30. How many times can a bottle holding gallon, be filled from a demijohn containing gallons? of } of a of 1 Ans. 7. 31. I bought of 7 cords of wood for of $32. What did 1 cord cost? 32. A man bought 728 pounds of candles at 163 cents a pound. Had they been purchased for 37 cents less a pound, how many pounds could have been bought for the same money? Ans. 95317. 33. What number, divided by 13, will give a quotient. of 91? Ans. 128. 34. The product of two numbers is 6, and one of them is 1846. What is the other? Ans. 3 35. A stone mason worked 113 days, and after paying his board and other expenses with of his earnings, he had $20 left. How much did he receive a day? 36. If of 4 tons of coal cost $51, what will tons cost? of 2 Ans. $5. 37. A man gave 6 pounds of butter, at 12 cents a pound, for of a gallon of oil. What was the oil worth a gallon? Ans. 100 cents. 38. In an orchard of the trees are apple trees, peach trees, and the remainder are pear trees, which are 20 more than of the whole. How many trees are there in the orchard? Ans. 800. acres of land, sold of What was the value of 39. A man, having 271 it, and gave of it to his son. the remainder, at $574 per acre? Ans. $45773 80⚫ 40. A horse and a wagon cost $270; the horse cost 11 times as much as the wagon. What was the cost of the wagon? 41. What number taken from 2 times 12 will leave 203? Ans. 114. 42. A merchant bought a cargo of flour for $21731, and sold it for 2 of the cost, thereby losing of a dollar per barrel. How many barrels did he purchase? Ans. 126. 43. A and B can do a piece of work in 14 days; do as much as B. In how many days can each do it? Ans. A in 32 days; B in 24 days. yards of cloth of a yard wide, are of a yard wide? Ans. 111. 44. How many equal to 12 yards 45. A, B, and C can do a piece of work in 5 days; B and C can do it in 8 days. In what time can A do it? 46. A man's family expenses are $24651 a year, which is of his income. What does he save? 47. A man put his money into 4 packages; in the first he put, in the second, in the third, and in the fourth the remainder, which was $24 more than of the whole. How much money had he? Ans. $720. 48. A man bequeathed to his son $35,000, which was of what he left his wife. How much did he leave his wife? cords of wood, how many cords Ans. 44 cords. 50. How many times is of of 27 contained in 7 of of 423 ? 49. If $74 will buy 3 can be bought for $10!? 51. A boy lost of his kite string, and then added 30 feet, when it was just of its original length. What was the length at first? Ans. 100 feet. 52. A man bought of a box of candles, and having used of them, sold the remainder for 1 of a dollar. How much would a box cost at the same rate? Ans. $573 53. A post stands in the mud, in the water, and 21 feet above the water. What is its length? 54. A father left his elder son of his estate, his younger son of the remainder, and his daughter the remainder. She received $1723§ less than the youngest son. What was the value of the estate? Ans. $211141. DECIMAL FRACTIONS. 143. Decimal Fractions are fractions which have for their denominator 10, 100, 1000, or 1 with any number of ciphers annexed, and are generally written like the orders of integers. 1. The word decimal is derived from the Latin decem, which signifies ten. 2. Decimal fractions are commonly called decimals. 3. Since 100, 180 = 1880, etc., the denominators of decimal fractions increase and decrease in a tenfold ratio, in the same way as simple numbers. = DECIMAL NOTATION AND NUMERATION. 144. Decimal Fractions are the decimal divisions of a unit into tenths, hundredths, etc., just as common fractions are the common divisions of a unit into any number of equal parts, as halves, fifths, etc. Thus, a unit is divided into ten equal parts, called tenths; each of these tenths is divided into ten other equal parts, called hundredths; each of these hundredths into ten other equal parts, called thousandths; and so on. Since the denominators of decimal fractions increase and decrease by the scale of ten, the same as in simple numbers, in writing decimals the denominators may be omitted. In simple numbers, the unit, 1, is the starting point of notation and numeration; and so also is it in decimals. We extend the scale of notation to the left of units' place in writing integers, and to the right of units' place in writing decimals. The first place at the left of units is tens, and the first place at the right of units is tenths; the second place at the left is hundreds, and the second place at the right is hundredths; the third place at the left is thousands, and the third place at the right is thousandths; and so on. The Decimal Point is a period (.), which must always be placed before or at the left hand of the decimal. Thus, place is left hand place. The decimal point is also called the Separatrix. it only when it stands between the integral and number. .5 is 5 tenths, .05 is 5 hundredths, .005 is 5 thousandths, 1 is expressed .6 6 To which =116 of 5 units; 66 of 5 tenths; = 10 of 5 hundredths. 10 Universally, the value of a figure in any decimal the value of the same figure in the next Hundred-thousands. Ten-thousands. 7 6 The relation of decimals and integers to each other is clearly shown by the following table : NUMERATION TABLE. INTEGERS. 66 = This is a correct name for decimal parts of the same 5 6 7 8 9 DECIMALS. By examining this table we see that Tenths are expressed by one figure. Hundredths are expressed by two figures. Thousandths are expressed by three figures. Ten thousandths are expressed by four figures. And any order of decimals is expressed by one figure less than the corresponding order of integers. 145. Since the denominator of tenths is 10, of hundredths 100, of thousandths 1000, and so on, a decimal may be expressed by writing the numerator only; but in this case the numerator or decimal must always contain as many decimal places as are equal to the number of ciphers in the denominator; and the denominator of a decimal will always be the unit, 1, with as many ciphers annexed as are equal to the number of figures in the decimal or numerator. The decimal point must never be omitted. EXAMPLES. 1. Express in figures thirty-eight hundredths. 3. Write three hundred twenty-five thousandths. 4. Write four hundredths. Ans. .04. 5. Write sixteen thousandths. 6. Write seventy-four hundred-thousandths. Ans. .00074. 7. Write seven hundred forty-five millionths. 8. Write four thousand two hundred thirty-two tenthousandths. 9. Write five hundred thousand millionths. |