names, as Consols, Three per Cents. Reduced, &c. In France they are called Rentes.

In purchasing and selling the various kinds of shares and stocks, a broker is generally employed, who charges a certain percentage on the amount of stock sold or purchased.

All questions in shares and stocks resolve themselves into questions in Rule of Three.

Example 1. What is the value of £750 Metropolitan Railway stock when £100 stock is £132 ?

Example 2. What will be the annual dividend on £750 Metropolitan stock at 3 per cent. for the half-year?

Example 3. Purchasing £100 Metropolitan stock at £132, and receiving an annual dividend of 7 per cent., what rate of interest do I get for my money

P

Example 4. If I lay out £330 in the purchase of Metropolitan stock at £132, what amount of stock do I receive?

Example 5. If I lay out £330 in the purchase of Metropolitan stock at £132, which produces a dividend of 7 per cent., what will be the amount of my dividend?

Example 6. A person has stock in the 3 per cent. consols, which produces him £300 a year; he sells out at 92, and invests the proceeds in the South Devon Railway, when a £50 share is worth £23. What percentage ought the South Devon Railway to pay, that he may increase his income £50 by the transaction?

The amount of stock is

300
3

X £100 £10000.

This sold at 92 brings in £9200.

This invested in South Devon shares as in the question produces

If this produces £350 dividend, £100 stock will produce

86. Ratio is the relation between two quantities of the same kind with respect to magnitude, when the first quantity is considered as containing the other a certain number of times, whether integral or fractional.

The ratio of £7 to £9, or of 7 men to 9 men, or of 7 tons to 9 tons, is represented thus, 7 : 9 (read 7 is to 9), or by the fraction, since this fraction measures the number of times £7 contains £9, and so on.

In the ratio of £7 : £9, the £7 and £9 are called the terms of the ratio; the £7 is called the antecedent, and the £9 the consequent.

It is evident that to find what ratio one quantity bears to another, is the same as finding what fraction one is of the other. (Art. 31.)

The ratios £7: £9, 14 tons: 18 tons, 21 men: 27 men, 28 yards 36 yards, £1 15s.: £2 5s., are all equal to one another, represents them all.

because the same fraction

When two ratios are equal, the fractions representing them, when reduced to their lowest terms, must be the same.

A ratio is said to be inverted when the antecedent and consequent are interchanged: thus £9: £7 is the ratio £7: £9 inverted.

A ratio is said to be compounded of two other ratios, when its antecedent is the product of their antecedents, and its consequent of their consequents. Thus, 6: 35 is compounded of 2 : 5 and 3:7.

EXERCISE LXXIII.

87. The equality of two ratios is called a proportion. Thus £7: £9 14 tons: 18 tons, is a proportion.

The quantities £7, £9, 14 tons, 18 tons, are called the 1st, 2nd, 3rd, 4th, terms of the proportion.

If two equal ratios are inverted they are still equal and form a proportion: thus £9: £7 = 18 tons: 14 tons.

It is frequently required to find the 4th term of a proportion when the three other terms are given.

Example. To find the 4th term of the proportion, of which the first three terms are £7, £9, 14 tons.

Now £7: £9 14 tons

represented by the fraction 7.

of 14 tons, for each of these ratios is

Hence to find the 4th term of a proportion, we express (if necessary) the first two terms in the same denomination, and then multiply the 3rd term by the 2nd, and divide by the Ist.

This is exactly the same process as is performed in Rule of Three; and solving a question in Rule of Three is the same as finding the 4th term of a proportion, and vice versa.

In a similar way we can find any term of a proportion when the other three are given.

The 1st and 4th terms of a proportion are called the extremes, and the 2nd and 3rd the means. The student will see from any example that numerically the product of the extremes is equal to the product of the means.

EXERCISE LXXIV.

88. One quantity is said to vary as another, when if the second quantity is changed, the first is changed in the same proportion,

i. e. if the second is halved, the first is also halved; if the second is trebled, the first is also trebled, and so on.

Thus the price of any article varies as the quantity of the article, the wages of a labourer vary as the time he has been working, the income-tax varies as the amount of income. So also the quantity of an article varies as the price, the time as the wages, the income as the tax; that is, if one quantity varies as another, the second also varies as the first.

When one quantity varies as another a proportion can always be formed by them, thus, "the price of one quantity of coal: the price of another quantity, as the first quantity: the second quantity."

Ex. Price of 15 tons : price of 17 tons = 15 tons : 17 tons. 89. One quantity is said to vary inversely as another, when as one quantity is increased the other is diminished in the same proportion, i. e. if the second quantity is halved, the first is doubled; if the second is trebled, the first is divided by 3, and so on. Thus the time of doing a piece of work varies inversely as the number of men employed to do it, the length of carpet required to cover a given room varies inversely as its breadth.

In such a case a proportion will be formed thus:-time occupied by II men time occupied by 15 men = 15 men : 11 men.

And if carpets A, B are each capable of covering a given room : Length of carpet A: length of B = breadth of B: breadth of A. The second ratio in these proportions have their terms corresponding to the terms in the first ratio in an inverted order, whence the expression, vary inversely.

In all cases of Rule of Three four quantities are involved, the three given quantities and the answer: two of these quantities are of one kind, and two of another: the quantities of the first kind must in the nature of the case vary directly, or inversely, as those of the second, or a proportion cannot be found amongst them, and the process of Rule of Three is not applicable.

EXERCISE LXXV.

90. When one quantity varies as another it is said to depend upon it: : one quantity may, however, depend upon several other quantities. In the cases above we have only considered one of such

quantities: thus when we say that the price of an article varies as the quantity, we neglect the quality, or tacitly assume that it is unchanged: when we say that the income-tax varies as the amount of income, we assume that the rate of the tax is unchanged.

One quantity may vary as two or more quantities multiplied or divided together. Thus the amount of income-tax varies as the income, multiplied by the rate: the contents of a cistern varies as the length breadth x depth: the number of men required to dig a rectangular field varies as the length multiplied by the breadth, and divided by the time in which the work is to be done. In these cases the first term varies as each of the quantities composing the second term separately: thus the number of men required to a rectangular field varies as the length, if the breadth and time of performing the work are fixed: the number of men varies as the breadth, if the length and time are fixed, and varies inversely as the time if the length and breadth are fixed.

91. The converse of this will also be true, viz. that when one quantity varies as several others, independently of each other (that is, when all but one are fixed, it varies as that one), then it varies as their product. If, however, it varies inversely as any of the other quantities, such quantity will divide the other quantities instead of multiplying them.

And, in the same way, as when one quantity varies as one other, a proportion can be formed thus

Number of men required to dig a field A: Number relength of A x breadth of A quired to dig a field B =

length of B x breadth of B.

time of digging B.

time of digging A.

In such cases, if all the quantities composing the proportion but one are given, this one can be found in a manner similar to that in Art. 87.

Example. If 12 men can dig a field 70 yards long, 45 broad, in 7 hours, how many men can dig a field 80 yards long, 75 broad, in 8 hours ?