XVIII.-1. To find the time by means of Table IV.

Divide the given sum by the given annuity; the result will be the amount of £1 per annum in the required time.

In how many years will an annuity of £30 amount to £557 19 4, reckoning compound interest at 3 per cent. per annum?

557.966666÷30=£18.598888, the amount of £1 per annum at 3 per cent. in the time £18.598914 corresponds to 15 years, the time re

required;

And in Table IV., under 3 per cent., quired.

£1074 16 4 is the amount of an annuity of £40 laid up and constantly improved, during a certain number of years, at the rate of 3 per cent. per annum. How many years was the annuity forborne ? Answer, 20 years.

A person going abroad was constrained to forbear an annuity of £100 for a certain number of years; but on his return home received £315 5s. as the full amount of his claim, including 5 per cent. per ann., comp. interest, for the use of his money. For what period was his pension in abeyance? Answer, 3 years.

In how many years will a debt of £3949 2 11 be discharged, by investing annually for that purpose £500, to be improved at the rate of 4 per cent. per annum, compound interest? Answer, 7 years.

A farmer on quitting his farm being in arrears for rent, to the amount of £167 10 4, his landlord was glad to accept an arrangement for paying him £30 6 4 per annum, until the whole of the debt should be finally liquidated. In how many years would this be effected, supposing even on these terms the landlord allowed at the rate of 5 per cent. per annum, compound interest, upon every payment? Answer, 5 years.

2. The rate per cent. is found in the same manner as the above.

At what rate per cent., compound interest, will £780 10s. per annum amount, in ten years, to £9370 15 4?

9370.7666666780.5=12.006107, the amount of £1 per ann. in 10 years at the rate re

quired;

And in Table IV., at 10 years, £12.006107 corresponds to 4 per cent., the rate required. In forty-five years £3000 per annum amounted to £479100 9 4; at what rate per cent. per annum was the money improved, compound interest being allowed? Answer 5 per cent. per annum.

A tenant owed his landlord £1200 for arrears of rent, which he agreed to liquidate in nine equal annual payments, on condition that these payments should be allowed to increase at a certain per centage. The amount of the annual payment agreed on was £113 7 10; what rate per cent. was demanded?

Answer, 4 per cent. per annum.

An agent being intrusted with the investment of £500 per annum, increased it in seven years to £3949 2 11. What rate per cent. per annum was he enabled to obtain, compound interest being allowed? Answer, 4 per cent. per annum.

At what rate per cent. per annum will a pension of £100 amount to £315 58., supposing the money to be improved at compound interest during a period of three years? Answer, at 5 per cent. per annum.

XIX. To find the present value of £1 per annum for any number of years at any given rate per cent.

From unity, or 1, subtract the present value, as found by Table III., of £1 for the given time at the given rate:

Divide this difference by the given rate for £1, the result will be the required present value of £1 per annum.

What is the present value of £1 per annum for three years, money being discounted at the respective rates of 3, 4, and 5 per cent. per annum, compound interest?

OPERATION.

The interest of £1 at 3, 4, and 5 per cent. is denoted respectively by the decimals .03, .04, and .05; and the present value of £1, by Table III., for three years, is, at 3 per cent. £.915142; at 4 per cent. £.888996; and at 5 per cent. £.863838:

Making the present value of £1 per annum for three years, at the above rates, £2.8286, £2.7751, and £2.72324, respectively.

What is the present value of £1 per annum for four years, money being discounted at the respective rates of 3, 4, and 5 per cent. per annum, compound interest?

Making the present value of £1 per annum for four years, at the above rates, £3.7171, £3.6299, and £3.54596, respectively.

What is the present value of £1 per annum for five years, money being discounted at the respective rates of 3, 4, and 5 per cent. per annum, compound interest?

Making the present value of £1 per annum for five years, at the above rates, £4.5797, £4.4518, and £4.32948, respectively.

Continue these operations until the following Table V. be completely constructed at the several rates of 3, 4, and 5 per cent. per annum, compound interest, for the term of fifty years.

Shewing the present value of £1 per annum for any number of years, not exceeding fifty, at the several rates of 3, 4, and 5 per cent. per annum, compound interest.

XX.-TEMPORARY ANNUITIES.

To find by means of Table V. what sum paid down now is equal to the value of a given annuity for a given number of years, not exceeding fifty, at the several rates of 3, 4, and 5 per cent. per annum, compound interest.

Multiply the present value of £1 per annum, as found by Table V., for the given number of years at the given rate by the given annuity; the result will be the required present value.

Required to find what sum paid down will purchase an immediate annuity of £30, to continue twenty years, reckoning interest at 3 per cent. per annum? The pres. value of £1 per ann., by Table V., at 3 per cent., for 20 years=£14.877475; And 14.877475 x £30 £446.32425, or, £446 6 6, the present value required.

=

What sum would be required for the purchase of an annuity of £600 for fifteen years, supposing interest at 4 per cent. per annum? Answer, £6671 0 8. What is the present value of an annuity of £100 for the next seven years, at the rate of 5 per cent. per annum, compound interest? Answer, £578 12 9. of which the clear Ans. £1964 1 8.

