nomination mentioned in either, then add them together, viz.-10s. 3d., £3. 15s., £9. 4s. 6d. A. £13.4875. 19. Add together 6yd. 1qr., 5qr. 3 na., 175yd., and 256yd. 21qr. A. 281.8625 yards. 3 16 20. Find the sum of 605 years, 1902 years, 6753 years, 45 years, and 203 years. A. 998.3375 years. 21. Add decimally 19 miles, 3 furlongs, 20 rods; 31 miles, 5 furlongs; and 98 miles, 2 furlongs, 15 rods. A. 149.359375 miles. SUBTRACTION OF DECIMALS. RULE. LXXI. 1. Write tenths under tenths, hundredths under hundredths, &c.; then subtract as in whole numbers and point off as in Addition. 2. Suppose a merchant bought 14.25 barrels of flour, and sold 8,375 barrels; how many barrels has he on hand? 1 4 2 5 3. Say, 5 from 10 leaves 5, 1 to carry to 7. For 14.25-14,25%-8 375 (by LXIV. 27.)= 5,875 barrels, Ans. 1000 0 0 4. From $90.025 take $8.6285. 5. From $38.036 take $4.0375. A. $81.3965. A. 417.9875 barrels. A. 801.9632. 6. Bought 513.025 barrels of flour, and sold 95.0375 barrels; how much had I left? 7. From 891.037 take 89.0738. 8. From 376,683 take 47.0931. 9. From 83.12 take 5.3758. 10. From 835.2 take .1234567. 11. Subtract $53.008 from $100. 12. From $5 take 5 dimes and 5 mills. 13. From 8 take 9 thousandths. 14. From 1 take 1 hundredth. 15. From 1 take 1 millionth. 16. How much greater is 4 than 3.99999999? A. 329.5899. A. 77.7442. A. 835.0765433. A. .999999999999 A. 52.275. A. 8.475bl. A. £94.185. A. 406.015A. LXXI. Q. What is the rule for subtracting decimals? 1. What is the dif ference between .3 and .25?-between 1 and .1?-between .5 and .18?-be tween 2.5 and 3?-between .9 and unity? ΤΟ 24. From 10 eagles and of a mill take $99.9998. 25. From an unit subtract 5 millionths. 27. From £8. 17s. 63d. take £3. 19s. 6d. A. .999995. A. 335.13225lb. A. £4.901. MULTIPLICATION OF DECIMALS. LXXII. 1. Every decimal has as many places as its denominator has ciphers, as is .5; is .05, &c. 5 To 5 100 2. The product of any two decimals, then, must have as many decimal places as the product of their denominators has ciphers. 3. Thus, .5×.7=.35, for 5×35.35. 7 4. The product of the denominators of any two decimals has as many ciphers as both its factors. (XVIII.) 6. Hence the product of any two decimal expressions must have as many decimal places as both its factors. 7. Thus .5X.87.435; for 586 87 10 190 43 5 1000 =.435. 8. When the product has not as many places of figures as its factors have places of decimals, we must supply the deficiency by prefixing ciphers. 9. Thus, .2×.4-.08 because 2×14=18.08. GENERAL RULE. 10. Multiply as in whole numbers, and point off so many places for decimals in the product as are equal to the decimal places in both the factors; but if the product has not so many places, prefix ciphers to make up the number. 12. Multiply 81.235 wine gallons by 35. 13. Multiply 90,325 puncheons by .45. 14. Multiply 3.25 1 by 7 A. 2. 2 7 5 7 A. 2843.225. A. 40,646.25. Here are 4 decimal places in both the factors, therefore make 4 decimal places in the product. 15. Multiply 2.345 ale gallons by.15. A. .35175 of an ale gallon. 100 LXXII. Q. How is 5 written decimally? What is the rule for it? 1. Why does .7 multiplied by .5 make .35? 4. What is the rule for ascertaining the decimal places? 2. How many ciphers have all such products? 4. What is the inference in respect to pointing off the product in decimals? 6. When are ciphers to be prefixed? 8. Why then is .08 the product of .2 by .4? 9. General Rule? 10. Multiply 8 by .6, by .06, by .006,-.8 by .5, by 05. hv 0005;-.08 by .08 by .008. A. $.03315. A. .00045 of a gallon. 18. Multip.y .085 of a dollar by .39. 21. Multiply .618 of a hogshead by .312. 22. Multiply .521 of a bushel by .48. 23. Multiply .235 of a century by .45. 25 A. $18.59ct. 6,35m. A. .192816hhd. A. .25008bu. A. .10575C. 24. Multiply .375 of a square inch by .00027. A. .00010125sq.in. 25. Multiply 8.165 of a minute by .00089. A. .00726685min. 26. What will 800 trees cost at $.375 a-piece? 27. Multiply 800 and .008 together. 28. Multiply 5 and .0005 together. 29. Multiply .16 and 500 together. 30. Multiply .003 by 8500000. A. $300. A. 6.4. A. .0025. A. 80. A. 25500. 31. Since decimals increase from the right to the left in a tenfold proportion, therefore,— 32. To multiply by 10, or 100, or 1,000, &c.—Merely remove the separatrix one place farther towards the right for every cipher, anu it is done. 33. Multiply .3621 by 10;-by 100;-by 1,000 ;-by 10,000 ;—by 100,000. A. 3.621; 36.21; 362.1; 3621; 36210. 34. What would one million of flax seeds cost at $.000001 for each seed? A. .31598. A. $1. 35. Multiply 25 millionths by 18 thousandths. A. .000000045. 36. Multiply .02562 into 121. 37. What will be the cost of 3331 Rohan potatoes at $.0645 a-piece? A. $21. 