Theorem 3.

{RA+u-A-R. If this Equation be continually

divided by R, until nothing remain, the Number of those Divifions will be t. See Page 255.

into Numbers, according to the Method proposed in Selt. 3. Chap. 10. the Root will shew the Value of R.

Question 1. If 301. Yearly Rent, or Annuity, &c. be forborn (i. e. remain unpaid) Nine Years; what will it amount to, at 6 per Cent, per Annum Compound Interest?

Here is given u=30, t=9, and R = 1,06; to find A. per Theorem I.

R91,689479 By the Table of Amounts for Years

30=1

Ru=50,684370

-и=30,

R-1=0,06) 20,684370 (344,7395-3441 145. 9d.=A the Amount required.

Question 2. What Yearly Rent or Annuity, &c. being forborn or unpaid Nine Years, will raise a Stock of 344 1. 145. 9d. =344,7395, at 6 per Cent. &c.

Here is given A=344,7395, 19, and R=1,06; to find u. per Theorem 2.

AR = 344,7395×1,06 = 365,42387
-A=344-7395

R-11, 689479-1 = 0,689479) 20,68437 (30=u

Question 3. In what Time will 301. Yearly Rent raise a. Stock or Amount to 3441. 14s. 91d. allowing 6 per Cent. for the Forbearance of Payments?

Here is given u = 30, A=344,7395, and R=1,06; to find t.

per Theorem 3.

First

AR+u-A=365,42387+30-344,7395-50,68437.

And u=30) 50,68437 (1,689479=R'. Then R=1,06) 1,689479 (1,593848. And 1,06) 1,593848 (1,50363; and so on until it become 1,06) 1,06 (1. which will be at

the Ninth Division; therefore t=9.

Or R1,689479, being fought in the Table of Amounts for Years, will be found to stand overagainst 9 Years, which is the Time required.

Question 4. If 301. per Annum, being unpaid Nine Years, will amount to 3441. 145. 9d. allowing Compound Interest for every Payment as it becomes due, What must the Rate of Interest be per Cent. &c.

Here is given u=30, A=344,7395, and t=9; to find R

by the last of the Four Æquations, Viz.

A

{습

A-u

R-R-

A-u

3447825 11,491317. And = 10,491317.

Hence there is this Equation; 11,491317 R-R9=10,491317

Let/1/+e = R, and supposer=1

1929+gre +36r7 ee = R9

IX-in Numb. 3 11,491317+11,4913178=11,491317R

2 in Numb. 4 1,000000+9,000000e+36ee=R9

3-4510,491317+2,4913178-36ee=10,491317

Whence 6

2,491317e

636ee
636 780,06 &c.

First r=1

+ = 0,06

As may be easily try'd by invol} = 1,06=R<ving 1, it, and ordering it, as the

Æquation above directs,

Section 3. To find the Present Worth of Annuities, Penfions, or Leafes, &c. at Compound Interest.

Let P=the present Worth of any Annuity, or Leafe, &c. and the reft of the Letters as before.

Then, from what has been faid in Section 3. Chap. II. about Purchasing of Annuities, &c. at Simple Interest, it will be easy to form the like Theorems here at Compound Interest, viz. by Combining Theorem 1. Page 266. and Theorem 1. Page 254. into one

Theorem.

The Amount of any Yearly Rent being unany Number of Years. Per Theo

AS paid

= A

{

rem 1. of the last Section. Page 266. The Amount of any Principal or Sum being put to Interest, for the fame Number of Years. Theorem 1. Page 254.

Per

Hence

Viz. PR-PR = uR - u being the very same Æquation with that of my Compendium of Algebra, Page 112. which is there raised from the Confideration of purchasing Annuities, or taking of Leafes, &c. to be grounded upon a Rank or Series of Geometri

u

R

cal Proportionals continually decreasing. Thus is the First and Greatest Terms; R the common Ratio of all the Terms; and Pis the Sum of all the Series.

RRR

и

R

น be the Sum of all R

the Sum of all the Confequents.

