7. What is the sum of 6.15768; 1.713458 and .6573128? 8. What is the sum of .0256; 15.6941; 3.856 and .00035? 9. Add together 256.31; 29.7; 468.213; 5.6 and .75. 10. Add together 25.61; 78.003; 951.072 and 256 .3052. 11. Add together .567; 37.05; 63.501; 76.25 and .63. 12. Add together .005; 1.25; 6.456; 10.2563 and 15.434. 13. Add together 256.1; 10.15; 27.09; 35.560 and 2.067. 14. Add together 5.00257; 3.600701 and 2.10607. 15. Add together 5 tenths, 25 hundredths, 566 thousandths, and 7568 ten thousandths. 16. Add together 34 hundredths, 67 thousandths, 13 ten thousandths, and 463 millionths. 17. Add together 7 thousandths, 63 hundred thousandths, 47 millionths, and 6 tenths. 18. Add together 423 ten millionths, 63 thousandths, 25 hundredths, 4 tenths, and 56 ten thousandths. SUBTRACTION OF DECIMAL FRACTIONS. 188. Ex. 1. From 25.367 substract 13.18. Operation. 25.367 13.18 12.187. Ans. Having written the less number under the greater, so that units may stand under units, tenths under tenths, &c., we proceed exactly as in subtraction of whole numbers. (Art. 40.) Thus, 0 thousandths from 7 thousandths leaves 7 thousandths. Write the 7 in the thousandth's place. As the next figure in the lower line is larger than the one above it, we borrow 10. Now 8 from 16 leaves 8; set the 8 under the column, and carry 1 to the next figure. (Art. 38.) Proceed in the same manner with the other figures in the lower number. Finally, place the decimal point in the remainder directly under that in the given numbers. 189. Hence, we deduce the following general RULE FOR SUBTRACTION OF DECIMALS. Write the less number under the greater, with units under units, tenths under tenths, hundredths under hundredths, &c. Subtract as in whole numbers, and point off the answer as in addition of decimals. (Art. 187.) PROOF-Subtraction of Decimals is proved in the same manner as Simple Subtraction. (Art. 39.) Note. When there are blank places on the right hand of the upper number, they may be supplied by ciphers without altering the value of the decimal. (Art. 183.) EXAMPLES. 2. From 15 take 1.5. 3. From 256.0315 take 5.641. 5. From 63.25 take 50. 6. From 201.001 take 56.04037. 7. From 1 take .125. 8. From 11.1 take .40005. 9. From .56078 take .325. 10. From 1.66 take .5589. 12. From 1 take .000001. 13. From 256.31 take 125.4689301. 14. From 8960.320507 take 63.001. 15. From 57000.000001 take 1000.001. Ans. 13.5. 16. From 75 hundredths take 75 thousandths. 17. From 6 thousandths take 6 millionths. 18. From 3252 ten thousandths take 3 thousandths. 19. From 539 take 22 thousandths. 20. From 7856 take 236 millionths. QUEST.-189. How are decimals subtracted? How point off the answer? How is subtraction of decimals proved? MULTIPLICATION OF DECIMAL FRACTIONS. 190. 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 or, is taking half of the multiplicand once. Now .48, or ÷2=124. (Art. 138.) But 2.24. (Art. 179.) Operation. .48 We multiply as in whole numbers, and pointing off as many decimals in the product as there are decimal figures in both factors, we have .240. But since ciphers placed on the right of decimais do not affect their value, the O may be omitted. (Art. 183.) But .24-12, which is the same result as before. .240 Ans. 191. From the preceding illustrations we deduce the following general RULE FOR MULTIPLICATION OF DECIMALS. 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 figures as there are decimals in both factors, what is to be done? PROOF-Multiplication of Decimals is proved in the same manner as Simple Multiplication. (Arts. 53, 74.) OBS. The reason for pointing off as many decimal places in the product as there are decimals in both factors, may be illustrated thus: Suppose it is required to multiply .25 by .5. Supplying the den ominators .25-12, and .5. (Art. 180.) Now X=1. (Art. 135.) But .125; (Art. 179;) that is, the product of 25x.5, contains just as many decimals as the factors themselves. In like manner may be shown that the product of any two or more decimal numbers, must contain as many decimal figures as there are places of decimals in the given factors. EXAMPLES. 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 barrels ? 3. In one rod there are 16.5 feet: how many feet are 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 can 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? 10. What is the product of 50.0065 into 1.003? 192. When the multiplier is 10, 100, 1000, &c., 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. 12. Multiply 2.6501 by 1000. 13. Multiply .5678 by 10000. 14. Multiply .000781 by 2.40001. 15. Multiply 1.002003 by .0024. 16. Multiply .58001 by .0001003. 17. Multiply 8.001502 by .00005. 18. Multiply 85689.31 by .000001. 19. Multiply .0000045 by 69.5. 20. Multiply .0340006 by .000067. 21. Multiply .5 by 5 millionths. 22. Multiply .15 by 28 ten thousandths. Ans. 460.51. 23. Multiply 25 hundredth thousandths by 7.3. 24. Multiply 225 millionths by 2.85. 25. Multiply 2367 ten millionths by 3.0002. DIVISION OF DECIMAL FRACTIONS. 193. Ex. 1. Divide .75 by .5. Operation. .5).75 We divide as in whole numbers, and point 1.5 Ans. off 1 decimal figure in the quotient. OBS. 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. 2.4) 289(.12+Ans. 24 49 48 1 rem. Since the divisor contains two figures, we substitute long division for short, and point off the quotient as before. |
