37. Thirty yards of cloth and 20 yards of silk together cost $70; and the silk costs twice as much per yard as the cloth. How much does each cost per yard?

38. In a company of 180 persons, composed of men, women, and children, there are twice as many men as women, and three times as many women as children. How many are there of each?

39. Two trains traveling, one at 25 and the other at 30 miles an hour, start at the same time from two places 330 miles apart, and move toward each other. In how many hours will the trains meet?

40. Twelve persons subscribed for a new boat, but two being unable to pay, each of the others had to pay $4 more than his share. Find the cost of the boat.

41. A tree 84 feet high was broken so that the part broken off was five times the length of the part left standing. Required the length of each part.

42. At an election there were two candidates, and 2800 votes were cast. The successful candidate had a majority of 160. How many votes were cast for each?

43. Divide 20 into two parts such that four times the greater exceeds three times the smaller by 17.

44. The sum of two numbers is 50, and seven times the smaller number exceeds three times the greater number by 10. Find the numbers.

45. Divide 19 into two such parts that twice the smaller part exceeds the greater by two.

46. Three times the excess of a certain number over 6 is equal to the number plus 18. Find the number.

47. Thirty-one times a number exceeds 80 by as much as nine times the number is less than 80. Find the number.

48. Find the number whose double diminished by 3 ex. ceeds 80 by as much as the number itself is less than 100. 49. Divide 19 into two parts such that the greater part exceeds twice the smaller part by 1 less than twice the smaller part.

50. A man is now twice as old as his son; 20 years ago he was four times as old as his son. Find the age of each. 51. A man is four times as old as his son; in 20 years he will be only twice as old. Find the age of each.

52. A man was four times as old as his son 7 years ago, and will be only twice as old as his son 7 years hence. Find the age of each.

53. A man has 8 hours for an excursion. How far can he ride into the country in a carriage that goes at the rate of 9 miles an hour so as to return in time, walking back at the rate of 3 miles an hour?

54. A man was hired for 26 days. Every day he worked he was to receive $3, and every day he was idle he was to pay $1 for his board. At the end of the time he received $62. How many days did he work?

55. A, walking 4 miles an hour, starts two hours after B, who walks 3 miles an hour. How many miles must A walk to overtake B?

56. A river runs 1 mile an hour. A man swims a certain distance up the river in 3 hours, and the same distance down in 1 hour. Find his rate of swimming in still water.

57. A man bought 12 yards of velvet. If he had bought 1 yard less for the same money, each yard would have cost $1 more. What did the velvet cost a yard?

58. A and B have together $8; A and C, $10; B and C, $12. How much has each?

59. I have in mind a certain number. If this number is diminished by 8 and the remainder multiplied by 8, the result is the same as if the number were diminished by 6 and the remainder multiplied by 6. What is the number?

60. A man having only ten-cent pieces and five-cent pieces wished to give some children 15 cents. each, but found that he had not money enough by 25 cents; he, therefore, gave them 10 cents each and had 30 cents left. How many children were there?

61. A sum of money was divided among A, B, and C in such a way that A received three times as much as B, and B twice as much as C. If A received $6 more than C, how much did each receive?

62. The sum of the ages of a man and his son is 80 years; and the father's age is 2 years more than twice the age of his son. What is the age of each?

63. Two casks contain equal quantities of vinegar. From one cask 37 gallons are drawn, and from the other 7 gallons are drawn. The quantity now remaining in one cask is 7 times that remaining in the other. How much did each cask contain at first?

64. A merchant has two kinds of tea; one worth 50 cents a pound, and the other 75 cents a pound. He makes a mixture from these of 100 pounds, worth 60 cents a pound. How many pounds of each kind does he take?

65. A had $7 and B had $5. B gave A a certain sum; then A had 3 times as much as B. How many dollars did B give A?

66. A boy bought 9 dozen oranges for $2.50. For a part he paid 25 cents a dozen, and for the remainder 30 cents a dozen. How many dozen of each kind did he buy?

NOTE. In the following examples express in cents all money values. 67. How can $2.25 be paid in quarters and ten-cent pieces so as to pay twice as many ten-cent pieces as quarters?

68. I have $1.80 in ten-cent pieces and five-cent pieces, and have four times as many five-cent pieces as ten-cent pieces. How many have I of each?

69. I have $6 in silver half-dollars and quarters, and I have 20 coins in all. How many have I of each?

70. I have five times as many half-dollars as quarters, and the half-dollars and quarters amount to $11. How many have I of each?

71. A man has $65 in ten-dollar bills and one-dollar bills. He has three times as many one-dollar bills as tendollar bills. How many bills has he of each kind?

72. A sum of money is divided among three persons, A, B, and C, in such a way that A and B together have $6, A and C $6.50, and B and C $7.50. How much has each?

73. A purse contains 27 coins which amount to $11.25. There is a certain number of silver dollars, and three times as many half-dollars as dollars; the remaining coins are quarters. Find the number of each.

74. A man bought 10 yards of calico and 20 yards of cloth for $30.60. The cloth cost as many quarters per yard as the calico cost cents per yard. Find the price of each per yard.

75. A man has a certain number of dollars, half-dollars, and quarters. The number of quarters is twice the number of half-dollars and four times the number of dollars. If he has $15, how many coins of each kind has he?

CHAPTER III.

POSITIVE AND NEGATIVE NUMBERS.

61. Positive and Negative Quantities. If a person is engaged in trade, his capital will be increased by his gains, and diminished by his losses.

Increase in temperature is measured by the number of degrees the mercury rises in a thermometer, and decrease in temperature by the number of degrees the mercury falls.

In considering any quantity whatever, a quantity that increases the quantity under consideration is called a positive quantity; and a quantity that decreases the quantity under consideration is called a negative quantity.

62. The Natural Series of Numbers. If from a given point, marked 0, we draw a straight line to the right, and beginning from the zero point lay off units of length on this line, the successive repetitions of the unit will be expressed by the natural series of numbers, 1, 2, 3, 4, etc. Thus:

0

1 2 34 5 67 8 9 10 11

In this series if we wish to add 2 to 5, we begin at 5, count 2 units forwards, and arrive at 7. If we wish to subtract 2 from 5, we begin at 5, count 2 units backwards, and arrive at 3. If we wish to subtract 5 from 5, we count 5 units backwards from 5, and arrive at 0. If we wish to subtract 5 from 2, we cannot do it, because when we have counted backwards from 2 as far as 0, the natural series of numbers comes to an end.