Or (in the fame Ratio) u :

น R

::P:P-
R

which produces PR+ + '-uR=PR-u. As above. From this Æquation may be deduced the following Theorems.

Theorem 4.

P

P+-PR

=

P

= R {Which

R

being continually divided

by R, will give t.

R+RR+'. The Resolving of

which Æquation will discover the Value of R.

Question I. What is 30 l. Yearly Rent, to continue Seven Years, worth in ready Money, allowing 6 per Cent. Compound Interest to the Purchaser?

Here is given u = 30. t = 7. And R = 1,06 to find P.

Then R-1=0,06) 10,0483 (167,4716=P=1671.9s.5d. being the Answer required.

Question 2. What Annuity or Yearly Rent, to continue Seven Years, may be purchased for 167 1. 9 s. 5 d. allowing 6 per Cent. Compound Interest to the Purchaser?

In this Question there is given P=167,4716. 17:
And R=1,06 to find u . By the Second Theorem.
First PR XR = 251,8153 × 1,06=266,9242
And - PR=167,47 16 × 1,50363)=251,8153

Then R-1 = 0,50363)

That is u = 30 1. the Answer required.

15,1089 (30=u

Question 3. How long may one have a Lease of 30 1. Yearly Rent, for 167 1. 9 s. 5 d. allowing 6 per Cent. Compound Interest to the Purchaser?

Then 19,9517)30=u(1,50363=R

If this 1,50363=Rt be either continually divided by 1,06=R until nothing remain (As before in Page 255.) Or if it be fought in the Table of Amounts for Years, &c. it will discover t=7 which is the true Anfwer required.

Question 4. Suppose one should give 167 l. 93. 5d. for the Purchase of a Pension, or Annuity of 30 1. per Annum, to continue Seven Years; At what Rate of Interest, per Cent. would that Purchase be made, aliowing Compound Intereft to the Purchaser?

In this Question there is given, P=167,4716. u=30 and

12

1=7 to find R. Per Theorem 4 in this Equation {1=블

R+RR+which being brought into Numbers, and its Root extracted, as in the fourth Question of the last Section; the Value of R will be found 1,06, and then it will be 1: 0,06:: 100:6 the Rate per Cent, as was required,

a

Thefe

:

These Four Questions include all the Varieties that can be propofed about purchasing Annuities or Leafes, &c. which are to be either immediately enter'd upon, or in Poffeffion at the Time when the Purchase is made.

But such Questions as relate to Annuities, or a taking of Leafes, &c. in Reversion, must be parted or divided into two distinct Questi ons, each to be separately confider'd by itself (See Page 252.) As in the following Examples.

Example 1. Suppose it were required to compute the present Worth of 751. Yearly Rent, which is not to commence or be enter'd upon, until Ten Years hence; and then to continue Seven Years after that Time: at 6 per Cent. &c. Compound Interest?

The First Work in this Question is, to find what 751. per Annum, to continue Seven Years, is worth in ready Money; as if it were to be immediately enter'd upon: And to perform that, there is given u=75. R=1,06. and 1=7. to find P. as in the First Question of this Section.

Then, R-1=0,06) 25,1207=418,6783=4181. 145. 64d. the Answer to the First Part of the Question.

Then the next Work will be, to find what Principal or Sum being put out Ten Years, at 6 per Cent. &c. will amount to 4187. 14s. 6d. Here is given A=418,6783, R= 1,06, t = 10. to find P. Per Theorem 2. Page 254.

Thus R° 1,790847) 418,6783= A (233,7884 = 2331. 15 s. 9 d. the present Worth of 75 l. per Annum in Reversion, &c. As was required.

Example 2. What Annuity or Yearly Rent to be enter'd upon Ten Years hence, and then to continue Seven Years, may be purchased for 2331. 155. 9 d. Ready Money, at 6 per Cent. &c. Compound Intereft ?

In the ist Work of this Question there is given, P=233,7884 R=1,06. And t=10 (the Time which the Annuity is not to be enter'd upon) to find A. Per Theorem 1. Page 254.

Thus, PR = 233,7884 × 1,790847 = 418,6783 A the

Amount

.