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Hence x - 2x = 2, or x2 – 2r + 1 = -1, and r- I =İV-1; r =l+-1 or = I - V -1, the two other sought. Ex. 2. To find the roots of x? - 9.72 +28.7 = 30.
Ans. r = 3, or = 3+1 -1, or =3-v-1. Ex. 3. To find the roots of r -- 7.x2 + 14x = 20.
Ans. x = 5, or = 1 +1 -3, or = 1-V- 3.
OF SIMPLE INTEREST.
As the interest of any sumy, for any time, is directly proportional to the principal sum, and to the time; therefore the interest of 1 pound, for 1 year, being multiplied by any given principal sum, and by the time of its forbearance, in years and parts, will give its interest for that time. That is, if there be put
= the rate of interest of 1 pound per annum, P = any principal sum lent, t = the time it is lent for, and a = the amount or sum of principal and interest; then
= the interest of the sum p, for the time t, and conseq. P + prt or p X(1 + rt) = d, the amount for that time.
From this expression, other theorems can easily be deduced, for finding any of the quantities above mentioned: which theorems, collected together, will be as below:
1st, a = p + prt, the amount,
1 + rt
P 3d, r =
For Example. Let it be required to find, in what time any principal sum will double itself, at any rate of simple interest.
In this case, we must use the first theorem, a = p + prt, in which the amount a must be made = 2p, or double the principal, that is, p + prt = 2p, or prt = P, or rt = 1; and hence to
Here, r being the interest of il. for 1 year, it follows, that the doubling at simple interest, is equal to the quotient of any, sum divided by its interest for 1 year. So, if the rate of interest be 5 per cent. then 100 * 5 = 20, is the time of doubling at that ráte. Or the 4th theorem gives at once a-P 2p - P
2 - 1 1
the same as before. pro pr
BESIDES the quantities concerned in Simple Interest, namely;
p = the principal sum,
t = the time,
Ř = i ir, the amount of it. for 1 time. Then the particular amounts for the several times may be thus compüted, viz. As ll. is to its amount for any time, so is any proposed principal sum, to its amount for the same time; that is, as 1l. : R::P
: PR, the 1st year's amount, il. : R:: PR : PR', the 2d year's amount, 11. : R:; PR : PR', the 3d year's amount,
and so on.
Ist, a = pR', the amount,
the ratio, 4th, ts VOL.I.
log. of a – log. of p, the time.
log. of R
From which, any one of the quantities may be found, when the rest are given.
As to the whole interest, it is found by barely subtracting the principal p from the amount a.
Example. Suppose it be required to find, in how many years any principal sum will double itself, at any proposed rate of compound interest.
In this case the 4th theorem must be employed, making A = 20; and then it is log. a- log. P _log. 2p – log. p log. 2
log. R So, if the rate of interest be 5
per cent. per annum; then R=1+05 = 1.05; and hence log. 2
= 14.2067 nearly ; log. 1.05 •021189 that is, any sum doubles itself in 14 years nearly, at the rate of 5 per cent. per annum compound interest,
Hence, and from the like question in Simple Interest, above given, are deduced the times in which any sum doubles itself, at several rates of interest, both simple and compound; viz.
The following Table will very much facilitate calculations of compound interest on any sum, for any number of years, at various rates of interest.
The Amounts of 11. in any Number of Years.
1 1'0300 1.0350 1.0400 1.0450 1.0500 1.0600 2 1.0609 1.0712 1:0816 1:0920 1.1025 1:1236 3 1.0927 1:1087 1.1249 | 1:1412 1:1576 1.1910 4 11255 1•1475 | 1:1699 1.1925 1.2155 1'2625 5 1.1593 1:1877 | 1.2167 1.2462 1.2763 1•3332 6 1:1941 | 1.22931•2653 1•3023 1•3401 1.4185 7 1.2299 1.2723 1.3159 1.3609 1.4071 2:5036 8 1.2668 | 1:3168 1:3686 1.4221 1.4775 1.5939 9 1•3048 1•3629 104233 1.4861 1:5513 1.6893 10 1.3439 | 1'4106 1.4802 1.55301002891.7909 Il 1.3842 1:4600 1.5895 1.6229 | 107103 1.8483 12 1.4258 1.5111 | 1:6010 1.6959 | 1.7959 2:0122 13 1:4685 1.5610 1.6651 1.7722 18856 2.1329 144 1:5126 1.6187 | 107317 | 1.8519 | 1.9799 2.2009 15 1.5580 1.0753 1.8000 1.9353 2.0789 2.3965 16 1.6047 | 1•7340 1.8730 2:0224 2:1829 2.540+ 17 1.65281.7917 1.9479 2:1134 2:2920 2.6928 18 1.7024 | 1.8575 2:0258 2:2095 2:4006 2.8543 19 | 1.7535 1.9225 | 2:106 2:3070 2:5270 3:0256 20 1.8061
1.9898 2:1911 | 2:4117 | 2.6533 | 3.2071
The use of this Table, which contains all the powers, R', to the 20th power, or the amounts of 11, is chiefly to calculate the interest, or the amount of any principal sum, for any time, not more than 20 years.
For example, let it be required to find, to how much 5231. will amount in 15 years, at the rate of 5 per cent per annum compound interest. In the table, on the line 15, and in the column 5
cent. is the amount of 11, viz.
2.0789 this multiplied by the principal
gives the amount
1057.2647 10871. 55. 30. 5641. 55. 3 d.
and therefore the interest is
Note 1. When the rate of interest is to be determined to any other time than a year ; as suppose to a year, of a year, &c; the rules are still the same ; but then twill
express that time, and r must be taken the amount for that time also.
Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year ; it may be determined in the following manner :
1st, For any time which is some aliquot part of a year :Find the amount of ll. for 1 year, as, before ; then that root of it which is denoted by the aliquot part, will be the amount of it. This amount being multiplied by the principal sum, will produce the amount of the given sum as required.
2d, When the time is not an aliquot part of a year :Reduce the time into days, and take the 365th root of the amount of il. for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before. Ånd in these calculations, the operation by logarithms will be very useful.
ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual payments.
Annuities are divided into those that are in Possession, and those in Reversion : the former meaning such as have commenced ; and the latter such as will not begin till some particular, event has happened, or till after some certain time has elapsed.
When an annuity is forborn for some years, or the payments not made for that time, the annuity is said to be in Arrears.
An annuity may also be for a certain number of years ; or it may be without any limit, and then it is called a Perpetuity.
The Amount of an annuity, forborn for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due.