14. Add together 394.61; 81.928; 3624.8103; 640.203; 6291.302; 721.004; and 3920.304. 15. Add together 25 hundredths, 8 tenths, 65 thousandths, 16 hundredths, 142 thousandths, and 39 hundredths. 16. Add together 9 tenths, 92 hundredths, 162 thousandths, 489 thousandths, and 92 millionths. 17. Add together 45 thousandths, 1752 millionths, 624 ten millionths, and 24368 millionths. 18. Add together 29 hundredths, 7 millionths, 62 thousandths, and 12567 ten millionths. 19. Add together 95 thousandths, 61 millionths, 6 tenths, 11 hundredths, and 265 hundred thousandths. 20. Add together 1 tenth, 2 hundredths, 16 thousandths, 7 millionths, 26 thousandths, 95 ten millionths, and 7 ten thousandths. 21. Add together 96 hundred thousandths, 92 millionths, 25 hundredths, 45 thousandths, and 7 tenths. 22. Add together 85 thousandths, 17 hundredths, 36 ten thousandths, 58 millionths, 363 hundred thousandths, 185 millionths, and 673 ten thousandths. SUBTRACTION OF DECIMAL FRACTIONS. 321. Ex. 1. From 425.684 subtract 216.96. Operation. 425.684 216.96 208.724. 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. 72.) Thus 0 thousandths from 4 thousandths leaves 4 thousandths. Write the 4 in the thousandths' place. Since the figure 9 in the lower line is larger than the one above it, we borrow 10. Now 9 from 16 leaves 7; set the 7 under the column and carry 1 to the next figure. (Art. 72.) 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 number. 322. 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. 320.) PROOF.-Subtraction of Decimals is proved in the same manner as Simple Subtraction. (Art. 73.) 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. 315.) EXAMPLES. 2. From 456.0546 take 364.3123. Ans. 91.7423. 3. From 1460.39 take 32.756218. 4. From 21.67 take .682349. 6. From 100.536 take 19.36723. 7. From .076345 take .009623478. 9. From 10 take .000001. 10. From 65.00001 take .9682347. 11. From 24681 take .87623. 12. What is the difference between 25 and .25? 13. What is the difference between 3.29 and .999 ? 14. What is the difference between 10 and .0000001 ? 15. What is the difference between 9 and .999999? 16. What is the difference between 4636 and .4636 ? 17. What is the difference between 25.6050 and 567.392? 18. What is the difference between 76.2784 and 29.84234 ? 19. What is the difference between .0000001 and .0001 ? 20. What is the difference between .0000004 and .00004 ? 21. What is the difference between 32 and .00032 ? QUEST.-322. How are decimals subtracted? How point off the answer? How is sub raction of decimals proved? 22. What is the difference between .00045 and 45 ? 23. What is the difference between .00000099 and 99 ? 24. From 1 thousandth take 1 millionth. 25. From 7 hundred take 7 hundredths. 26. From 29 thousand take 92 thousandths. 27. From 256 millions take 256 thousandths. 28. From 46 hundredths take 46 thousandths. 29. From 95 thousandths take 909 ten thousandths. 30. From 1 billionth take 1 trillionth. 31. From 2874 millionths take 211 billionths. 32. From 6231 hundred thousandths take 154 millionths. 33. From 7213 ten thousandths take 431 hundred thousandths. 34. From 8436 hundred millionths take 426 ten billionths. MULTIPLICATION OF DECIMALS. 323. Ex. 1. If a man can reap .96 of an acre in a day, how much can he reap in .5 of a day? - Analysis. Since he can reap 96 hundredths of an acre in a whole day, in 5 tenths of a day he can reap 5 tenths as much. But multiplying by a fraction we have seen, is taking a part of the multiplicand as many times as there are like parts of a unit in the multiplier. (Art. 210.) Hence, multiplying by .5, which is equal toor, is taking half of the multiplicand once. Now .96, or 2. (Art. 227.) But .48. (Art. 311.) Operation. .96. .5 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 480. But .480 Ans. ciphers placed on the right of decimals do not affect their value; the 0 may therefore be omitted; and we have .48 for the answer. 324. 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. PROOF.-Multiplication of Decimals is proved in the same manner as Simple Multiplication. OBS. The reason for pointing off as many decimal places in the product as there are decimals in both factors, may be illustrated thus: 100 100 Suppose it is required to multiply .25 by .5. Supplying the denominators .25=25, and .5. (Art. 312.) Now 251125. (Art. 219.) But 25.125; (Art. 311;) that is, the product of .25.5, contains just as many decimals as the factors themselves. In like manner it 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 rod there are 16.5 feet: how many feet are there in 41.3 rods? 2. In 1 degree there are 69.5 statute miles: how many miles are there in 360 degrees? 3. In 1 barrel there are 31.5 gallons: how many gallons in 65.25 barrels ? 4. In 1 inch there are 2.25 nails: how many nails are there in 60.5 inches? 5. In 1 square rod there are 30.25 square yards: how many square yards are there in 26.05 rods ? 6. In one square rod there are 272.25 square feet: how many square feet are there in 160 rods ? QUEST.-324. How are decimals multiplied together? How do you point off the prod act? When the product does not contain so many figures as there are decimals in both factors, what is to be done? How is multiplication of decimals proved? 7. How many square rods are there in a field 60.5 rods long and 40.75 rods wide? Multiply the following decimals: 8. 1.0013X.25. 9. 44.046 X.43. 10. 3.6051X4.1. 11. 0.1003 X 6.12. 12. 8.0004 X.004. 13. 35.601 X 1.032. 14. 213.02 X 4.318. 15. 0.0006X.00012. 16. 0.3005X.3005. 17. 10.2106 X 38.26. 18. 164.023X1.678. 19. 9.40061X25.812. 20. 7.31042 X 10.021. 21. 40.4369 × 1.2904. CONTRACTIONS IN MULTIPLICATION OF DECIMALS. CASE I. 325. 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. 99, 324.) 1. Multiply 85.4321 by 100. Ans. 8543.21. 2. Multiply 42930.213401 by 10. 3. Multiply 1067.2350123 by 100. 9. Multiply 75 hundredths by 100000. 10. Multiply 65 ten thousandths by 1000. 11. Multiply 48 hundred thousandths by 100000. QUEST.-325. How proceed when the multiplier is 10, 100, &c.? |