EXAMPLES TO BE PERFORMED BY ANALYSIS. 1. If / of a bushel of corn cost 63 cents, what cost a bushel ? what cost 15 bushels ? Ans. $ 10.80. ILLUSTRATION. — If 7 eighths of a bushel cost 63 cents. 1 eighth will cost 1 seventh of 63 cents 9 cents; and 8 eighths will cost 8 times 9 cents 72 cents, and 15 bushels will cost 15 times 72 cents $ 10.80. 2. If 47lb. of pepper cost $ 2.15, what cost 1 pound? what cost 30lb. ? Ans. $ 13.50. ILLUSTRATION. — In 47lb. there are 43lb. Then, if 43 ninths lb. cost $ 2.15, 1 ninth will cost 1 forty-third of $ 2.15 $ 0.05, and 9 ninths or llb. will cost 9 times $ 0.05 - $ 0.45, and 30lb. will cost 30 times $ 0.45 $ 13.50. 3. When $ 1728 are paid for 3045 tons of iron, what cost 1 ton? what cost 7 37 tons ? Ans. $ 432. 4. When $ 432 are paid for 737 tons of iron, what quantity should be received for $ 1728? 5. For 737 tons of iron there were paid $ 432; what sum will it require to pay for 304 tons ? 6. For 304 tons of iron $ 1728 were paid ; what quantity should be received for $ 432 ? 7. Gave 7 10 bushels of rye for a barrel of flour; how much rye will it then require to purchase 64 barrels of flour? Ans. 494 bushels. 8. Divide $ 1728 among 17 boys and 15 girls, and give each boy 11 as much as a girl ; what sum will each receive ? Ans. Each girl, $ 6691 ; each boy $ 424 9. If [ of a ton of hay cost $14.49, what cost 43 tons ? 10. If 43 tons of hay cost $ 82.50%, what part of a ton will $14.49 buy? 11. If $ 14.49 will buy / of a ton of hay, how much hay can be obtained for $ 82.503 ? 12. When $ 82.50; are paid for 4 tons of hay, what will be the cost of } of a ton ? 13. When 147 tons of copperas are sold for $ 500, what is the value of 1 ton ? what is the value of 9 11 tons ? 14. When 911 tons of copperas are sold for $ 333.33], what is the value of 147 tons ? 15. Gave $333.334 for 911 tons of copperas ; what quantity of copperas should be received for $ 500 ? 16. For 143 tons of copperas $ 500 were paid ; how much might be purchased for $ 333.331? 17. Purchased 977 gallons of molasses for $31.32; what cost 1 gallon ? what cost 763 gallons ? 18. Sold 763ž gallons of molasses for $ 244.36; what should I receive for 97% gallons ? 19. If $244.36 will buy 7633 gallons of molasses, what quantity can be obtained for $ 31.32 ? 20. Gave 1975lb. of flax for 40 barrels of four; how many pounds were given for 1 barrel ? how many pounds would it require to buy 144 barrels ? Ans. 7110lb. 21. If 17 bushels of rye cost $ 15.75, what cost 1 bushel ? what cost 94 bushels ? Ans. $ 8.5663. 22. If 9 barrels of flour cost $ 504, what cost 1 barrel? what cost 87} barrels ? Ans. $ 4925 23. If 13 boarders consume a barrel of pork in 78 days, how long would it last, if 7 more boarders were added to their number? Ans. 5070 days. 24. If a man by laboring 10 hours a day can, in 9 days, perform a certain piece of work, how many days would it require to do the same work, were he to labor 15 hours a day? 25. If a man, by laboring 15 hours a day, in 6 days can perform a certain piece of work, how many days would it require to do the same work by laboring 10 hours a day? 26. If a man, by laboring 10 hours a day, can in 9 days perform a certain piece of work, how many hours must he labor each day to perform the same work in 6 days ? 27. Sold 17 11 bushels of corn for $5%; what was received for 1 bushel ? what should I have charged for 97ş bushels ? Ans. $ 30315 28. Bought 93 tons of hay at $ 197 per ton; for what must it be sold per cwt. to gain $ 7 on my bargain ? Ans. $13730 29. If I sell hay at $ 14 per cwt., what should I give for 9 tons, that I may make $ 7 on my bargain ? Ans. $ 329. 30. How many bushels of corn at $ 0.75 per bushel will it require to purchase 47 & bushels of wheat at $2 per bushel ? Ans. 1689.bu. 31. If 15 cords of wood cost $ 57 11, what cost 1 cord? what cost 197 cords? 32. If 198 cords of wood cost $76-67, how many cords may be obtained for $ 578 ? 33. At 7 shillings per yard, what cost 474 yards ? Ans. 17£. 5s. 6 d. 34. When 172£. 15s. Oşd. are paid for 47} yards of broadcloth, what is the value of 1 yard ? Ans. 3£. 