Divide the propofed Stock (viz. 634,4) by the given Principal (viz. 375,5) and the Quotient will fhew the tabular Number that ftands over against the Time fought. Thus 375,5) 634,4 (1,689479 &c. this Number being fought in the Table, will be found to ftand against 9 Years, which is the Time required. But if the Quotient cannot be truly found in the Table of Amounts for Years, as above; then take out of that Table the neareft Number that is lefs, and make it a Divifor, by which you muft divide the first Quotient; and then feek the fecond Quotient in the Table of Amounts for Days (which is inferted a little further on) and it will affign the Number of Days: as in this Example. In what Time will 5631. amount to 8601. at 6 per Cent. per Annum, Compound Intereft? Anfwer. In 7 Years and 99 Days.. Thus 563) 860 (1,52753 which fhews the Time to be more (or above) feven Years; for over againft 7 Years is 1,50363 which being made the new Divifor: Viz. 1,50363) 1,52753 (1,01589 &c. this Number is the nearest Amount to 99 Days. Note, If the Stock, Principal, and Time be given; the Rate of Intereft will be beft found by extracting the Root, &c. as before in the fourth Question. The next Thing that I fhall here propofe, is to make this Table (which is only calculated for the Rate of 6 per Cent.) univerfally useful for all the Rates of Compound Intereft, which I may prefume to say, is a new Improvement of my own, being well fatisfied it never was published before; and not only fo, but I have heard feveral very good Artists affirm it was impoffible to be done. The Method of performing it is briefly thus, Let x= the Difference between 1,06 R, the Amount of 17, for one Year (in the Table) and any other proposed Amount of 1 l. for one Year; which admits of two Cafes. Cafe 1. If the propofed Rate be greater than the 1,06 = R, then will R+the true Amount of 1 I, for one Year at that Rate. Cafe 2. But if the propofed Rate be lefs than 1,06R, then it will be Rx the Amount of 1 1. &c. Make {t-1=b, t-2 = c, t = 3 =d, t—4=f, &c. 2} + b = 8,468 m, 4 dmn, ±ƒn=s&c. LI Then the Then will R+ t Rb x + g Rc x2 + m Rd x* &c. = the Amount of 11. at the given Rate, for any Time denoted by t, in Cafe 1. And R − t Rb x + g R¢ x3 — m Rd x3 &c. Amount of 11. in Cafe 2. Which is no more but this: Let R+x or R. — * (which foever it is) be involved (as directed in Sect. 5. Chap. 2.) to the fame Power or Height as the Index t the given Time in the Que. ftion denotes: rejecting all the Powers of x above x ×× or ×××× at moft, as ufelefs. Then multiply that Power of R+x or Rx into the given Principal, and their Product will be the Amount required. 7 An Example or two in each Cafe will render all easy. Suppofe it were required to find what 2561. would amount to in fifteen Years, at 81. per Cent. per Annum Compound Intereft? Here t=15. First 100: 108 :: : 1,08 the Amount of 11. at 8 per Cent. Next 1,08-1,06=0,02=x. And R+x=1,08 as in Cafe 1. Then R15 R14x + 105 R13 x x + 455 R12 xxx &c, the Amount of i. for 15 Years, at 8 per Cent. 12 Here x 0,02. xx 0,0004.. and x x x 15 R14 x And 105 R13 xx ,000008 By the Table R15 = 2,396558 2,260904 x 15 x,02 2,132928 x 105 x,0004 0,678271 =0,089583 Sum 3,171736 455 R12 xxx= 2,012196 x 455 ×,0000080,007324 Then 3,171736 x 256 811,964416 A That is, 811. 95. 3 d. fere. Which is the Anfwer required. What will 3651. amount to in feven Years at four and a half per Cent, &c. First 100 104,5 : 1,045 the Amount of 1 l. at 4% per Cent. Next 1,06-1,045=0,015=x. Confequently R-x=1,045 as in Cafe 2. The R-7R6x + 21 R3 xx— 35 R4xxx &c. the Amount of 1 1. for 7 Years, at 4 per Cent. Here Here x =,015; xx=,000225; and xxx000003375 7 R6 x =0,148944 And+21 R3 x x = +0,006323 R' — 7 R6 x + 21 R3 x x — 35 R+ xxx=1,360868 Then 1,360868 x 365 496,71682=A. That is, 496. 