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chased, varies as the sum given for it, if the price remain the same; therefore the amount can be found by a Simple Proportion, as thism

P: S 100 : M. Thirdly, let it be required to find what rate of interest may be obtained by purchasing in stock bearing a given rate of interest. Here we have given the interest on the price of £100 stock, and the interest on £100 is required. Since the interest varies as the sum invested, other circumstances being the same, the solution of this question is one of Simple Proportion, thus

P : 100 :: : R,
T, R, being the rates of interest on £100 stock, and £100 sterling.

Fourthly, let it be required to find in which of two kinds of stock it is more advantageous to invest. Evidently the answer is, that in which the higher rate of interest is obtained. The question therefore might be solved by finding these rates in the two cases, and comparing them. But since, from last case, it appears that R= x 100, it follows that the rate of interest is greater in that kind of stock, in which the ratio r : P is greater; so that it is requisite only to compare these ratios; whence the Rule.

Fifthly, let it be required to find what annual income will be realized by investing a given sum in stock at a given rate. The income of course varies directly as the sum invested, other circumstances remaining the same; hence, since the interest on £100 stock is the income arising from the investment of the price of £100 stock, we have the proportion,

P S :: go

; I Sixthly, let it be required to find what sum must be invested to produce a given income. The sum varies directly as the incoine, other circumstances remaining the same: hence, since the price of £100 stock is the sum required to produce an income equal to the interest on £100 stock, we have the proportion,

I P : S Seventhly, let it be required to find how much of one kind of stock can be purchased with the proceeds of the sale of a given amount of other stock. Here we consider that the amount of stock, which can be purchased with a sum of money, varies inversely as the price of the stock, therefore this question may be solved by a Simple Proportion thus,

pi : P :: M ; M1 Eighthly, let it be required to find the difference in a person's income caused by transference of money from one stock to another. Here it is to be considered, that the income derivable from the investment of a given sum varies directly as the rate of interest, and inversely as the price of the stock. Hence the solution of the question is obtained by the proportion,

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Prop. 62.- To explain the Rules of Commission, Brokerage,

and Insurance,

Commission and Brokerage are allowances made to agents for assisting in the transference of propertyThese allowances are at the rate of so much for every £100 of value of the goods disposed of, or bought; except in the case of “ Stock,” where the calculation is made on the nominal value of the Stock.

Insurance (or rather the premium for insuring) is a sum paid by one person to another, or to a company, in consideration of which the latter agrees to make good the loss of the former to a stipulated amount. The fremium is calculated at so much for every £100 insured.

It is evident therefore that the Commission, Brokerage, or Insurance on a given sum varies as the sum. Hence the amount of each of these is obtainable by a proportion thus :As £100 : the given sum £P :: the rate per cent. £R : the ans. £A. whence 100 P R A

100 : R :: P А £P : £A

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{£P X R} = 100. Whence the Rule.

Sometimes an insurance is effected to an amount sufficient to cover not only the value of the goods, but also the cost of insurance. To find what sum must be insured to effect this, we consider that the insurance of £100 will actually recover in the event of loss only the excess of £100 over the cost of insurance. Also the sum which must be insured varies as the sum to be covered ; hence the solution of the question is obtained by a proportion thus:-As the sum covered by £100 : the whole sum to be covered :: £100 : ans.

Prop. 63.— To explain the several cases in Profit and Loss.

1st. If it be required to find the profit or loss per cent. made in a given transaction, it is to be considered that, if the circumstances under which

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the transaction is effected remain the same, but the amount of the transaction be doubled, the profit or loss would also be doubled. Hence the profit or loss varies as the value of the goods on which it is made: so that the gain or loss per cent. will be obtained by the proportion. As the price of the goods : £100 :: the given gain or loss :

2nd. If it be required to find at what price an article must be sold to produce a gain or loss of so much per cent. it is to be considered, that the produce of a given sum varies as the sum, the rate of gain or loss being the same. For instance, if £5 be gained or lost on every transaction of £100, then £10 will be gained or lost on a transaction of £200, so that £100 will produce £105 or £95, and £200 will produce £210 or £190, which is double the produce of £100. Hence the solution of this question also is by Simple Proportion thus: As £100 the cost of the article :: the produce of £100 :

3rd. If the prime cost of an article be required, it being given that by selling it at a certain price a given gain or loss per cent is made, it is to be considered that the prime cost varies as the produce, the rate of gain or loss being the same; for it was shewn in the last case that the produce varies as the prime cost. Hence this question is solved by a proportion thus :

As the produce of £100 the given price :: £100 : 4th. If it be given that by selling goods at a certain price a given gain or loss per cent. is made, and it be required to find the selling price at which another gain or loss will be made, then we know from the last two cases that the ratio of the produce of £100 to the selling price of an article is equal to the ratio of £100 to the prime cost. Hence this last ratio being the same, when the rate of gain or loss is altered, it follows that if the produce of £100 be doubled the selling price must also be doubled, i.e. the selling price varies as the produce of £100. Hence the solution of the question is effected by a Simple Proportion thus : As the 1st produce of £100 : the 2nd :: the 1st selling price : the 2nd.