12s. 1133;d. 35. If ilb. of sugar cost 13 of a dollar, what is the value of 439 lb.? Ans. $23.6173 36. If 173lb. of sugar cost $ 211, what cost 50lb. Ans. $ 7.581919. 37. Bought 874 yards of broadcloth for $ 612; what was the value of 1410 yards ? Ans. $ 102.90. 38. If [ of an acre of land cost $ 43.75, what cost 10 acres ? 39. When $ 500 are paid for 10 acres of land, how much might be obtained for $ 43.75 ? 40. If 9 hogsheads of sugar cost $ 71.87, what cost of a hogshead? 41. Paid $ 4.56%: for 4 of a hogshead of sugar; what ought to be given for 9 hogsheads ? 42. If 19 men can grade a certain road in 111 days, how long would it require 47 men to perform the same labor ? 43. When 47 men can grade a certain road in 4444 days, how long would it require 19 men to perform the same labor ? 44. If t1 of a ton of hay cost $ 9.20, what cost 17 tons ? 45. When $ 430.10 are paid for 17 tons of hay, what cost of a ton ? 46. If io of a tub of butter cost $ 7.15, what cost 7 tubs ? 47. When $ 114.40 are paid for 7 tubs of butter, what cost To of a tub ? 48. If a horse eat 19% bushels of oats in 874 days, how many will 7 horses eat in 60 days? Ans. 93} bushels. 49. Henry Smith can reap a field in 10 days, by laboring 8 hours a day. His son John can reap the same field in 9 days, by laboring 12 hours a day. How long would it take both to reap the field, provided they labored 8 hours a day? Ans. 547 days. DECIMAL FRACTIONS. Toso 259. A DECIMAL FRACTION is a fraction whose denominator is some power of ten. It, therefore, originates from dividing a unit, first into 10 equal parts, and then each of these parts into 10 other equal parts, and so on indefinitely, so that its fractional units are tenths, hundredths, thousandths, or some like order of parts. 260. Decimal fractions, their denominators being obvious, are commonly expressed by writing the numerator only, with the decimal point (.) before it, care being taken to put a cipher in any decimal place not requiring a digit ; thus, 19 may be written .9 and be read 9 tenths. 10% .13 13 hundredths. .005 5 thousandths. 1080 .0105 105 ten-thousandths. 261. By examining the foregoing fractions, it will be seen that, 1. The denominator of a decimal fraction is 1 with as many ciphers annexed as the numerator has places of figures. 2. In writing a decimal fraction without its denominator, every decimal place not having a significant figure mus be filled by a cipher. 3. The first figure or place of a decimal fraction on the right of the decimal point is tenths ; the second, hundredths ; the third, thousandths ; the fourth, ten-thousandths ; &c. 4. Each figure in the expression of decimal fractions, as in whole numbers, represents value, according to its distance from. the place of units. 262. A whole number and a decimal fraction, in a single expression, constitute a mixed number. Thus, 17.63 is a mixed number, and is read seventeen, and decimal sixty-three hundredths ; 150.302, read one hundred and fifty, and decimal three hundred and two thousandths. NOTE. — For the sake of brevity, especially in reading mixed numbers, as in the instance just given, a decimal fraction is commonly called simply a decimal. 10, 263. If ciphers are placed on the left of decimal figures, between them and the decimal point, those figures change their places, each cipher removing them one place to the right, and thus diminishing the value represented tenfold. Thus,.9 = 30 but .09 T&o, and .009 Ισσσ: : 264. If ciphers are placed on the right of decimal figures, or are taken away, since their places remain the same, the value represented is not changed. Thus, .7 = 1o, and .70 = 70% To = .7. 265. By regarding the dollar as a unit, we may consider cents and mills of United States money as fractional parts of a decimal character. Thus, 3 dollars and 25 cents, is 3 dollars and 25 hundredths of a dollar, or $ 3.25; also, 10 dollars 12 cents and 5 mills, is 10 dollars and 125 thousandths of a dollar, or $ 10.125. NOTATION AND NUMERATION OF DECIMALS. 266. The relation of decimals to whole numbers and to each other, and also the names of their different orders and places, are shown by the following |