14. In a fire B lost twice as much as A, and C lost 3 times as much as A. If their combined loss was $300, what was the loss of each?

15. A house and lot cost $3000.

If the house cost 4 times as

much as the lot, what was the cost of each?

16. In a business enterprise the joint capital of A, B, and C was $2100. If A's capital was twice B's, and B's was twice C's, what was the capital of each?

17. John, William, and George together had 120 marbles. If William had twice as many as John, and George had 3 times as many as John, how many had each ?

18. In an orchard of apple, pear, and cherry trees, containing 1690 trees in all, there were 4 times as many cherry trees as pear trees, and twice as many apple trees as cherry trees. How many trees were there of each kind?

19. A number plus itself, plus twice itself, plus 4 times itself, is equal to 72. What is the number?

20. Charles is twice as old as his younger brother, and half as old as his older brother. If the sum of the ages of the three brothers is 28 years, what is the age of each?

21. A farmer had twice as many sheep as horses, and twice as many hogs as sheep and horses together. If there were in all 360 animals, how many were there of each kind?

22. A tract of land containing 640 acres was divided into three farms, such that the first was 3 times as large as the second, and the third 4 times as large as the first. How many acres did each farm contain?

23. Three boys divided 160 marbles among themselves so that one of them received twice as many as each of the others. How many did each receive?

24. Divide 30 into two parts, one of which is 14 times the other.

25. Divide 18 into three parts, such that the first is twice the third, and the second is 3 times the third.

26. Divide 21 into three parts, such that the first is twice the second, and the second is twice the third.

27. Divide 36 into three parts, such that the first is twice the second, and the third is equal to twice the sum of the first and second.

28. Three newsboys sold 60 papers. If the first sold twice as many as the second, and the third sold 3 times as many as the second, how many did each sell?

29. Henry earned a certain number of dollars per week. With 4 weeks' earnings he purchased a rifle, and with 20 weeks' earnings, a bicycle. If both together cost $72, how much did he earn per week? How much did the rifle cost? the bicycle?

30. A man sold some ducks for 50 cents each, and the same number of geese for 75 cents each. If for all he received $12.50, how many of each did he sell?

31. John has 5 times as much money as James. James has 24 cents less than John. How much has each?

32. A man had 675 sheep in three fields. If there were twice as many in the first field as in the second, and twice as many in the third field as in both of the others, how many sheep were there in each field?

33. A man bequeathed to his daughter twice as much money as to his son, and to his wife 3 times as much as to his daughter. If all together received $9000, how much did each receive?

34. A plumber and two helpers together earned $7.50 per day. How much did each earn per day, if the plumber earned 4 times as much as each helper ?

35. What number added to of itself equals 20?

36. If of a number is added to the number, the sum is 12. What is the number?

37. If of a number is added to twice the number, the sum is 35. What is the number?

38. The difference between 4 times a certain number and of the number is 30. What is the number?

39. The difference between of a certain number and of it is 16. What is the number?

40. After spending

$30.

of my money and losing of it, I had How much had I at first?

41. The difference between twice a certain number and of it is 20. What is the number?

42. The number 150 which is of the other. 43. One part of 45 is

can be divided into two parts, one of What are the parts?

of the other.

What are the parts?

44. Find two parts of 30 such that one is of the other.

45. To A, B, and C I owe in all $93. If I owe A ? as much as C, and B as much as C, how much do I owe each?

46. The length of a field is 13 times its width, and the distance around the field is 120 rods. If the field is rectangular, what are its dimensions?

47. A, B, C, and D buy $16,000 worth of railroad stock. How much does A take, if B takes 3 times as much as A, C twice as much as A and B together, and D as much as A, B, and C together?

48. In one season an orchard bore 650 bushels of fruit, consisting of as many bushels of pears as of peaches, and twice as many bushels of apples as of pears. How many bushels were there of each?

49. A horse, harness, and carriage cost $340. If the horse cost 3 times as much as the harness, and the carriage cost 14 times as much as the horse, what was the cost of each?

DEFINITIONS AND NOTATION

6. The ideas of number and the knowledge of the processes with abstract numbers that the student has gained in arithmetic are a proper and necessary introduction to his work in algebra; but since number is discussed in a more general way in algebra than in arithmetic, many arithmetical processes, terms, and symbols, as addition,' 'subtraction,' 'greater,' 'less,' 'exponent,' '+,' '-,' etc., must now be extended in meaning and application.

6

For example, in an arithmetical sense 8 cannot be subtracted from 5, nor does 8 have any meaning; but in an algebraic sense, as will be shown hereafter, 8 can be subtracted from 5 and 83 is as intelligible as 82.

Indeed, the processes and principles of arithmetic are but special cases of the more fundamental processes and principles of algebra.

7. A unit or an aggregate of units is called a Whole Number, or an Integer.

One of the equal parts of a unit or an aggregate of equal parts of a unit is called a Fractional Number.

Such numbers are called Arithmetical, or Absolute Numbers.

8. Arithmetical numbers have fixed and known values, and are represented by symbols called numerals; as 1, 2, 3, etc., Arabic figures, and I, V, X, etc., Roman letters.

9. It is often convenient, in solving a problem, to employ letters, such as x, y, z, to represent the numbers whose values are sought; and, in stating a rule, to employ letters to represent the numbers that must be given whenever the rule is applied.

Numbers represented by letters are called Literal Numbers.

For example, the volume of any rectangular prism is equal to the area of the base multiplied by the height. By using v for volume, a for area of base, and h for height, this rule is abbreviated

In each problem to which this rule applies a and h represent fixed, known values, but in consequence of being used for all problems of this class, a and h represent numbers to which any arithmetical values whatever may be assigned. Hence, the arithmetical idea of number is extended as follows.

10. A literal number to which any value can be assigned at pleasure is called a General Number.

11. A number whose value is known or a number to which any value can be assigned is called a Known Number.

The numerals, 3 and 43, and the general numbers a and h in v = a × h, in § 9, are known numbers.

Known literal numbers are generally represented by the first letters of the alphabet.

12. A number whose value is to be found is called an Unknown Number.

Unknown numbers are usually represented by the last letters of the alphabet.

ALGEBRAIC SIGNS

13. The Sign of Addition is +, read ‘plus.'

It indicates that the number following it is to be added to the

number preceding it.

a + b, read ‘a plus b,' indicates that b is to be added to a.

14. The Sign of Subtraction is read minus.'

It indicates that the number following it is to be subtracted

from the number preceding it.

a — b, read a minus b,' indicates that b is to be subtracted from a.