What is the lease of an estate for twenty-one years worth, annual rent is £140, reckoning interest at 4 per cent. per ann.? What sum would be required for the purchase of the lease of an estate for fifteen years, producing a clear annual rental of £250, if 5 per cent. per annum, compound interest, were allowed for the use of money? Answer, £2594 18 4.

A landlord agreed to grant a lease for fifty years of an estate estimated at £350 per annum, out of which the tenant would be required to pay various charges to the amount of £65 annually. What sum ought to be given for the lease, allowing the purchaser 4 per cent. per annum, compound interest, for his money?

In questions of this kind, the clear annual rent must, after every deduction for reserved rent, tithes, taxes, and all other charges, be first ascertained.

Answer, £6122 8 6.

A farmer holds for the term of thirty-three years an estate by lease, of the full value of £290 per annum, for which he pays an annual rent of £110, and also other charges for tithes, taxes, and similar expences, to the amount of £40 annually. What sum ought he to receive for the disposal of his lease, supposing 5 per cent. per annum the rate of interest allowed? Answer, £2240 7 2.

What is the present value of the lease of an estate for forty-five years, estimated at £167 10s. per annum, which is, however, charged with the payment of an annuity of £67 10s. for twenty years, assuming an allowance of 3 per cent. per annum to the purchaser for the interest of his money?

The present value of £1 per ann. for 45 years at 3 per cent. by Table V.= £24.518713, X 167.5=£4106.8844;

The pres. value of £1 per ann. for 20 years by Tab. V.=£14.877475; × 67.5=£1004.2295; Therefore 4106.8844-1004.2295=£3102.6549, or, £3102 13 1, the required value. What is the present value of an annuity of £500, payable at the beginning of each year, for the next twelve years, reckoning interest at 5 per cent. per annum? For the present value of annuities, payable at the beginning of each year, multiply the present value of £1 per annum, for the given number of years less one, as found by Table V., increased by unity, by the given annuity, for the required value.

The pres. value of £1 per ann. for 11 years, by Table V., =£8.306414; and 8.306414+1

=9.306414;

Therefore 9.306414 × 500 £4653.207, or, £4653 4 2, the required value.

A gentleman, entitled to an annuity of £60 for nineteen years to come, the first due, being suddenly called abroad, is desirous to dispose of the same immediately, and is willing to allow the purchaser interest at the rate of 4 per cent. per annum. What sum ought he to receive as the present value of his title? Ans., £819 11 2.

XXI. TEMPORARY ANNUITIES continued.

1. To find by means of Table V. the annuity a given sum will purchase for a given number of years at a given rate per cent.

Divide the given sum by the present value of £1 per annum for the given number of years at the given rate per cent., for the annuity required.

What annuity, to continue twenty years, may be purchased for £446 6 6, when interest of money is at 3 per cent. per annum?

The pres. value of £1 per ann. is, by Table V., at 3 per cent. for 20 years=£14.877475; And 446 6 6, or, 446.325, × 14.877475-£30, the annuity required.

A farmer being desirous to obtain a lease of ground for seven years is asked to pay down a fine, or grassum, of £578 12 9. Being, however, unprepared with ready capital, he is willing to submit to an annual charge in compensation of the fine. How much additional rent ought to be levied upon him as an equivalent for the same, allowing his landlord at the rate of 5 per cent. per annum, compound interest, for his money? Answer, £100 per annum. What annual rent ought an estate to produce, the lease of which for fifteen years is valued at £6671 0 8, in order that the purchaser may realise 4 per cent. per annum interest? Answer, £600 per annum.

2. To find the time a temporary annuity ought to continue for a given sum at a given rate per cent.

Divide the given sum by the given annuity, the result will be the present value of £1 per annum in the time required.

For how many years may an annuity of £250 be purchased for £2594 18 4, when interest of money is 5 per cent. per annum?

2594 18 4, or, 2594.9166,÷250-£10.37966, the present value of £1 per ann., at 5 per cent., in the time required;

And in Table V., under 5 per cent., 10.37966 corresponds to 15 years, the time required.

The lease of an estate, estimated at the clear annual rent of £285, was sold at the auction mart for £6122 8 6. How many years had the lease to run, supposing the purchaser was allowed at the rate of 4 per cent. per annum, compound interest, for his money? Answer, 50 years.

A farmer being desirous to retire from agricultural pursuits disposes of his lease for £2240 7 2, the estate being estimated at the clear annual rent of £140. How many years was the purchaser entitled to possession, allowing him interest for the use of his money at the rate of 5 per cent. per annum ? Answer, 33 years.

3. The rate per cent. is found in the same manner as the above. For the lease of an estate for forty-five years, producing £167 10s. per annum, the sum of £4106 17 8 was given. Required what rate per cent. per annum was allowed the purchaser in the transaction?

4106 17 8, or, 4106.8833,÷167.5=£24.5187, the present value of £1 per annum at the rate required;

And in Table V., at 45 years, 24.5187 corresponds to 3 per cent. per annum, the rate required.

The lease of an estate for twelve years, estimated at the clear rent of £500 per annum, sold for £4431 13s.; what rate of interest did the purchaser make of his money? Answer, 5 per cent. per annum. The creditors of a bankrupt sold, for the benefit of his estate, an annuity of £60, to which he was entitled for nineteen years, for £788 0 8. What rate of interest was allowed? Answer, 4 per cent. per annum.