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CHA P. XII.
Of Compound Interest, and Annuities, &c. COMPOUND Intereft is that which arises from any Princi
pal and it's Intereft put together, as the Interest fo becomes due; so that at every Payment, or at the Time when the Payments became due, there is created a new Principal; and for that Reason it is called Intereft upon Interest, or Compound Intereft.
As for Instance ; Suppole 100l. were lent out for two Years, at 6 per Cent. per Annum, Compound Interest: then at the End of the firft Year, it will only amount to 1061. as in Simple Intereft. But for the second Year' this 106). becomes Principal, which will amount to 112l. 75. 2 1 d. at the second Year's End, whereas by Simple Intereft it would have amounted to but 112 l.
- And altho’ it be not lawful to let out Money at Compound Interest; yet in purchasing of Annuities or Penfions, &c. and taking Leases in Reverfion, it is very usual to allow Compound Intereft to the Purchaser for his ready Money; and therefore it is very requisite to understand it.
Sect. 1. Of Compound Interest.
P= the Principal put to Intereft.
as before. Let A=the Amount of the Principal and Intereft.
S the Amount of I I. and it's Interest for 1 Year, at
any given Rate, which may be thus found. Viz. 100 : 106 :: 1 : 1,06=the Amount of il. at 6 per cent. Or 100 : 105 :: 1 : 1,05 = the Amount of il, at 5 per cent. and so on for any other afligned Rate of Intereft. Then if R = the Amount of 1 l. for one Year, at any Rate.
R? the Amount of l for two Years.
R=the Amount of 1 l. for five Years.
As one Pound : is to the Amount of one Pound at one That is Year's End :: fo is that Amount: to the Amount of one Pound at two Year's End, &c.
Whence it is plain, 'that Compound Interest is grounded upon a Series of Terms, increasing in Geometrical Proportion continued ; wherein t (viz. the Number of Years) does always assign the Index of the last and highest Term : Viz. the Power of Ry which is R.
Again, As 1 : R::P:P R = A the Amount of P for the Time, that R the Amount of 1l.
As one Pound : is to the Amount of one Pound for any That is given Time :: fo is any proposed Principal (or Sum) to it's
Amount for the same Time.
From the Premises (I prefume) the Reason of the following Tbeorems, may be very easily understood.
Tbeoren 1. P R S A, as above.
? From hence the two following Theorems are easily deduced.
=P. Theorem 3. =R.
By these three Theorems, all Questions about Compound Intereft may be truly resolved by the Pen only, viz. without Tables; tho' not so readily as by the Help of Tables, calculated on Purpose; as will appear farther on.
Question 1. What will 256l. 10 s. amount to in seven years, at 6 per Cent. per Annum, Compound Interest?
Here is given P=256,5; t=7; and R=1,06 which being involved_until it's Index = t (viz. 7.) will become R* = 1,50363. Then 1,50363 x 256,5=385,681151=3852 135. 7 1 d. which is the Answer required.
Question 2. What Principal or Sum of Money must be put (or let) out to raise a Stock of 385 !. 135. 7 d. in seven Years, at 6 per Cent. per Annum, Compound Interest?
Here is given A=385,6811;. R=1,06; and t=7; to find P. by Theorem 2. Thus R=1,50363) 385,6811=Å (256,5=P. That is, P = 2561. 10 s. which is the Principal or Sum, as was required.
Question Question 3. In what Time will 2561, 10$. raise « Stock of (or amount to) 385 l. 135. 7 i d. allowing 6 per Cent. per Annum, Compound Intereft? Here is given P=256,5; A= 385,6811; R=1,06; to
A find by the third Theorem R =
385,6811 == = 1,50363,
256,5 which being continually divided by R=1,06 until nothing remain, the Number of those Divifions will be 7=t. Thus 1,06) 1,50363 (1,41852. And 1,06) 1,41852 (1,338225. Again 1,06) 1,338225 (1,262477. And so on until it become 1,06) 1,06 (1. which will be at the seventh Division. Therefore it will be r = 7 the Number of Years required by the Question.
Question 4. If 2561. 10 s. will amount to (or raise a Stock of) 385 1. 139.7 1d in seven Years Time; what must the Rate of Intereft be, per Cent. 'per Annum? Here is given P=256,5; A= 385,6811, and t=7, Quære
Put! ir te=R, then
273 +7906+ 210cc=R' = 1,50363=G
D 4 51 1+3 letr=, then D = 0,07197
1 Operation r = 1,00 +3=0,18
1,00 =r Divisor 1,18) 0,0719 0,06=
1,06 =rte R II to be rejected.
