it require to purchase 47-?T bushels of wheat at $ 2§ per bushel? Ans.,168g85bu. 31. If 15 cords of wood cost $ 57T9X, what cost 1 cord? what cost 191 cords? 32. If 191 cords of wood cost $ 76T6T7iT, how many cords may be obtained for $ 57T9T? 33. At 7T3jj shillings per yard, what cost 47 £ yards? Ans. 17£. 5s. 6fd. 34. When 172£. 15s. Ofd. are paid for 47^ yards of broadcloth, what is the value of 1 yard? Ans. 3£. 12s. llfffd. 35. If l1b. of sugar cost t7tj of a dollar, what is the value of 43f1b.? "Ans. $23.61T73. 36. If 17g1b. of sugar cost $ 2T7T, what cost 501b. Ans. $ 7.58iai|. 37. Bought 87^ yards of broadcloth for $ 612; what was the value of 14^ yards? Ans. $ 102.90. 38. If J 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 f of a hogshead? 41. Paid $ 4.5611} for f 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 44J| days, how long would it require 19 men to perform the same labor? 44. If T\ 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 f*T of a ton? 46. If T75 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 r7g of a tub? 48. If a horse eat 19£ bushels of oats in 87 f days, how many will 7 horses eat in 60 days? Ans. 93J bushels. 49. Henry Smith can reap a field in 10 days, by laboring 8 hours a clay. His son John can reap the same field in 9 day^ by laboring 12 hours a day. How long would it take both to reap the field, provided they labored 8 hours a day? Ans. 5f$- days. DECIMAL FRACTIONS. 259. A Decimal Fraction is a fraction whose denominator is some power often. 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, ,-fij may be written .9 and be read 9 tenths. ".13 " 13 hundredths. ".005 " 5 thousandths. ".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 must 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; fyc. 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. 263i 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 = T90, but .09 = and .009 = Ti&„. 261. 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 = -fa, and .70 = Tvct = T7?! = ,'• 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 Table. £ A II ill Of the mixed number expressed in the table, the part on the left of the decimal point is the whole number, and that on the right the decimal. The decimal part is numerated from the left to the right, and the value represented is expressed in words thus: Two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety-two billionths. And the mixed number thus: Seven million six hundred fifty-four thousand three hundred twenty-one, and decimal two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety-two billionths. 267. From the table we deduce the following rules: 1. Read a decimal as though it were a whole number, adding the name of the right-hand order. 2. Write a decimal as though it were a whole number, supplying with ciphers such places as have no significant figures. Examples. Express orally, or write in words the following numbers: — 25. Three hundred twenty-five, and seven tenths. 26. Four hundred sixty-five, and fourteen hundredths. 27. Ninety-three, and seven hundredths. 28. Twenty-four, and nine millionths. 29. Two hundred twenty-one, and nine hundred-thousandths. 30. Forty-nine thousand, and forty-nine thousandths. 31. Seventy-nine million two thousand, and one hundred five thousandths. 32. Sixty-nine thousand fifteen, and fifteen hundred-thousandths. 33. Eighty thousand, and eighty-three ten-thousandths. 34. Nine billion nineteen thousand nineteen, and nineteen hundredths. 35. Twenty-seven, and nine hundred twenty-seven thousandths. 36. Forty-nine trillion, and one trillionth. 37. Twenty-one, and one ten-thousandth. 38. Eighty-seven thousand, and eighty-seven millionths. 39. Ninety-nine thousand ninety-nine, and nine thousand nine billionths. 40. Seventeen, and one hundred seventeen ten-thousandths. 41. Thirty-three, and thirty-three hundredths. 42. Forty-seven thousand, and twenty-nine ten-millionths. 43. Fifteen, and four thousand seven hundred-thousandths. 44. Eleven thousand, and eleven hundredths. 45. Seventeen, and eighty-one quadrillionths. 46. Nine, and fifty-seven trillionths. 47. Sixty-nine thousand, and three hundred forty-nine thou, sandths. 268. Decimals, since they increase from right to left, and decrease from left to right, by the scale of ten, as do simple whole numbers, may be added, subtracted, multiplied, and divided in the same manner. ADDITION OF DECIMALS. 269. Ex. 1. Add together 23.61, 161.5, 2.6789, and 61.111. Ans. 248.8999. We write the numbers so that figures of the same decimal place shall stand in the same column, and then, beginning at the right hand, add them as whole numbers are added, and place the decimal point in the result directly under those above. Rule. — Write the numbers so that figures of the same decimal place shall stand in the same column. Add as in whole numbers, and point off in the sum, from the right hand, as many places for decimals as equal the greatest number of decimal places in any of the numbers added. 2 3.6 1 |