4. The value of a certain house in 1880 has increased by 35 per cent. since 1877. The house was rated in 1877 at two-thirds of its value, and in 1880 it is rated at three-fifths of its value, the rate in the £ remaining the same. Compare the rate paid in 1877 with that paid in 1880.

5. A merchant buys certain goods at £35 per ton. At what price per lb. must he sell them in order to gain 10 per cent. on his outlay? 6. By selling 10 acres of land for £4699. 8s. 3d. a man gained 5 per cent.: what was the original price per acre?

7. A tradesman forms a mixture of tea by adding 1 lb. at 3s. and lb. at 4s. 6d. to every 2 lb. at 2s.: what must he sell it at per lb. in order to gain 10 per cent.?

8. If the annual increase in the population of a state is 25 per thousand, and the present number of inhabitants is 2,624,000: what will the population be in 3 year's time? And what was it a year ago?

9. If a man buys eggs at 10 for a shilling, and sells them at 8 for a shilling, what rate per cent. profit does he make on his outlay? 10. If 4 per cent. be lost by selling silk at 10s. per yard, at what price per yard should it be sold in order to gain 5 per cent.?

11. One gallon of spirit which contains 11 per cent. of water is added to three gallons containing 7 per cent. of water, and to this mixture half a gallon of water is added: find the percentage of water in the mixture.

12. If a grocer gains 10 per cent. by selling tea at 2s. 3d. per lb., what will he gain per cent. by selling it at 2s. 9d.?

13. Define ratio. Does it follow from your definition that it would be wrong to speak of the ratio of 5 tons to 3 miles, and, if so how does it follow?

Summary.

Involution.—The continued multiplication of a number by itself is called Involution. The number is called the root or first power, the second power is called the square, the third the cube. Thus 22= square of 2, 23=cube of 2, etc.

Evolution.-Given a power of any number the process of finding the root is called Evolution. The sign indicates the square root. Thus 9 means the square root of 9. This is also written 9. Similarly the cube root of nine is written 3/9 or 93.

Cube Root. If a number is multiplied three times by itself the product is called the cube of the number, and the number itself is called the cube root of the product. Thus 33=27 and 3/27, or

27=3.

Ratio. A comparison of one number with another, obtained by dividing one by the other, is called Ratio. Hence the ratio of 3 to 93 = 3.

Proportion is the equality of two ratios.

The four numbers 3, 4, 15, and 20 form a proportion, or the ratio of the first two is equal to the ratio of the last two terms,

When three terms are known the fourth can be found. The product of the means, or, in the above example, 4 × 15, is equal to the product of the extremes, or 3 × 20.

CHAPTER VI.

ALGEBRA.

ADDITION, SUBTRACTION, MULTIPLICATION,

DIVISION. SIMPLE OPERATIONS.

AND FACTORS. SIM

PLIFICATION OF ALGEBRAICAL EXPRESSIONS.
INDICES.

Explanation of Symbols.-In operating with numbers or digits as the numerals 1, 2, 3 are called, accurate results are obtained whatever be the unit employed. Thus, the digit 7 may refer to 7 shillings, ounces, yards, or other units. In adding two digits, such as 7 and 5, together-taking care that the quantities are of the same kind-we obtain the sum 12, whatever the unit employed may be.

The signs already made use of in Arithmetic are also employed in Algebra, but in Algebra representations of quantities are utilised which have a further generality. Both letters and figures are used as symbols for numbers or quantities. These numbers may be known numbers and are then usually represented by the first letters of the alphabet a, b, c, etc., or they may be numbers which have to be found, called unknown numbers, and these are often denoted by a', Y, Z.

A more general meaning is given to the signs + and

If a distance AC measured along a line is said to be positive, the distance CA measured in the opposite direction would be negative.

The result of the first measurement would be indicated by +a, while the same distance CA, but measured in the opposite direction, would be indicated by a.

Again, if a length AB be measured in the same direction from left to right and be denoted by +b, the length BA measured from right to left would be indicated by -b.

Hence, +a+b would mean the sum or addition of the two lines and so a line of length equal to AD is obtained, where

AD=AC+AB.

Similarly, +a-b would denote the length BC obtained by measuring a length a in the positive and a length b in the negative direction.

When writing down an expression it is usual, where possible, to place the positive quantity first and to dispense with the + sign. The above expressions would, therefore, always be written as a+b and a-b. The signs placed between the numbers indicating in the first case the sum of two positive quantities, and in the second case the subtraction of one positive quantity from another; in the latter case, the quantity a-b could also be described as the addition of a negative quantity b to a positive quantity a, this is called the algebraical sum of the two quantities, or the algebraical sum of two quantities is the result after carrying out the operation indicated by the signs before the two quantities.

The algebraical sum of +10 and -18=-8.

- b.

The

In the quantity a-b, if a represents a sum of money received, then -b will represent a sum of money paid away. algebraic sum is represented by the balance aAgain, a may represent the height of a point P above a certain level AB, then -b may represent the distance of another point D below the same level.

It will be seen that in Algebra the word sum is used in a different and a wider sense than in Arithmetic. Thus, in Arithmetic ab indicates that b is to be subtracted from a, but in Algebra it also means the sum of the two quantities.

The arithmetical symbols of operation, +, −, ×, and ÷, are used in Algebra, but are varied according to circumstances; the general sign for the multiplication of quantities is ; but the product of single letters may be expressed by placing the letters one after another, thus the product of a and b may be written a × b but is usually written as ab. In a similar manner the product of 4a, x, and y is expressed by 4axy.

The product of two quantities a+b, and c+d may be expressed as (a+b)x(c+d), or usually as (a+b)(c+d).

Multiples of the quantities a, b, c, etc., may be expressed by placing numbers before them as, 2a, 3b, 5x; the numbers 2, 3, and 5 thus prefixed are called the coefficients of a, b, and x.

The product of a quantity multiplied by itself any number of times is called a power of that quantity and is indicated by writing the number of factors on the right of the quantity and above it. Thus :

a×a is called the square of a and is written a2;
bxbxb is called the cube of b and is written b3.

Similarly, cxcxc... n factors is written c" and indicates c to the power n.

The number denoting the power of a given quantity is called its index or exponent.

It is very important that the distinction between coefficient and index be clearly understood. Thus 4a and a1 are quite different terms.

Ex. 1. Let a 2, then 4a=8; but a1= 21= 16.

=

The use of signs may be exemplified in the following manner :
Ex. 2. In the expression a2+b− c.

when a = 1, b=2, c=3, d=0.

Substituting these values in the given equation we obtain

(−3-0)√1 × 2 × 3+4×3×0+(9× − 1 × 0) − 2

== 3√6 2= -6.