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Here, , being the interest of 1l. for 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 rate. Or the 4th theorem gives at once a-p 21 - 1 2-1 1
the same as before. fir pr
Besides the quantities concerned in Simple Interest namely,
t = the principal sum,
t = the time,
R=1+ r, the amount of 11. for 1 time.
Il.:R:: :PR, the 1st year's amount,
and so on.
Ist, a = fRt, the amount,
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.
Erample. 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= 2; and then it is,
log. a-log. log. 2p-log.
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.
10300 1 0609 1.0927 11255 1 1593 11941 1.2299 1.2668 13043 1 3439 1:3842 1.4258 1.4685 1:5126 1.5580 1.6047 1.6528 1.7024 1.7535 1 8061
7 8 9 10 11 12 13 14 15 16 17 18 19 20
1:0350 1 0400 1.0450 10500 1 0600
1.1412 1.1576 1.1910
2.1899 | 2:5404
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 com pound interest. In the table, on the line 15, and in the column 5 per cent. is the amount of 11. viz.
2.0789 this multiplied by the principal
gives the amount
10871. 58. 3 d. and therefore the interest is
5641. 58. 3 d. Nove 1.. When the rate of interest is to be determined to any other time than a year; as suppose to 1 a year, or 1. a year, &c; the rules are still the same ; but then = will
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 1l. This amount being multiplied by the prin. cipal 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 ll. 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.--And 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 pay ments 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 annuiiics for that number of years, together with the interest due upon cach after it becomes due,
The Present Worth or Value of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off, or paid all at once. Let a = the annuity, pension, or yearly rent;
n = the number of years forborn, or lent for ;
v = its value, or its present worth. Now, I being the present value of the sum r, by proportion the present value of any other sum a, is thus found:
the present value of a due 1 year hence.
is the present value of a due 2 years,
hence ; for R :1::
R* R 5 be the present values of a, due at the end of 3, 4, 5, &c. years respectively. Consequently the sum of all these, or
1 1 1 1 -+-+ -t. + &c. = (-+-t
&c.) XQ, R R3
R% continued to n terms, will be the present value of all the n years' annuities. And the value of the perpetuity, is the sum of the series to infinity. But this series, it is evident, is a geometrical progression,
1 having — both for its first term and common ratio, and the number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be 1 I 1
When the annuity is a perpetuity ; n being infinite, R* is
1 also infinite, and therefore the quantity becomes 0,
1 X - also =0; consequently the expression