SCHOLIU M. Although it be according to the Laws and Custom of England, to compute Intereft at the Proportion of 6 per Cent. (as above) yet he that takes up Money at Intereft for any Time lefs than even or compleat Years, pays more Intereft than feems reasonably due, according to the Rules of Art. As for Instance; if 100%. be forborn at Intereft one whole Year, it amounts to 106/. But (I fay) if it be paid at the half Year's End, it should not amount to 103; as appears from this following Proportion. Let the Amounts due at the half Year's End; then it will be 100 a::a: 106 the Amount at the Year's End. Ergo aa= 10600, and a✔10600102,9563= 102/. 19s. 1d. which is lefs than 103. by 10 d. And if it be paid in lefs than half a Year's Time, the Error muft needs be the greater. Sect. 2. Of Annuities, or Pensions in Arrears, computed at Simple Intereft. ANNUITIES, or Penfions, &c. are faid to be in Arrears, when they are payable or due, either Yearly, or Half-yearly, &c. and are unpaid for any Number of Payments. Therefore the Bufinefs is, to compute what all thofe Payments will amount unto, allowing any Rate of Simple Intereft for their Forbearance, from the Time each particular. Payment became due: Now in order to that, Put u the Annuity, Penfion, or Yearly Rent, &c. the Time of it's Continuance, or being unpaid. R= the Ratio, or Intereft of 11. for 1 Year, as before. A the Amount of the Annuity and it's Intereft, Then if u the first Year's Rent, due without Interest. Ru the Intereft 2= the Rotere 2u 2 Ru the Interest 3%= the Rent. 3u 3 Ru the Intereft 4u the Rent. 4 Ru the Intereft 5u= the Rent, due at the End of the fecond Year. due at the End of the third Year. due at the End of the fourth Year. due at the End of the fifth Year. And so on for any Number of Years. Hence it is evident, that Ru+2Ru+3Ru+4Ru+5u=A the Sum of all the Rents and their Intereft, being forborn 5 Years. From From whence it follows, that Ru+2Ru+3Ru+4Ru=A—tu. Here t5. Divide by u, then R+2R+3R+4R= A-tu Next to find the Sum of this Progreffion (See Page 185) thus, Let R+2R+3R+4R &c. =z, then 1+2+3+4 &c. = Here the Sum of the firft and laft Terms are 4+15=1, and the Numbers of all the Terms is 41. Therefore hence x tt t= the Sum of all the Terms; that is *** 2 ttRtR Atu 2 Now from this Equation it will be eafy to deduce the following Theorems. Question 1. If 2501. yearly Rent (or Penfion, &c.) be forborn or unpaid feven Years; what will it amount to in that Time, at 6 per Cent. for each Payment, as it becomes due? Here is given u250, 17, and R=0,06; to find A. Per Th. 1. Firft 250 × 71750=tu, 1750 × 7 = 12250 ttu. Again 12250-1750 10500 ttu-tu, and 1000 x 0,06—315. Laftly 315+1750=2065=A; Viz. 2065 l. is the Anfw. required. But if the Annuity, Rent, or Penfion, is to be paid by Quarterly or half Yearly Payments, &c. Then 0,06 0,03 R for 2 =0,015 R for quarterly; 4 half yearly Payments: and Example of half Suppofe 2501. per Annum, to be paid by half yearly Payment's were in Arrears, or unpaid for feven Years; what would it amount to, allowing 6 per Cent. per Annum for each Payment, as it be comes due ? In this Example there is given u=125=220;t=14 the Number of Payments; and R= 0,03= ; thence to find A. 0,06 First 125 x 14 = 1750=tu; 1750 x 14 = 24500=ttu: again 24500 1750=22750=ttu-tu; then 22750=11379, and 11375x0,03=341,25. Laftly 341,25+1750=2091,25; that is, A2091 l. 5 s. the Answer required. N. B. Hence it may be obferved, that half yearly Payments are more advantageous than yearly. For 2091 l. 5 s. 7 2065 l. by 261. 