.25. 14. Divide 1 by 562.5. 15. Divide .012 by .005. 16. Divide 2 by .0002. 17. Divide 5 by .000001. 18. Divide 13.2 by .75. 19. Divide .0248 by .04. 20. Divide 2071.31 by 65.3. Quest.-195. When the divisor is 10, 100, 1000, &c. how may the division be performed ? |