5th. If it be given that by selling goods at a certain price a certain gain or loss is made, and it be required to find what will be the gain or loss when the goods are sold at another given price, it is considered that since, as was shewn in the last case, the selling price varies as the produce of £100, therefore also the produce of £100 varies as the selling price. Hence a proportion stated thus : As the 1st selling price : the 2nd :: the 1st produce of £100 : the 2nd. will give the produce of £100 in the event of the goods being sold at the second given price. The difference between this and the second price will be the gain or loss per cent. required.

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Prop. 64.—To explain the Rules of Fellowship. 1st. Let the times, for which each partner has been engaged, be the same. It is evident that the gain or loss of each ought to vary as the stock of each; so that the several gains or losses will bear the same ratio to one another, as. the several stocks. Hence we must divide the whole gain or loss into a number of parts in these ratios. Now the ratios of the stocks are the same as the ratios of the numbers of pounds in the stocks, or of the numbers of shares; therefore (by Prop. 46) the several shares of gain or loss are found by dividing the whole gain or loss by the sum of these numbers, and mnltiplying the result by each in succession ; or by finding the ratio of the whole gain or loss to the whole stock, and multiplying by it each stock.

2nd. Let the times be different. Here it is plain, that the gain or loss ought to vary directly as the amount of stock, when the time is the same ; and also directly as the time, when the stock is the same: therefore when both time and stock are altered, the gain or loss will vary as the product of these. Hence the whole gain or loss must be divided into parts, which shall have the same ratios as the products of the units in the several stocks by the units in the several times. So that the several shares of gain or loss will be obtained by dividing the whole gain by the sum of these products, and multiplying by each product separately. If the numbers of shares in the concern be given instead of the amounts of stock, these may be used for the numbers of units, the unit being the amount of a share.

Prop. 65.- The product of different powers of the same

number is a power of a degree equal to the sum of the

degrees of the several powers. Every power is composed of factors, equal to the number, and in number equal to the degree of the power. Thus in the 3rd, 4th, &c. powers there are 3, 4, &c. factors. Hence if two or more powers be multiplied together, the result is a number of as many factors as there are in the several powers together, i.e. a power of a degree equal to the sum of the degrees of the several powers. Thus if the 3rd and 5th powers of a number be multiplied, the result is a number composed of 8 factors, or a power of the 8th degree. And generally if a denote a number, am xan =(a Xa xa X &c. to m factors) X (a Xaxa X &c. to n factors)

= a Xa xa X &c. to (m + n) factors

= am+n.

Prop. 66.- A power of a power is another of a degree equal

to the product of the degrees of the two.

If a power of a number be raised to a given power, the result is a number composed by the product of the first power, repeated a number of times, equal to the degree to which it has to be raised; i.e. (by last Prop.) is a power of a degree equal to the sum of the degree of the first power repeated a number of times equal to the degree of the second, or of a degree equal to the product of the two degrees. Thus if the 8th power is to be cubed, the result is the product of three 8th powers, i.e a power of a degree equal to three times 8 or 24. And generally if a be any number, then

(am)n = cm x cm x am X &c. to n factors

=a mtm tm + &c. to n times = amn Cor. Hence the nth root of the (m n)th power is the mth power; and therefore the mth root of the nth root is the first power or the (m n)th root of the (m n)th power. So that the extraction of two roots in succession is equivalent to the extraction of a root of a degree equal to the product of the degrees of the roots.

Prop. 67.— The power of a product is equal to the product

of the powers of the factors. For in the power each factor will be repeated a number of times equal to the degree of the power; so that the product of all the factors is equal to the product of the powers of the factors.

Cor. Conversely, the root of a product is equal to the product of the roots of all the factors. Prop. 68.—The square of a number is equal to the sum of

the squares of any two parts into which it may be divided,

together with twice the product of these parts. In multiplying one number by another, we may separate the multiplicand and multiplier into any parts we please, then multiply each part of the multiplicand by each part of the multiplier, and add the products. Let then the number, whose square is required, be separated into two parts; the square will be found by multiplying both these parts by each of them, and adding the products. These products will be in order, the square of the 1st part, the product of the two parts, the product of the two parts, the square of the 2nd part. Hence the whole square is equal to the sum of the squares of the two parts together with twice the product of the parts. Thus the square of 16 = (12 + 4)X(12+4)

= (12 + 4) X 12+ (12 + 4) X 4
= 12 X 12+ 4 X 12 + 12 X 4 + 4 X 4

= 122 + 2 X 4 X 12 +42 Algebraically expressed the Prop. will stand thus :

(a + b)2 = a2 + 2 ab +62.

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