5 s. confequently, quarterly Payments are more advantageous than half yearly Payments. Question 2. What yearly Rent, Penfion, &c. being forborn or unpaid feven Years, will raife a Stock of 20651. allowing 6 per Cent. per Annum for each Payment, as it becomes due? Here is given A= 2065, t=7, and R=0,06; to find u. Per Theorem 2. Firft 7 x 0,06 0,42 t R, and 0,42 x 7 0,060,42 2,94tt R. Then ttRtR 2,52. Laftly R —¿R+21= 16,52) 4130 = 2 A (250=u; that is, 2507 per Annum, &c. will raise 2065 7. the Stock required. Question 3. In what Time will 2501. yearly Rent raife a Stock of 2065 1. allowing 6 per Cent, &c. for the Forbearance of the Payments as they become due? Here is given = 250, A= 2065, and R=0,06; to find t. Per Theorem 4. First 2 33,3333; and 33,3333 2 R ,06 =32,3333== I. Then 16,16666 &c. = x; R A 261,3605 &c. = 4 xx. Again 4132 = 275,333 = 2 A÷Ru, =275,333=2 2 A Ru and 275,3333+261,3605=536,6938 = **. Then ✓536,693823,1666. Laftly, 23,1666 — 16,1666—7 the Time required. Question 4. If 2501. yearly Rent, being forborn seven Years, will amount to 2065 I. allowing Simple Interest for every Payment as it becomes due; what muft the Rate of the Intereft be per Cent. &e. Here is given u=250. A= 2065, and 7; to find R: Per Theorem 3. Thus ttu - tu= 10500) 630=2A-2tu (0,06 R. Then I 0,06 :: 100: 6 the Rate required. Sect. Seet. 3. The Pielent Worth of Annuities or Penfions, &c. computed at Simple Intereft. HE Bufinefs of purchafing Annuities, or taking of Leafes, &c. for any affigned Time, depends upon the true equating of the Principal or Money laid out on the Purchase, with the Annuity or Yearly Rent, by allowing (or difcompting) the fame Rate of Intereft to both Parties. Which may be easily performed by duly applying the respective Theorems of the two laft Sections together; as will fully appear by the following Question, Question 1. What is 751. yearly Rent, to continue nine Years, worth in ready Money, at 6 per Cent. per Annum Simple Intereft? 1. Per Theorem 1. of the last Section, find what the proposed yearly Rent would amount to, if it were forborn 9 Years, at 6 per Cent. Thus u 75, t=9, and R= 0,06: ttu=6075 itu -tu=5400 Quære A. Then 2) 5400 (2700 Multiply R=0,06 162, +tu=673; } = 837=4 2. Then by Theorem 2. Section 1. find what Principal, being put to Intereft for the fame Time, and at the fame Rate, will amount to 837 1.=A. Thus t R=0,549 x 0,06; R+1 =1,54) 837 (543,5064 =P : that is, P=5431. 10 s. 11⁄2d, which is the Worth of 75 1. a Year, as was required. From the Work of these two Operations (duly confidered) it muft needs be easy to conceive, how the two Theorems by which they were performed, may be combined in one. For 1. tt Ru-tRu+21u 2 Confequently PR+P= =A; and 2. PtR+P≈A. tt Ru-tRu+2tu 2 this Equation may be deduced the following Theorems. Theorem I. And from tt Ru-tRu+2tu =P, or 2tR+2 21 R+2 xu=P. By this Theorem all Questions of the fame Kind with the last (viz. that above) may be eafily and readily answered at one Operation. Ru + xx น 2 P Ri By the fecond and fourth Theorems, two very ufeful Queftions may be easily answered. 1. As for Inftance: If it be required to find what Annuity, er yearly Rent, &c. may be purchased, for any propofed Sum, to continue any affigned Time, allowing any Rate of Intereft? This Queftion may be answered by Theorem 2. 2. Again: If it be required to find how long any yearly Rent, Penfion, or Annuity, &c. may be purchased (or enjoyed) for any pro pofed Sum, at any given Rate of Intereft? All Queftions of this Kind are easily answered by Theorem 4. In thefe Queftions it is fuppofed, that the Purchase or yearly Rent, is to commence or be immediately entered upon. But if it be required to find the Value or Purchase of an Annuity or yearly Rent, &c. in Reverfion; that is, when it is not to be entered upon until after fome Time, or Number of Years are paft; then you muft fuft find what the Sum propofed to be laid out in the Purchase, would amount to, if it were put to Intereft, during the Time the Annuity, &c. is not to be put in prefent Poffeffion; and make that Amount the Sum for the Purchafe, proceeding with it as in either of the two laft Questions, &c. Note, From the first Question of this Section it will be eafy to conceive how to perform the Equation of Payments, between Debtor or Creditor, at any Rate of Intereft, without doing any Damage to gither Party. That is, when feveral Sums of Money are to be paid, at several different Times, to find the Time when all the Payments may be truly difcharged at once: as if one Sum were to be paid at the End of two Months, another at fix Months, and perhaps a third Sum at eight Months End, &c. And if it were required to find the Time when all thofe Sums may be truly discharged at one Payment without Lofs, &c. CHAP. |