145. 34 d. is the Anfwer required. If the Reason of these two Operations be but well understood, it will be very easy to conceive how to find P, the Principal, by having A, t, and x given (because R and it's Powers are always given by the Table.) For R2±t Rbx+g R¢ x x + m Rd x x x × P = A (as above.) Therefore A R2 + t Rb x + g R¢ x x + m R4 x x x Or if A, P, and t, be given x may be found. A P For R2 ± 1 Rb x + 8 R‹ × × ±m Rdxxx=2. This Equation g P being folved (as in Chap. 10.) the Value of x will be found; and then either R+x, or R -x will fhew the Rate of Intereft, &c. But I fhall leave the numerical Operations to the Learner's Practice, fuppofing enough done to fhew how all Questions of this Kind that are limited by whole Years may be computed. And if the Time given or fought be not terminated by whole Years, but by Weeks, Months, Quarters, or Half-Years, &e. for refolving fuch Questions, the beft Way will be to reduce those Parts of a Year into Days; that done, find an Anfwer according to the Demand of the Queftion (and agreeing to 1 l. as before) for that Number of Days; and in order to that, it will be requifite to find the Amount of 1. for one Day (as in my Compendium of Algebra, Page 110) which I fhall here infert. Put a the Amount fought, then it will be 1:a :: a: aa¦ ¦ aa: aaa :: aaa : à a a a÷to a365. As one Pound is to it's Amount for one That is Amount to the Amount of two Days of two Days to that of three Days. to 365 Days. L12 Doy :: fo is that and fo is that And fo on in Then Then the laft of the Terms will be a365 — 1,06 1365 z in Numb. 3- 1 2μ365 +365r36 +e+66430r363ee=a365=1,06 4 365 e66430 ee = 0,06 466430 5,00549 e +ee = 0,0000009032 D 5 ÷ 6e= D ,00549 +e ift Divifor ,00559) 0,0000009032 +e=,00015 2d Divifor,00574 2870 and only too much by New r1,00016 for a fecond Operation. Then 2 in Numb. 71,06013401407 +386,887 e +70402,172 €8 1,06. Hence it appears that r — e = a. 81,06013401407 — 386,887 e +70402,172ee Therefore 1,06 - 89386,887 e — 70402,172 e e = 0,0001 3401407 Which being further pursued to a third Operation will give a = 1,000159653587453 &c. This I This Value of a is the Amount of 11. for one Day, from which, if 1. be fubtracted, the Remainder =,000159653587 &c. will be the Intereft of 11. for one Day. Confequently, if any proposed Principal be multiplied into either of these, the respective Product will be the Amount or Intereft of that Principal for one Day, at 6 per Cent, &c. And that the Amount (or Intereft) of any Principal or Sum may be eafily computed for any Number of Days lefs than a Year; I have here inferted the following Table, which with a great deal of Care (and I believe Exactnefs) is calculated from the laft found (1,000159653587453) Amount of 11. for one Day. To which alfo is annexed a Table of the Amounts of 1 l for Months. 1.0019175262 37 1.0059241901 62 1.0099468767 15 13 1.0020774859 38 1.0060847895 63 1.0101081184 1.0102693858 1.0104306789 1.0065667416 66 1.0105919978 17 19 1.0027175803 42 1.0067274436 67 1.0028776677 43 1.0030377808 44 1.0070489245 69 1.0031979193 45 21 1.0033580850 46 22 1.0035182732 47 23 1.0036784885 48 24 1.0038387294 49 25 1.0039989958 1.0107533424 1.0078881712 68 1.0109147128 1.0110761090 1.0072097035 70 1.0112375309 1.0115604521 1.0073705082 71 1.0113989786 1.0118834764 50 1.0080139835 75 1.0120450272 Days |