X. To find, by means of Table II., the amount of any given sum for any given number of years, not exceeding fifty, improved at compound interest, at the several rates of 3, 4, and 5 per cent.

per annum.

Find, by the Table, the amount of £1 for the given number of years, and multiply it by the given sum; the product will be the amount required.

Required the amount of £1000, invested at comp. int. for 30 yrs., at 3 per ct. per an.? The amount of £1 for thirty years, at 3 per cent., is, by the Table, £2.427262; £1000, the given sum, (II. 3) £2427.262, or £2427 5 3,

Therefore £2.427262

amount required.

=

the

What is the amount of £100, improved for 5 years, at 4 per cent, per an., comp. int.? The amount of £1 for five years, at 4 per cent., is, by the Table, £1.216652;

Therefore £1.216652 X £100, the given sum, (II. 3) £121.6652, or £121 13 4, the

amount required.

Required the amount of £150 10, improved for 10 yrs. at comp. int., 5 per ct. pr. an.?
The amount of £1 for ten years, at 5 per cent., is, by the Table, £1.628894;
The given sum, £150 10,=£150.5 (VII.);

=

Therefore £1.628894 × £150.5, the given sum, £245.1485, or £245 3 0, the amount required.

The sum of £355 15 was invested for four years, at the rate of 4 per cent. per annum, compound interest; required the amount at the expiration of the term? Answer, £416 3 7.

A father deposited, at their respective births, the sum of £250 for each of three children. intending the same to be accumulated at compound interest. The portion for the first child was to be improved at the rate of 3 per cent. per annum; for the second, at 4 per cent.; and for the third, at 5 per cent. per annum. To what amount would each child be entitled on attaining the age of twenty-one?

Answer, the first to £465 1 6; the second to £569 13 10; and the third to £696 9 10.

To what sum will the present payment of £120 entitle a person, at the expiration of twenty years, supposing money to accumulate at the rate of 4 per cent. per annum, compound interest? Answer, £262 18 9.

A legacy of £500 was bequeathed to each of three persons, of the respective ages of thirtyfive, forty, and forty-five, on condition that the several amounts should be paid on each life attaining the age of fifty, and that the principal should, in the mean time, be improved, at the rate of 4 per cent. per annum, compound interest. The parties having survived the term, what amount was due to each?

Answer, £900 9 5 to the youngest; £740 2 5 to the next; and £608 6 6 to the eldest.

A gentleman received, at the age of thirty, a legacy of £3000, half of which he invested for accumulation, at 3 per cent. per annum, compound interest. Supposing he should attain the age of fifty-eight, to what sum would his investment then amount?-Answer, £3431 17 10.

Assuming money to increase at the rate of 5 per cent. per annum, compound interest, what amount ought to be received, six years hence, equivalent to a sum of which the present value is £45 17 6? Answer, £61 9 7.

Two several sums of £2800 each were invested at 4 and 5 per cent. per annum, compound interest; required the difference of the amounts at the expiration of five years?

Answer, £166 19 2.

XI. 1. To find, by means of Table II., the time in which a given sum will increase to a given amount.

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

How long must the sum of £1000 be invested, at the rate of 3 per cent. per annum, compound interest, to amount to £2427 5s. 3d.? 2427.262÷1000=(III.2)2.427262, the amount of £1 at 3 per cent.in the time required; And in the Table under 3 per cent. 2.427262 gives 30 years, the time required.

The sum of £100 being improved, at 4 per cent. per annum, compound interest, amounted to £121 13s. 4d. What period was required for this accumulation?

121.666÷100=(III. 2) 1.21666, the amount of £1 at 4 per cent. in the time required; And in the Table under 4 per cent. 1.21666 gives 5 years, the time required.

In what time will the investment of £150 10s., at 5 per cent. per annum, compound interest, amount to £245 3s. Od.?

245.1458÷150.5=1.62871, the amount of £1 at 5 per cent. in the time required; And in the Table under 5 per cent. 1.62888 gives 10 years, the time required.

Assuming that money improves at the rate of 4 per cent. per annum, comp. interest, in what time will £100 amount to £564?—Ans. in 44 years.

A. owed B. £355 15s., which they mutually agreed should be paid off when the amount reached £416 3s. 7d., at 4 per cent. per annum, compound interest. What extension of time did B. allow to A. to liquidate the debt? Answer, 4 years.

How many

A young lady having deposited £120 to accumulate at compound interest, lived to receive £262 18s. 9d. in lieu of the original sum. years had she to wait for this improvement to her money, supposing it to have increased at the rate of 4 per cent. per annum?-Answer, 20 years.

2. The rate per cent. is found in the same manner as the above. In thirty years £1000 amounted to £2427 5s. 3d.; at what rate per cent. per annum, compound interest, was the money invested? 2427.262÷1000=(III. 2) 2.427262, the amount of £1 in 30 years, at the rate required; And in the Table at thirty years 2.427262 gives 3 per cent., the required rate.

An investment of £100 amounted in five years to £121 13s. 4d.; required the rate per cent. per annum, compound interest?

121.666÷100=(III. 2)1.21666, the amount of £1 in 5 years, at the rate required; And in the Table at five years 1.21666 gives 4 per cent., the required rate.

