For In calculating compound interest by this table, take the multiplier opposite the number of years and under the given rate, multiply the principal by it, and the product will be the amount. most cases it will be sufficient to employ four decimal places in the multiplier, increasing the last figure by 1, if the figure omitted is 5 or more. NOTE. If the time be greater than the extent of the table, take any two or more multipliers, whose times, added together, will equal the given time, and multiply by them successively; that is, multiply the principal by one of them, and that product by another, and so on-thus, 11. Find the amount of $1,000.00 at 5 pr. ct. Compound interest, for 36 years. Multiplier, by table 3.22509, for 24 years. Then 1,000×3.22509 =3,225.09 and 3,225.09×1.79585 (multiplier for 12 years)=$5,791. 7778+ Ans. NOTE. The pupil will understand why we multiply successively by these multipliers, if he considers that the product $3,225.09 is the amount for 24 years. This, by the rule for compound interest, must be made the principal for the succeeding time; and, as 12 years remain, the question becomes, what is the amount of $3,225.09 for 12 years. We must therefore multiply $3,225.09 by the multiplier corresponding to 12 years. 12. Find the amount of $7,500.00 at 6 pr. ct. Compound interest for 4 years. Ans. $9,468.529. NOTE. This answer is strictly correct, as will be seen by performing the example by rule I. The table will give $9,468.536. Thus it will be seen that, in ordinary cases, the error by table is of no importance. 13. Find the amount of $5,000.00 for 18 yrs. 6 mo. at 5 pr. ct.. Compound interest. Ans. $12,333.876. 14. Find the amount of $1,108.092 at 6 pr. ct. Compound interest for 11 years. 15. Find the amount of $63,700.00 for 19 years, at 5 pr. ct. Compound interest. 16. Find the amount of $47,753.18 for 17 yrs. 6 mo. 9 d. at 5 per ct. Compound interest. If a principal be multiplied by a number taken from the table, an amount is obtained. Of course, if this amount be divided by the same number, the quotient will be the principal again. In calculating present worths, therefore, at compound interest, on sums due at a future day, we may divide the debt by the number in the table, corresponding to the given time. The present worth, thus obtained, being subtracted from the whole debt. will give the discount. 17. What is the present worth of $304.900096 due 3 years from the present time, when compound interest is allowed at 6 per ct. ? A. $256. 18. Find the present worth of $1,593.30, for 20 yrs., compound interest being allowed at 5 per cent. A. $600.50- Of $1,925.881, for the same time, at 6 per cent. A. $600.50 19. Find the discount at 5 per cent. on $607.753, for 4 years. A $107.7531. 20. Find the discount on $1,642.992 for 21 years, at 4 per cent. A. $921.992. This divisor must be calculated, as it is not in the table. 21. Find the present worth of $875.00 for 6 years at 5 per cent. Of $947.00 at 6 per cent. for 9 years. Of $1,785.00 for 16 years at 5 per cent. Of $8,963.25 at 5 per cent. for 11 years. At simple interest, a sum is 16 yrs. 8 mo. in doubling itself. At compound, it is 11 yrs. 10 mo. and between 21 and 22 days. But as simple interest only doubles the original principal, every 16 yrs. 8 mo., and compound interest doubles its amount, every 11 yrs. 10 mo. and 21 or 22 d., is evident that a sum will increase vastly more rapidly at compound, than at simple interest. In a few centuries the increase of a sum of money at compound interest becomes almost incredible. § LXXXIX. TO CALCULATE INTEREST ON NOTES WHEN THERE ARE ENDORSEMENTS. We have hitherto used the term NOTE without defining it. A NOTE is a written promise to pay a sum of money, either on demand (that is, when it is required,) or after a certain space of time has elapsed. Notes are given with or without interest. Unless the words, "with interest" are expressed, a note is understood to be without interest. If a note without interest given for a specified time, be not paid when it is due, it draws interest afterwards, till paid. Payments in part, or partial payments are sometimes made, and, in this case, a written acknowledgement is made on the note, called an ENDORSEMENT. As the debtor is not obliged to make payments in part, before payment is demanded, or before the note falls due, it is evident that he ought to be allowed interest on the payment, up to the time of settlement, if the note is given on demand, or till it is due, if given. for a specified time. For if he had kept the money in his own hands, he might have had the use of it for that t.me. And, after he has paid it, the creditor has the same advantage. Hence, the following rule, 1. FIND THE AMOUNT OF THE PRINCIPAL OF THE NOTE FOR THE WHOLE TIME. II. FIND THE AMOUNT OF EACH PAYMENT, FROM THE TIME IT WAS PAID, TILL THE TIME OF SETTLEMENT. III. ADD THE AMOUNTS OF THE PAYMENTS, AND DEDUCT THE SUM FROM THE AMOUNT OF THE PRINCIPAL. NOTE. When the note is for a specified time, it should be recollected that the payments ought to draw interest no longer than till the note is due. $500.00 EXAMPLES. Hartford April 1, 1828. 