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This Prop. can be readily proved only algebraically:-Let B, C, D, &c. B1, C1, D1, &c. be given values of two sets of quantities, connected with another, whose corresponding value is A, in such a manner that this latter varies directly as each of the first set, and inversely as each of the second, when all the rest remain unchanged; then, if all be changed, the value of

B x C x D x &c. A will be changed in the same ratio as

Bix C?x Dix &c.
First let B be changed to b, then A will be changed to some value Ai,

А
Ai

B 5
Next let B1 be changed to 67, then A1 will be changed to some value Aji

1
Ai
: Aii

B1 61
Simly. if c,ct; d, d1; &c. be other values of C, C1; D, D1, &c. and
Aiii, Aiv, Av, Avi, &c. the corresponding values of A, we shall have the
proportions,

Aii

с

1
Aiii : Aiv

Ci ci
Aiv AV D: d

1 1
AV
Avi

Di di Whence compounding all these proportions, and striking out common means and extremes, we have

BXCXD Хc Хd
A : Avi

BXCXD1 bix cixd which expresses the fact that A has been changed in the same ratio as that

BXCXD in which

BiXC1XD 1 has been changed ; that is, that the value of A varies directly as the product of B, C, D, and inversely as that of B2,C,D1.

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A iii ::

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с

::

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Prop. 56.- To explain the Rule for Compound Pro

portion. In questions of Compound Proportion, several quantities of different kinds are involved, which are so connected with another quantity, that this latter varies directly or inversely as each of them, when all the rest are unchanged. Certain values of the first-mentioned quantities, and the corresponding value of the last-mentioned are given, and also other values of the first, from which it is required to find the corresponding value of the last. Let B, C, D; B1, C1, D1, be given values of two sets of quantities, and A the corresponding value of another quantity, which varies directly as each of the former, and inversely as each of the latter quan

:

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с

:

Bi ::

Aiii :

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tities; also let b, c, d; 61, c', di, be other given values of these quantities, from which it is required to find the corresponding value of A. If each of the quantities B, C, D; B1, C1, Di, be changed in succession, the others remaining unaltered, and if Ai, Ali, Aii, Aiv, AV, Avi, be the successive values of A, we have the following proportions,

B : b :: A Ai
с

Ai

: Ali D : d :: Aii :

A ii bi

Aiv ci Ci :: Aiv : Av

: Di :: Av: Avi in each of which the first and second terms are precisely those which would have arisen from considering the question as one of Simple Proportion, depending only upon the quantities in those terms.

Now if these proportions be compounded, all the A's except A and Avi will disappear; and we have the result that the ratio compounded of the several ratios B : 6 &c. is equal to the ratio of A to Avi. Hence if the ratios B : 6 &c. be formed, and compounded, Avi, or the value of the quantity required may be found by solving a simple proportion in which the third term is A, and the first and second are those of the compound ratio.

Note. This rule might have been deduced immediately from the last Prop.; for

BXCXD bXcxd
Since A : Avi ::

BiXC1XD1 31x cixdi .: A : Avi :: BXCXDX61Xcxdl : 6XcXdX BixCXD1 which is the proportion compounded of the above proportions.

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Prop. 57.To explain the Rule for finding the Simple

Interest on a given sum. Questions in Simple Interest are questions in Compound Proportion, the quantities involved being cash, time, and interest; interest varying directly as each of cash and time, when the other is unaltered. Hence, if P, R, 1, denote the numbers of pounds in the given sum, in the interest per cent. per annum, and in the required interest, and n the number of years, we have

100 1 : PXn :: Ꭱ : I
100

R :: PXn : I

:: £ P xn : £I ... £I= { £PXnX R }=100 Whence the Rule.

Note. P is the number of pounds in the principal, but if there be shillings and pence in the given sum, it manifestly will not be necessary to

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n

n

n

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360

n

n

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express them actually as fractions of a pound, because they are virtually so expressed, since instead of the denominations shillings, pence, might be used 2.£, 21,£. Cor. 1. If n be the number of months in the time, then since n months 12 years, we have only to substitute for n in the above formula;

12 i.e. we have to divide the interest for n years by 12.

Cor. 2. If n be the number of days in the time, we have only to substitute for n in the above formula; by which it becomes 365

PXn XR PXR
I=

Х

(A)
365 X 100 100 365
PXR

1
Xп
100

73 X 360
PXR
Х
nearly

(B)
100 360
PXR

1 1
Х 1 +

nearly (C)
100 360 72 72 X 72
PXR
Х Х

(D)
73
PXR
x Х

nearly (E) 100 5 72 72 Formula A gives the exact method for finding interest for a number of days; formula B gives a first approximation to exactness; C gives a result nearer the truth; D is the formula applicable to decimal operations, giving the exact result; E is also applicable to decimals, giving an approximate result.