At what rate per cent. per annum, compound interest, will £150 10s. amount, in ten years, to £245 3s. Od.?

245.1485÷150.5=1.62888, the amount of £1 in 10 years, at the rate required ; And in the Table at ten years 1.62888 gives 5 per cent., the required rate.

Three several sums of £250 each were invested for twenty-one years, and amounted, at the expiration of that term, to £465 ls. 6d., £569 13s. 10d., and £696 9s. 10d., respectively. At what rate per cent. per annum, compound interest, were the several investments improved?

Answer, at 3, 4, and 5 per cent. per annum, respectively.

D

XII. To find what sum paid down now is equal to the value of £1 to be received at the end of a given number of years, according to a given rate per cent. per annum.

Divide £1 by the amount of £1, as found by Table II., at the given rate per cent. for the given term: that is, the reciprocals (V.) of Table II. will be the required present values.

What sums should be paid down now as the present values of £1 to be received three years hence; the rate of discount being 3, 4, and 5 per cent. per annum, respectively?

By Table II. the amount of £1, improved for 3 years, at 3 per cent., is £1.092727; at 4 per cent., £1.124864; and at 5 per cent., £1.157625:

Therefore, by dividing £1 by each of these amounts, the required present values will be obtained.

1.092727)1.0000000(.915141 1.124864)1.0000000(.888996 1.157625)1.0000000(.863837

Making the present value of £1, due three years hence, at 3 per cent. discount= £.915141; at 4 per cent. discount=£.888996; and at 5 per cent. discount: £.863837.

=

Proceed now to find the present values of £1, due four years hence, at 3, 4, and 5 per cent. per annum discount.

By Table II. the amount of £1, improved for 4 years, at 3 per cent., is £1.125508; at 4 per cent., £1.169858; and at 5 per cent., £1.215506:

Therefore, by dividing £1 by each of these amounts, the required present values will be obtained.

1.125508)1.0000000(.888487 1.169858)1.0000000(.854804 1.215506)1.0000000(.822702

Making the present value of £1, due four years hence, at 3 per cent, discount= £.888487; at 4 per cent. discount=£.854804; and at 5 per cent. discount= £.822702.

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

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

XIII. To find, by means of Table III., what sum paid down now is equal to the value of a given sum, due at the end of a given number of years, not exceeding fifty, at the several rates of 3, 4, and 5 per cent. per annum, discount.

Multiply the present value of £1, as found by Table III., due at the end of the given number of years at the given rate, by the given sum forborne: the result will be the required present value.

Supposing £1000, which will not be due until thirty years hence, were required to be converted into ready money now, what is the present value of it, allowing discount at the rate of 3 per cent. per annum?

The present value of £1 for 30 years, at 3 per cent., is, by the Table, £.411987; Therefore .411987 × 1000=(II. 2) £411.987, or, £411 19s. 9d., the present value required.

What principal will amount to £121 13s. 4d. in five years, presuming a rate of 4 per cent. per annum, compound interest; that is, what is the present value of £121 13s. 4d., forborne for five years, discount being allowed at the rate of 4 per cent, per annum?

The present value of £1 for 5 years, at 4 per cent., is, by the Table, £.821927; Therefore .821927 × 121.666=£100, the present value or principal required.

What sum should be paid down now as the present value of £150 10s. due ten years hence, reckoning discount at the rate of 5 per cent. per ann.? The present value of £1 for 10 years, at 5 per cent., is, by the Table, £.613913; Therefore .613913 × 150.5= £92.394, or, £92 7s. 11d., the present value required. A person requiring present capital desires to dispose of £3000 to which he is absolutely entitled twenty years hence; what is the present value of his reversion, reckoning discount at the rate of 5 per cent. per annum, compound interest? Answer, £1130 13s. 4d. What sum laid up now, and improved at compound interest during twenty-four years, will amount to £500, at the rate of 4 per cent. per annum? Answer, £195 ls. 3d.

A party, on obtaining a legacy of £1500, appropriated only so much of it to present uses as enabled him, by investing the remainder, to receive the full amount of the legacy seven years afterwards. What sum did he lay up to accumulate, and how much did he apply to immediate purposes, assuming that money improved at the rate of 3 per cent. per annum, compound interest? Answer, £1219 12s. 9d., the sum invested; and £280 7s. 3d., the sum used.

A. being under bond to pay to B., or his executors, £250 at the end of twenty-one years, is willing to discharge the claim at once, upon being allowed discount at the rate of 4 per cent. per annum. What present sum is B. entitled to receive? Answer, £109 14s. 2d.

At the expiration of six years a certain sum forborne amounted to £175; required to know what that sum was, supposing it to have accumulated at the rate of 5 per cent. per annum, compound interest?-Ans. £130 11s. 9d.

What principal will amount to £360 10s. in fourteen years, at 3, 4, and 5 per cent. per annum, respectively, compound interest?

Ans., at 3 per cent. £238 6s. 8d.; at 4 per cent. £208 3s. 7d.; and at 5 per cent. £182 1s. 7d. DISCOUNT is the difference between the principal sum forborne and its present value. The discount therefore in the last question is, At 3 per ct. £122 3s. 4d., at 4 per ct. £152 6s. 5d., at 5 per ct. £178 8s. 5d.