16. If a man's daily wage had been $1 less, it would have taken him 15 days longer to earn $180. How many days did he work to earn $180?

17. A picture 10" x 14" is placed in a frame of uniform width. If the area of the frame is equal to half the area of the picture, how wide is the frame?

Draw a figure.

18. Find two consecutive even numbers whose product is 528.

19. Find two consecutive odd numbers whose product is 8099.

20. If the difference between the parallel sides of a trapezoid is 5 ft., and the altitude of the trapezoid is equal to the longer of the parallel sides, find the lengths of the parallel sides when the area is 2375 sq. ft.

21. A flower bed is 15' x 20'.

How wide a walk must surround the bed, to increase the total area by 770 sq. ft.?

22. A tinner makes a square box 3 in. deep, with a capacity From each corner of a square sheet of tin a cut and the four rectangular parts of the tin What are the dimensions of the square sheet

of 1587 cu. in. 3-inch square is are turned up. of tin?

Draw a figure.

23. If a square has its length reduced by 7 in. and its width by 10 in., what are the linear dimensions of the resulting rectangle, if its area is 8370 sq. in. ?

24. An oil tank can be filled by one pipe in 2 hours less time than by another pipe. If both pipes are open 1 hours, the tank will be filled. In what time can the tank be filled by each pipe?

25. A number of postage stamps can be arranged in a rectangle, each side containing 60 stamps. If the same number

of stamps be arranged in two rectangles so that each side of one rectangle will contain 12 more stamps than each side of the other, how many stamps does a side of each of the latter rectangles contain?

26. A boat's crew can row at the rate of 9 miles an hour. What is the speed of the current in the river if it takes them 2 hours and 15 minutes to row 9 miles up stream and back?

27. Divide $ 1248 among three persons, so that the second shail have $ 3 more than the first, and the third shall have as many times the share of the second as there are dimes in the first person's share.

28. The population of a city increases from 20,000 to 20,808 in two years. What is the annual rate of increase per hundred?

29. A sum of $2000 drawing interest that is compounded annually, amounts to $ 2142.45 in two years. Find the rate of interest.

CHAPTER II

MORE ADVANCED THEORY AND OPERATIONS

FUNDAMENTAL LAWS OF ALGEBRA

76. The operations of algebra obey certain fundamental laws which we have not formulated thus far. Nevertheless we have so accustomed ourselves to follow them, that we find it difficult to see how a new algebra might be made, in which a different set of laws would prevail. We shall now explain the laws which underlie our algebra.

If several positive and negative numbers are added or subtracted, it matters not in what order the operations are performed; the numbers may be commuted at pleasure.

This is called the commutative law for addition.* ters, the law may be stated thus,

a+b= b + a.

Using let

* In our algebra, addition and subtraction may be represented geometrically by the addition and subtraction of distances along a straight line. Let a and b represent distances measured off toward

the right, then a+b and ba both represent the same distance OC in Fig. 3; the O

a

C

a

FIG. 3.

above; suppose a means a rotation about the line OA as an axis, through 90 degrees, and b means a rotation about OB as an axis, through 90 degrees.

That is, the final result is the same, whether the 9 be assoIciated with the 5 or with the

2.

This is called the associative law for addition. Using letters, it may be expressed thus,

a+b+c= (a + b) + c

= a + (b + c).

If an expression contains two or more factors, it matters not in what order the multiplications are performed.

This is called the commutative law for multiplication.* Using letters, the law may be expressed thus:

a. b = b. a.

Let a rectangle be the figure rotated (Fig. 4). Then a+b (i.e. the rotation about OA, followed by a rotation about OB) brings the rectangle in a position where " Alg." is horizontal, as in Fig. 5. On the other hand, b + a (i.e. a rotation about OB, followed by a rotation about OA) brings the rectangle in

a position in which " Alg." is vertical (Fig. 6). Since the final positions of the rectangle are different, it follows that in this case, a+b+b+a. That is, the commutative law is not obeyed. (The symbol means "is not equal to.")

Whether the commutative law for addition is obeyed or not depends therefore upon the definitions given to a and b, and to the processes of addition and subtraction.

* There is an advanced algebra, called quaternions, in which ij #ji; that is, the commutative law for multiplication does not generally hold true. Quaternions are used in the study of mathematical physics.

Again, it matters not how the factors are associated or grouped, for

6.5.390.

6. (5.3)=90, (6.5). 390.

This is called the associative law for multiplication. Using letters, the law may be expressed thus:

Again, a factor placed before or after a parenthesis containing two or more terms may be distributed among the various terms without any change in the final result.

That is,

5(946) = 5 × 9-5 × 4+ 5 × 6 = 55.

This is called the distributive law for multiplication. Using letters, the law may be expressed thus:

a(b + c)= ab + ac.

HISTORICAL NOTE

=

77. It is a curious fact that, in the development of algebra, the fundamental laws were the last things to be explained. The beginnings of algebra can be traced back to about 2000 years before Christ, but not until the nineteenth century were the fundamental laws of algebra formulated. In the earlier treatment the laws were tacitly assumed to be true. For instance, it was assumed that ab ba, a(b + c) = ab + ac, (ab)c = a(bc), without special attention being called to this matter nor special names being given to the relations assumed. The need of an explicit statement of the fundamental laws came to be recognized when it was perceived that, besides the algebra which we are studying, in which, for example, ab is always equal to ba, there could be established other algebras in which ab is not always equal to ba. Among those who helped to perfect the science of algebra along these lines were the Englishmen, George Peacock, D. F. Gregory, Augustus De Morgan, and Sir William Rowan Hamilton; the Frenchmen, F. J. Servois, and A. L. Cauchy; the Germans, Martin Ohm, Hermann Grassmann, and Hermann Hankel; and the American, Benjamin Peirce. The names "commutative law," "distributive law," were first used by Servois in 1814. Among the first to use the name "associative law" was Sir William Rowan Hamilton.