19. If four times a number is subtracted from seven times the same number, the remainder will be 72. What is the number?

20. A traveller first rides his bicycle 9 miles an hour. He then rides the same number of hours in a car 35 miles an hour. If he travels 132 miles, how many hours did he ride his bicycle?

After

21. Two trains run out of New York in opposite directions. One runs 42 miles an hour, the other 34 miles an hour. how many hours will they be 228 miles apart?

22. Two trains run out of New York in the same direction. One runs 38 miles an hour, the other 34 miles an hour. After how many hours will they be 32 miles apart?

23. A boy has 75 cents in dimes and five-cent pieces. He has the same number of dimes as of five-cent pieces. How many coins of each kind has he?

24. A owes B $40. He pays his debt in ten-dollar bills, and receives in change the same number of two-dollar bills. How many ten-dollar bills did A pay B?

25. A cistern has two pipes. One lets in 8 gallons a minute, and the other 15 gallons a minute. If the cistern holds 207 gallons, how many minutes will it take the pipes to fill it?

26. A cistern has two pipes. One lets in 11 gallons a minute, and the other lets out 6 gallons a minute. How many minutes will it take the one pipe to let in 85 gallons more than the other lets out?

27. What is the sum of x, four times x, and seven times ? Of x, twice x, and five times x ?

28. The sum of a certain number, four times the number, and seven times the number is 96. What is the number?

29. Three boys, A, B, and C, together have 21 pencils. B has twice as many as A, and C four times as many as A. How many has A? How many has each ?

30. Divide 147 into three parts, so that the second part shall be four times the first, and the third part twice the first.

31. A merchant receives $ 64 in ten-dollar bills, five-dollar bills, and one-dollar bills. He receives the same number of each kind. How many of each does he receive?

32. At an election 726 votes were cast. A, B, and C were candidates. B received three times as many votes as C, and A twice as many as C. How many votes did each receive?

33. A cistern has three pipes. The first lets in 6 gallons a minute, the second 9 gallons a minute, and the third 12 gallons a minute. If the cistern holds 243 gallons, how long will it take the pipes to fill it?

34. A cistern has three pipes. The first lets in 5 gallons a minute, the second 14 gallons a minute, and the third lets out 10 gallons a minute. How many minutes will it take the two pipes to let in 108 gallons more than the third pipe lets out?

35. An estate of $9600 is divided among 2 sons and 2 daughters. The sons receive equal amounts, and a daughter receives three times as much as a son. How many dollars

does each receive?

36. What is twice 3 x? Seven times 5 x ? Four times 9 x ?

37. A receives x dollars, B receives three times as much as A, and C receives twice as much as B. How many dollars does C receive? How many dollars do all receive?

38. Three boys, A, B, and C, together receive $70. B receives three times as much as A, and C twice as much as B. How many dollars does each receive?

39. A merchant's profits doubled each year for three years. If his profits for the three years were $8750, what were his profits the first year?

40. In a company are 50 persons. The number of women is three times the number of men, and the number of children is twice the number of women. How many of each are in the

company?

41. What number is of x? of x?

42. If of a number is 16, what is the number?

43. The less of two numbers is of the greater. greater is x, what is the less? What is their sum?

difference?

If the

Their

44. The less of two numbers is of the greater. If their sum is 91, what are the numbers?

· 45. A and B together have $ 1133. If B has as much as A, how many dollars has each ?

46. A has $31 more than B. If B has as much as A, how many dollars has each ?

47. Two boys, A and B, catch 36 fish. If A catches as many as B, how many fish does each catch?

48. A workman pays of his wages for board. If he has left $8 each week, what are his wages?

49. Two boys together solve 65 problems. If the first solves as many as the second, how many problems does each solve?

50. A solves 21 more problems than B. If B solves as many as A, how many problems does each solve?

51. A tree 126 feet high is broken by the wind. If the part left standing is of the part broken off, how long is each part?

3 11

52. What is the sum of of x and of x? The difference? 53. If of a number is added to 3 of the same number, the sum will be 39. What is the number?

54. If of a number is subtracted from 3 of the same number, the remainder will be 3. What is the number?

55. If to a number is added of itself and of itself, the sum will be 50. What is the number?

56. Three boys, A, B, and C, together have 29 pencils. B has as many as A, and C has as many as A. How many pencils has each ?

57. Divide 104 into three parts, so that the first shall be three times the second, and the third of the second.

58. A man makes a journey of 69 miles. He goes by boat as by train, and as far by stage as by train. many miles does he go by each conveyance?

as far How

59. What is of three times x? Twice of x? of 1⁄2 of x?

60. The second of three numbers is three times the first, and the third is of the second. If the first number is x, what What is the sum of the three

is the second?

numbers?

The third?

61. The sum of three numbers is 99. The second is four times the first, and the third is of the second.

the numbers?

What are

62. The width of a field is of its length, and the distance around it is 88 rods. What is the width and the length of the field?

63. The sum of $420 is divided among A, B, and C. B receives as much as A, and C as much as A and B together. How many dollars does each receive?

64. A sells a number of apples at 2 cents apiece, and B sells as many at 3 cents apiece. If they receive together 87 cents, how many apples does each sell?

Parentheses.

23. Parentheses, (), and Brackets, [], are used to indicate that whatever is placed within them is to be treated as a whole.

E.g., 10(25) means that the result of adding 5 to 2, or 7, is to be subtracted from 10; that is,

But 10 2+5 means that 2 is to be subtracted from 10 and 5 is then to be added to that result; that is,

102 + 5 = 8 + 5 = 13.

In like manner, [27 − (3 + 2) × 5]÷2 means that the result of multiplying the sum 3+2 by 5 is first to be subtracted from 27, and the remainder is then to be divided by 2; that is,

[27 − (3 + 2) × 5] ÷ 2 = [27 — 25] ÷ 2 = 2 ÷ 2 = 1. Likewise, (a+b)c is the result of multiplying a+b by c, etc.

EXERCISES VI.

Find the value of each of the following expressions:

When a = 12, b = 6, c=3, find the values of:

30. a[a−(b+c)]. 31. [a—(b−c)]÷c. 32. [b+(a−c)]÷c.

POSITIVE AND NEGATIVE NUMBERS, OR ALGEBRAIC

NUMBERS.

24. In ordinary Arithmetic we subtract a number from an

equal or a greater number.

We are familiar with such opera