1. On demand I promise to pay PACKARD AND BUTLER, or order, five hundred dollars, with interest; value received. TIMOTHY BOOKWORM, On this note were the following endorsements. Jan. 16, 1829, rec'd $150.00 April 1, 1829, $150.00 July 13, 1830, 99 $100.00 What was due Aug. 1, 1830. Whole time, 2 yrs. 4 mo.; time for 1st payment, 1 yr. 6 mo. 15 d.; for 2d, 1 yr. 4 mo.; for 3d, 18 d. Hence we have the following Rates, .14; .0925; .08; .003-Then, to find amounts, 500×1.14= 570. 150X1.0925-163.875. 150×1.08-162. 100×1.003=100.3. Then, amt. of prin. $570.000 and $426.175 163.875+162+100.3= sum of amts. of payments, Hence there remains due $143.825 2. For value received, I promise to pay E. P. BARROWS, Jr. or bearer, one thousand dollars with interest. Hartford, March 10, 1827. WILLIAM MORRIS. On this note were the following endorsements. June 28, 1827, rec'd $500.00 Aug. 4, 1827, $200.00 Jan. 16, 1829, Nov. 22, 1828, What is due on the note Aug. 19, 1830. Amount of prin. Sum of amts. of payments, $1,206.50 $1,050.75 $155.75 Due 3. For value received, I promise to pay SELDEN W. SKINNER, OF order, six hundred and seventy five dollars. Hartford, April 17, 1816. SIMEON FAITHFUL. On this note were the following endorsements. May 7, 1817, rec'd $148.00 4. On demand, I promise to pay GEO. D. PRENTICE, or order, nine hundred and ninety nine dollars, ninety nine cents and nine mills; for value received. Hartford, May 19, 1823. $999.999 THOMAS PUNCTUAL On this note were the following endorsements. June 1, 1824, rec'd $300.00 NOTE. This rule has been commonly considered as incorrect. Accordingly we are told that "though a very common rule it is a very bad one," and that it may answer for short periods of time, but, in a long course of years will be found very erroneous." The objection made, is, that the interest allowed on the ments, will in time "expunge," or pay off the principal itself. For, it is said, that if a note be given for $100.00, and $6.00 be annually paid and endorsed on the note, then, as each of these $6.00 endorsements begins to draw interest from the time of its date till the settlement, it will be found in 10 years, that only $89.80 is due the creditor, according to the above rule. Now, it is said, these several annual payments were justly due the creditor on account of interest, since $6.00 is the annual interest of $100.00. It is therefore unfair that they should "eat up" the principal, "on which not one cent has been paid, but only its annual interest." We are further told, that, in 20 years, there would be due the creditor only $37.60, "not the least fraction of the $100.00" principal having been paid. "Extend it to 28 years and the creditor would become the debtor without receiving a cent of the $100.00." The fact that the debt is thus "expunged" we grant, and this will take place, allowing 365 days to the year, in 23 years, 27 days, 13 hours, 37 minutes, 53.1seconds. But we maintain that on the principles of simple interest, this ought to be the case. Let us take the instance mentioned above. Let the note for $100.00 run for 10 years. The amount at Simple Interest is $160.00. Then, if the note is settled at the end of 10 years, the debtor is only bound in law to pay $160.00, and this is all the creditor can claim. But suppose the annual 6 dollar payments to have been made, and suppose the creditor has put them at interest as soon as received. Then, on the first $6.00 he will gain 9 years interest; on the second, 8 years interest; on the third, 7 years interest, &c. For the nine payments, then, preceding settlement, he will receive the sum of these several amounts, which is found to be $70.20. But, as the legal interest is but $60.00, it is evident that he has received $10.20 more than the interest, and, consequently, the debtor must either give this amount to the creditor, or it must be deducted from the principal, leaving $89.80. If he gives it, nothing is more evident than that he gives 6 pr. ct. Compound Interest. Indeed, Simple Interest, annually paid, is Compound Interest. For, the creditor, on receiving his interest, may employ it in any profitable