38. What will 415 barrels of sugar weigh, the average weight of each being 495.00025 pounds? A. 205,524.1038. A. 166.375. A. 272.25 sq. ft. 39. Reduce the fractional parts of the following numbers to decimals, then multiply them together, viz. 301 by 51. 40. Multiply decimally 304 sq. yd. by 9 sq. ft. 41. Multiply 5.5 by 5 as a common fraction. 42. Multiply 5 by 5 decimally. A. 30.25-301. 43. What will 183 hogsheads of molasses cost at the rate of 15 dollars per hogshead? 44. What will 7 loads of hay cost at £3. Reduce both to decimals by LXIX.. Case IV. 3 A. $284.0625. 10s. Gd. per load? A. £26.4375. Q. How are decimals multiplied by 10 or 100, &c. easily? 32. Why has the process this effect? 31. What is the product of .1234 multiplied by 10?—by 100?-by 1,000? 45. Multiply £2. 3s. 84d. by 5 decimally. 46. Multiply 2m. 6fur. 303rd. by 5. A. £10.921875. A. 14.221875 miles. 47. What will 10 tons 15cwt. of Russia iron cost at £2. 3s. 6d. per ton? The product is £23.38125. (LXIX. Case v.) A. £23. 7s. 7d. 48. What will 14 hogsheads 18.9 gallons of molasses cost at £3. 15s. 9d. per hogshead? A. £54. 3s. 2 d. + 49. Suppose a certain farm to consist of 200A. 3R. 25rd.; what will be its value at $25.375 per acre. A. $5097.996.+ 1 50. If a man travels 30m. 3fur. 153rd. a day for 261 days, how far will he have traveled during that time? A. 806m. 1 fur. 303rd. 51. What will 102 bales of cotton cost, each bale weighing 3cwt. 2qr., at $10.62 per cwt.? A. $399.7612+. 52. How many solid feet in a stick of timber 40ft. 9in. long, Ift. 3in. wide, and Ift. 9in. deep? What will it cost at 25cts. a foot? A. 89ft. 243in. nearly; $22.285+. DIVISION OF DECIMALS. LXXIII. 1. We have seen that the factors in multiplication become the divisor and quotient in division. XXI. 53. 2. In multiplication of decimals, the product has as many decimal places as both its factors. 3. In division of decimals, therefore, the divisor and quotient must have as many decimal places as the dividend. 4. Thus .7.5=.35, product: then, .35.5=.7, quotient. 5. The same result will follow from considering the decimals as vulgar fractions. 7. From the above it follows, that when the divisor is a whole number, the quotient alone has as many decimal places as the dividend. 8. And when the divisor alone has as many decimal places as the dividend, the quotient is a whole number. GENERAL RULE. 9. Divide as in whole numbers, and point off figures enough to make the decimal places in the divisor and quotient just equal to those in the dividend, prefixing ciphers to the quotient when necessary to make out the number. LXXIII. Q. How many decimal places has any product? 2. How many have the divisor and quotient? 3. Why? 1. Why is .7 the quotient of .35 divided by.5 ? 6. When does the quotient have as many decimals as the dividend? 7. When is the quotient a whole number? 8. What is the general rule? 9, 10, 11. 10. When the dividend has not so many decimals as the divisor, make them first equal by annexing ciphers, in which case the quotient will be a whole number. 11. When any dividend, either with or without ciphers annexed, as above, does not contain the divisor, or when there is a remainder, annex ciphers, and for every such cipher reckon another decimal place in the quotient. A. 12. What is the quotient of .75 divided by 5? 5). 7 5 1 5 When the divisor is an integer, the quotient has as many decimals as the dividend. (See above, 7.) 13. Divide 87.745 barrels among 7 persons. 14. Divide 9.031455 into 9 equal parts. A. 12.535. A. 1.003495. When the divisor has as many decimals as the dividend, the quotient is a whole number. (See 8.) 15. How many times .108 in .972? 16. How many times .00009 in .00045? A. 9 times. A. 5 times. 17. When 8 barrels of flour cost $80.75, what does one barrel cost? 8) 8 0.7 5 0 0 0 See the rule (11) for annexing ciphers 10.0 9 3 75-$10. 9c. 3,5m. Answer. 18. Divide $50.5 into 16 equal parts. 100 100 A. $3.15c. 6,35m. 19. Divide 150.15 barrels into 32 equal parts. A. 4.6921875 20. At $.5 a yard, how many yards will cost $25? 5) 2 5 0 A. 5 See the rule (10) for annexing ciphers; 0 yd. which makes the quotient a whole number 21. Divide 30 hogsheads by .15 of a hogshead. A. 200. 22. How many times does 60.15 exceed .000005? A. 12,030,000. 23. At .5 of a dollar or 50 cents a yard, how much may be bought for .35 of a dollar or 35 cents? Make the 7 a decimal, then the decimals in .7 and .5 will equal those in .35. (Rule 9.) 24. Divide .192816 by .312. 25. Divide .25008 by .48. 26. What is the quotient of .00025 divided by .25? 25).000 25 A. 0 0 1 A. .618. A. .521. Prefixing 2 ciphers to 1 makes the decimal places in .001 and .25 equal to those in .00025. (Rule 9.) Q. Why is .75÷5.15? 12. Why is .75÷.25-3? 15. Why is 80.75÷ 8=10.09375? 17. Why is .25÷.5=50? 20. Why is .35÷5.7? 23. What is the quotient of 10.8 divided by 12?-10.8 by .12?-108 by 1.2?-.108 by .12?-108 by 12?-1.08 by 1.2?-1.08 by .12?-10.8 by 1.2?-108 by .12?108 by 1.2?--1.08 by 12? |