which (by the rule page 214,) is 1.05 (

a

-

Or, the square root of 1.340095640625 is 1∙157625, and the cube-root of 1.157625 is 1·05 as above. Hence the rate is 5 per cent.

(10.) At what. rate per cent. will 2757. amount to 318/. 6s. 11 d. in 3 years time?

(11.) At what rate per cent. will 7007. 15s. amount to 819l. 158. 63d-2504832 in 4 years?

(12.) At what rate per cent. will 8007. amount to 12418. 1s. 3'017467875d. in 9 years?

Examples to Prop. 4. Theo. IV.

(13.) In what time will 200l. amount to 2681. Os. 4d-363, at 5 per cent. per annum?

£268 0'41.363

£268 019128125 (= a.)

Then 268 019128125÷2001·340095640625

a

which, being divided by 1·05, (r,) and the quotient by 1'05, &c. till nothing remains, the number of divisions will shew the time, 6 years.

Note. This method of finding the time, by repeated divisions, is made use of by Mr. Ward, (see Ex. 3rd, page 255, 8th edit. of his Math Guide,) and several writers have followed his example; but it is far from being an eligible or correct method. It may serve to prove a question, when the time happens to be whole years. The best method of solving the questions in this and the preceding proposition is by logarithms.

(14.) In what time will 2754. amount to 3187. 6s. 11‡d., at 5 per cent. per annum?

(15.) In what time will 7007. 15s. amount to 819/. 15s. 62d 2504832, at 4 per cent. per annum ?

(16.) In what time will 800l. amount to 12411, 1s. 3 017467875d. at 5 per cent. per annum?

EQUATION OF PAYMENTS AT COMPOUND INTEREST.

Proposition. Having several debts, due at different times, from one person, to find the TRUE equated time for paying the whole at once, without loss either to the debtor or creditor, allowing compound interest.

Rule 1. Find the amount of each debt from the time it becomes due to the time of the last payment, [by Prop. 1. Compound Interest,] add these amounts, together with the last payment, into one sum.

2. Find in what time [by Prop. 4, Compound Interest,] the sum of the debts will amount to the sum of the amounts found above:-this time, subtracted from the time the last payment becomes due, will give the true equated time.

Note. This rule, which is Sir Samuel Moreland's, is founded on the same manner of reasoning as the common rule, Part I. and will bring out the same answer, allowing simple interest instead of compound.Were this a place for algebraical demonstrations, it might easily be shewn, that the above rule is universally true, allowing compound in terest, whether we argue from Burrow s, Kersey's, or Malcolm's principles, it being deducible from each.

Examples.

(1) A owes to B 1000l., 2001. of which will be due one year hence, 2001. two years hence, 150l. three years hence, 3007. four years hence, and 1507. five years hence; should these persons agree to have the whole discharged at once, what will be the true equated time, reckoning interest at 5 per cent. per annum?

105 4×200=243-10125 amount of the first payment.

200+200+150+300+150-£1000, sum of the debts. Now we have to find in what time £1000 will amount to £1105.00125, at 5 per cent. compound interest; and here we must be under the necessity of making use of logarithms, since the method made use of in Ex. 13, page 245, will by no means do.

0.0433628 (=log. a-log. p;) this divided by log. 1·05 (log. r) = 0.0211893 gives 2.0464, &c. for the quotient, (=t.) Hence 5 years 2·0464, &c. years=2.9535, &c. the true equated time.

(2.) A person has 3207. due to him; and, at the end of 5 years, 967, more will be due from the same debtor; now

both parties have agreed for the whole to be discharged The true equated time is required, reckoning interest at 5 per cent. per annum?

at once.

(3.) There is 100l. payable one year hence; and 1051, payable three years hence, what is the true equated time, allowing compound interest at 5 per cent. per annum?

years

(4.) There is 1007. payable one year hence, 2007. two years hence, 300l. three years hence, and 500l. five hence; required the true equated time for paying the whole at once, reckoning compound interest at 5 per cent. per annum?

ANNUITIES CERTAIN.

Definition 1, Annuities certain signify any interest of money, rents, or pensions, payable yearly, or from time to time, to some certain period, or for ever. They are divided into two parts, viz. annuities in possession, or such as are either entered upon, or are to be entered upon immediately; and annuities in reversion, or such as are not to be entered upon till some particular future event has happened, or till some given period of time has elapsed; and the time the purchaser holds the annuity, after he has entered upon it, is called the reversion.

2. An annuity is said to be in arrears when the debtor keeps it in his hands for any certain time after the term of payment; and the sum of all the single payments, together with the interest due upon each payment from the time of its becoming due to the time the whole is paid off, is called the amount of such annuity.

3. When an annuity, to be entered on immediately, or some time hence, is sold for ready money, the price which ought to be paid for it is called the present worth.

ANNUITIES AT COMPOUND INTEREST.

Let n the annuity, pension, rent, or payment, whether yearly, half-yearly, or quarterly.

the time, or number of payments,

↑ -- the ratio, or amount of £1. for a year, year, I year, &c. according as the payments are made yearly, halfyearly, quarterly, &c. by either of the methods or tables given in compound interest.

the amount,

ANNUITIES in Arrears, at Compound Interest.

Proposition 1. Given the annuity, payable in whole years, or at any equal number of payments, the rate per cent., and time, to find the amount.

Rule. Make an unit the first term of a geometrical series, the amount of £1 for 1 year, year, ‡ year, &c. the ratio, according as the payments are made yearly, half-yearly, quarterly, &c.-Carry the series to as many terms as there are payments, and find its sum, (by Prop. 2, of Geometrical Progression), which multiply by the annuity, and the product will be the amount.

1

Or, Theo. I. Xna, when n, r, and t are given.

r-1

Logarithmically. log. rt—1+log.n.—log, r—1—log. a.
Prop. 2.
Given a, r, and t, to find n.

Logarithmically. log. r—1+log. a—log. —1—log, n.
Prop. 3. Given a, r, and n, to find t.

Theo. III.

be found with logarithms.

+1=rt. If t be not a whole number, it cannot)

n

the value of r must be found as directed in the 2d note in Double Position. If t be a mixed fraction, the value of r cannot be found without logarithms.

Examples to Proposition 1.

(1.) What is the amount of an annuity of 1007. tocontinue 5 years, at 6 per cent. per annum, compound interest?

1+1·06+1·06 2+1·06]3+1·06] 4

0614. -1

+1.06 4=

1.06

5.63709296; this multiplied by 100, the annuity, gives 563-709296 =5631. 14s. 2.23104d. for the amount required.

Or, by Theorem I.

1.06×1.06 × 1·06 × 1·06 × 1·06—1.3382255776=rt and 1.3382255776-1-3382255776—rt-1; also, 1.06-106==r—1.

Hence

-3382255776

'06

·×100=563·709296, the amount as before.

(2.) What is the amount of an annuity of 801. unpaid, or in arrears, for 9 years, at 5 per cent?

(3.) Required the amount of an annuity of 560l. to continue 7 years, at 5 per cent. per annum ?

The following Examples are performed by Logarithms. (4.) If a pension of 3567. per annum, payable halfyearly, be unpaid for 9 years, what will it amount to at 6 per cent ?

Here n=178, r=1·03, (by Table I. p. 242), and t=18. log. r=log. 103=0·0128372

+2.0970248

log. r-1-log. 03-2·4771213

log of the amount +3.6199035, the number

answering to which is 4167-768_41671. 15s. 4·32d. answer.

(5.) What is the amount of an annuity of 350l., payable half-yearly, unpaid for 4 years, at 41 per cent. ?