Then 1: 0,06 :: 100 : 6 the Rate per Cent. required.
The firft three Questions may be much more easily performed by the following Table, which is only the Amounts of one Pound for thirty-ninc Years.
That is, of R. RR.RRRRR and fo on to R39.
The Amounts The Amounts The Amounts!
of rlat 6 per
pound Intereft. 11.06R 142.2609039557 27 4.8223459407 : A 21.1236 RR152.3965581931 28 5.1116866971 3.11.191016=R16 7.5403516847295-4183878990.4 1.26247696
17 2.692772785739 5:7434911729, 5 1.3382255776 182.85433915 29 31 6.0881006432 61.4185191122 1913.025599502132 6.4533866818 . 7 1.5036302590 20 3.2071354722 336.8405898828 8. 1.5938480745 21 3-3995636005 34 7.2510252757 9 1.6894789590 22 13.6035374166 35 7.6860867923. A 10 1.7908476965 |23 3.8197496616136 8.1472519998 11 1.8982985583 24 14.0489346413 37 8.6360871198 12 2.0127964718 254.2918707197 389.1542523470
. 132.1329282601 1264.5493829629 39 9.7035074878 The Title of this Table shews it's Construction, and it's Ule will easily appear by an Example or twoi
E X A M PLE 1,
What will 375 1. 10 amount to in nine years, at 6 per Cent. per Annum, &c?
The tabular Number against 9 Years is 1,689479 which being multiplied with the Principal 375,5 will produce 634,3993 &c. viz. 634 7.-8 s. ferè, being the Amount or Answer required.
EX A M P LE 2.
What Principal (or Sum) must be put to Interest to raife a Stock of 634 l. 8 s. in nine years Time, at 6 per Cent. per Annum, &c
If the proposed Stock (viz. 634; 4) be divided by the tabulat Number that is against the given Number of Years (viz. 9.) the Quotient will be the Principal (or Sum) required. Viz. againft 9 is 1,689479. Then 1,689479) 634,4 (375,5 = 375 1. 10$ the Principal (or Sum) required.
EX 4 MPLE 3. In what Time will 3751. 105. raise a Stock of för amount to) 634 1. 8 s. at 6 per cent. &68
Divide the proposed Stock (viz. 634,4) by the given Principal (uiz. 375,5) and the Quotient will thew the tabular Number that stands over against the Time fought. Thus 375,5) 634,4 (1,689479 &c. this Number being fought in the Table, will be found to stand against 9 Years, which is the Time required,
But if the Quotient cannot be truly found in the Table of Amounts for Years, as above; then take out of that Table the nearest Number that is less, and make it a Divisor, by which you muft divide the first Quotient; and then seek the second Quotient in the Table of Amounts for Days (which is inserted a little further on) and it will assign the Number of Days: as in this Example.
In what Time will 563 1. amount to 8601. at 6 per Cent. per Annum, Compound Interest?
Answer. In 7 Years and 99 Days. Thus 563) 860 (1,52753 which shews the Time to be more (or above) seven Years; for over against 7 Years is 1,50363 which being made the new Divisor: Viz. 1,50363) 1,52753 (1,01589&c. this Number is the nearest Amount to 99 Days.
Note, If the Stock, Principal, and Time be given; the Rate of Interest will be best found by extracting the Root, &c. as before in the fourth Question.
The next Thing that I shall here propose, is to make this Table (which is only calculated for the Rate of 6 per cent.) universally useful for all the Rates of Compound Interest, wbicb I may presume to say, is a new Improvement of my own, being well satisfied it never was published before ; and not only so, buc I have heard several very good Artists affirm it was impossible to be done.
The Method of performing it is briefly thus, Let x= the Difference between 1,06 = R, the Amount of il, for one Year (in the Table) and any other proposed Amount of il for one Year; which admits of two Cases. : .
Cafe 1. If the proposed Rate be greater than the 1,06 = R, then will R+*=the true Amount of I l. for one year at that Rate.
Cafe 2. But if the proposed Rate be less than 1,06 = R, then
si-i=b, 1-2=1, 1-3 =d, 1-4=f, &c.