adventure, or he may instantly put it out at interest. Thus, the interest itself may be made to draw interest, and that is all that is necessary to constitute Compound Interest. It is the creditor's own fault if he does not improve his advantage. Of course, on the ground of Simple Interest, when the principal vanishes, by the above rule, it has been actually and really paid off to the "least fraction." And when the creditor falls in debt, it is because he has received, in fact, more than both principal and interest of the note. That is, although the sum of the payments themselves may be less, the real advantage, derived from them, and the actual sum which they might have been made to produce, will be found to exceed the amount of the note. On the other hand, suppose that the debtor, instead of making these annual payments, should put $6.00, yearly, at interest. In 10 years he would have, aɛ before, $70.20, and this he might apply towards paying the note. The result is the same as before; for $160.00-$70.20-89.80 which he has still to pay. Now, if he be allowed no interest on his payments, it is best for him to make no payment at all till the time of settlement; but to keep his money rather at interest than to give it to his creditor. For in this way he obtains the interest, which in the other case he loses, or rather gives away to his creditor. The case we have been considering has been the strong ground of objection against this rule, and the argument drawn from it is very plausible, because at first thought it would seem, that the annual payments were nothing more than legal interest. But the pupil will by this time perceive, that 6 per cent. Simple Interest, for several years, is not worth as much as an annual payment of 6 dollars on the hundred. And he will perceive that the value of Simple Interest diminishes, as the number of years increases. This, then, is the evil, if there be any, and it is an evil which results from the nature of Simple Interest itself. Hence, the argument drawn from the case above, may be a very good one against Simple, and in favour of Compound Interest, but instead of invalidating the rule, on the ground of Simple Interest, it establishes, most completely, its correctness. When the payments are larger, the case is clearer still. Any one will grant, that when a part, manifestly, of the principal is paid, allowance ought to be made for it, so that the creditor shall not still draw interest on the whole debt. And for the reason given above, viz. that the payment may be made to produce, in the hands either of creditor or debtor, its legal interest, it is evident that the whole payment should be deducted from the whole principal, and interest, thenceforward, calculated on the remainder: or, which is the same thing in effect, the process in the rule should be employed. If it be granted, then, that the above rule is correct, on the ground of Simple Interest, it is plain that one which produces different results, must be incorrect. Hence, the rules established by law in Connecticut and Massachusetts are incorrect, since they seem to be constructed for the special purpose of preventing the "eating up" of the principal. Of course they both allow more than Simple Interest, that is, they allow Compound Interest, which is contrary to the statute laws of both states. Hence, we have one statute enjoining a practice, which is made penal by another. It will be necessary for the pupil to be acquainted with these rules. MASSACHUSETTS RULE. FIND THE AMOUNT OF THE PRINCIPAL TO THE FIRST TIME WHEN A PAYMENT WAS MADE, WHICH, EITHER ALONE, OR TOgether with THE PRECEDING PAYMENTS, (IF ANY,) EXCEEDS THE INTEREST THEN due. FROM THIS, SUBTRACT THE PAYMENT, OR THE SUM OF THE PAYMENTS, MADE WITHIN THE TIME FOR WHICH INTEREST WAS COMPUTED, AND THE REMAINDER WILL BE A NEW PRINCIPAL, WITH WHICH, PROCEED AS BEFORE; AND SO ON, TO THE TIME OF SETTLEMENT. EXAMPLES. 1. On demand, I promise to pay WILLIAM CARTER, or order, five hundred dollars, with interest; value received. Hartford, May 7, 1821. $500.00. On this note were the following endorsements. April 29, 1822, received THOMAS TRUSTY. $23.29 What is due June 1, 1824 ? Principal on interest from May 7, 1821, Interest to Sept. 19, 1822, (1 yr. 4 mo. 12 d.) Amount, Payment, April 29, 1822, less than int. due, $23.29 $500.00 41.00 $541.00 Do. July 13, 1822, do. 11.70 Do. Sept. 19, 1822, greater than int. due, 125.51 Due Sept. 19, 1822, forming a new principal, Payment, Jan. 7, 1823, greater than int. due, $380.50 6.849 Due Nov. 25, 1823, forming a new principal, $169.597 5.2574 Balance due June 1, 1824, $174.854 |