Cor. 3. Hence if a, b, c, &c. be several sums of money out at interest for numbers of days m, n, p, &c. the whole interest

a X m XR bXn XR CXPXR I=

+

+

+ &c. 36500

36500 36500 2 R

x

73000 which formula expresses the Rule 4 in the “ Practice."

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x{axm +oxn+cXp+&c.}

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Prop. 58.--To explain the Rules for finding any one of

the quantities, Time, Rate, Principal, when the others

and the interest are known. 1st. To find the time. The time evidently varies directly as the interest, if the principal and rate remain the same; hence if Il be the interest for 1 year, 1 the given interest, n the number of years,

I : 11 ::

: 1 that is, the ratio of the given interest to the interest for 1 year will be the number of years.

2nd. To find the rate. The rate evidently varies as the interest, when the principal and time remain the same; hence if Il be the interest for the given time at 1 per cent.;

I : 11 :: R: 1 that is, the ratio of the given interest to the interest at 1 per cent. is the number of pounds in the rate.

3rd. To find the principal. The principal evidently varies as the interest, when the time and rate are the same; hence if 11 be the interest on £l for the given time at the given rate,

I : 11 :: P: 1 that is, the ratio of the given interest to the interest on £l is the number of pounds in the principal.

If the amount be given, instead of the interest, since the principal varies directly as the amount, if M, M1 be the given amount, and that of £1,

M : Mi :: P : 1 that is, the ratio of the given amount to the amount of £1 is the number of pounds in the principal.

Prop. 59.-- To explain the Rule for finding the true dis

count on a sum of money. If A owe B a sum of money, which he is to pay at the end of a certain time, he will derive from the delay of payment an advantage equal to the interest of the debt for the given time. If then he pay the money at once, he ought to receive such a compensation as will leave him, at the end of the time, without loss. The question then is, what sum of money must A receive in order that, at the end of a given time, he may have gained a benefit from it equal to the interest on his debt? In other words, what is the sum which, put out to interest for the given time, will produce the interest on the debt? Thus the question of finding the discount becomes a question merely of finding the principal, when the amount, time, and rate are given : and hence the discount is the ratio of the interest on the debt to the amount of £1 for the given time.

Again: the creditor ought not to receive the whole of the debt, but such a sum as, put out to interest for the given time, will amount to the debt, ée. the present worth of the debt. Hence the discount is equal to the difference between the debt and its present worth.

Prop. 60.-To prove the Rule for finding the amount at

Compound Interest of a given principal; and conversely.

The amount at simple interest varies directly as the principal, the rate and time being the same: hence if r be the rate of interest per £1 for one year, Mi, M', Miii, &c. be the amounts at the end of the 1st, 2nd, 3rd, &c. years, since the amount of one year is the principal of the next, we have

Мі : 1tr P 1
Міі
1+1

1
Müi : 1tr

Mii : 1

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Mi :

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whence, compounding the proportions, and expelling common means and extremes,

Miii : (1+r)3 P 1 which gives

Müi = - P(1 + r) 3 and a similar formula may be obtained for any number of years.

Conversely, the principal required to produce a given amount at simple interest varies directly as the amount, the rate and time being the same; hence if Pii, Pii, Pi, be the principals required to produce the amounts at the end of the 3rd, 2nd, and 1st years, since the principal of one year is the amount of the year before, we have

piii : 1 M : 1tr
Pii: 1 Piii

1+1 Pi 1 рії : 1tr whence pi : 1 :: M : (1+r)3. now (1 + r)3 is the amount of £1 for 3 years, whence it appears that the ratio of the given amount to the amount of £1 is the number of pounds in the principal.

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Prop. 61.-To explain the Rules for the several cases of

Stocks. These are all applications either of Simple or Compound Proportion. Let P stand for the price of £100 stock, M the amount of stock, S the cost of the stock, I the income; and let P1, Mo, S, 1, stand for the same in another kind of stock.

First, let it be required to find the cost of a given amonnt of stock at a given price per cent. Evidently the cost varies directly as the amount to be purchased, if the price remain the same. Hence the cost required is found by a Simple Proportion, as this,

100 : M P : S.

MXP
whence SE which expresses the Rule.

100 Secondly, let it be required to find how much stock can be purchased for a given sum at a given price. Here the amount of stock, which can be pur

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