A's number amp B's number C's number ang mp+np+ng'mp+np+nq' mp+np+np 518. Four places are situated in the order of the letters A, B, C, D. The distance from A to D is 34 miles. The distance from A to B is to the distance from C to D as 2 to 3. And of the distance from A to B, added to half the distance from C to D, is three times the distance from B to C. What are the respective distances? Let 2 = the distance from A to B, and 3x = the distance from C to D, and (x+3x)=2x=3.BC, ... BC=x, and (2x+3x+ 3x)=x=34, .. }x=2, and x= = 6. Ans. From A to B=12; from B to C-4; from C to D=18. 519. Divide 10,000 in 2 such parts, that when each is divided by the other, the sum of both quotients will make 5. Let x and y denote the two numbers, s their sum, and q the sum x y of their alternate quotients; then x + y = s, and ~+ y Hence x2+y=qxy, and x2+2xy + y2=s2, or, by substitution, 9+2, { 1726.73164 8273.26835 520. Out of annuity of $1000 per annum, for 10 years, the first payment being due 1 year hence, the owner desires to know how much he may spend a year, so that his annual savings, with the simple interest arising therefrom, at 7 per cent. per annum, may at the expiration of this annuity, amount to a sum whose interest, at 7 per cent. per annum, shall be equal to the yearly expenditure. Put a the annuity = $1000, t the time of continuance = 10 years, r = the interest of $1 for a year =.07, the expenditures required; then, by the question, (1-1)r+2 2 and a Xtrx-a-x, and x = 2a= $2000 and = a x= $520,6977, x=479.3023 dollars, the annual expenditure sought. 521. Borrowed $5000, at 7 per cent. per annum, but I am, including the interest, to pay the lender annually 8 per cent. on the original sum. When will such payments discharge this debt? Put a = 5000, p X.08400 =α, r= 07, and x = time required; then, from the question, r = the rate of interest, a = annuity, and p its present worth, are given to find the time of continuance x = Log. a-Log. (a—pr) Log. (r+1) 30.734 years, the time. 522. The principal, time, and rate of interest being given, to find the amount, or money due at the end of that time, at simple interest; Let p=principal, t = time, r = rate of $1 for à certain time, as a year, &c. s = sum of all the arrears. then 1:r::p: rp, the interest of p for one year; 1 rpt: prt, the interest for the time t. and p+prt = the whole arrear at the end of the time t, and p+ prt: s, the arrear sought. See pages 316-17, or 215-16. Cor. 1. Hence p=rt+1; when s, r, and t are given. 523. The annuity, time, and rate of interest being given, to find the arrear, at the end of that time, at simple interest. = Put a annuity or yearly rent; t-time of forbearance; r= interest of $1 for a year; and s = the whole arrear; {0 interest due at one year's end; ra — interest at 2 years' end; 2ra interest at 3 years' end; 3ra: interest for 4 years; (t-1)ra interest for t years}; ta=rents due at the end of t years = (0+ 1+2+3....to t-1X into rata=s). By Arith. Progression, t(t-1) 0+1+2+3.... t − 1 = {{t(t — 1)}, and 2 -ra + ta= 524. Find the present worth of an annuity, to continue a given time, at a given rate of simple interest. Let p = present worth, a = annuity, t = time, r = interest of $1; then p+ prt = s, and ta (t-1)r+2 -ta=s= 2 (t-1)r+2 − 1) t= ra a 2p be found, by art. 70 case 1. p+ prt, or rt+1 } (t −1)r +1 × 2; 2ta 2p where t may 525. The principal, time, and rate of interest being given, to find the amount at the end of that time, at compound interest. Let p principal, t time, r interest of $1; R=1+", the amount of $1 and its interest, s = sum of money due at the end of that time; then 1+r or R money due at 1 year's end; = then, as before treated, pages 316-17, or 215-16, I have 1: R::R: R2= money due at 2 years' end; 1: R:: R2: R3 money due at 3 years' end, and R' money due at t year's end; 1: R'::p: pR2 the amount of p for the time t, and pR'=s; then I have, cor. 1, 2, and 3, p ort = Log.s-Log.p ; R =R Log.s-Log.p t Ρ = 526. The annuity, time, and rate of interest being given, to find the arrears due at the end of that time, at compound interest. Let a = annuity or yearly rent, t= time of forbearance, r interest for 1 dollar for a year, &c., R=1+r, s=sum of all the arrears, a = money due at one year's end. 2a+ra=a+Ra arrears at 2 years' end, a+aR+aR2= arrear in 3 years; a+aR +aR2+aR3= arrears for 4 years, and a+aR+aR2+aR3.......... to a.Ri arrears for t years; then, by pages 192-3, Geometrical R(R-1) Progression, 1+R+ R2+ R3....to R11= R-1 R-1 and a = money due at the end of t years; $R-R' = S a a a R-1 —R' ———, where R may be found, and then 7. 527. Find the present worth of an annuity, to continue a given time, at a given rate of compound interest. Let p - present worth, a = the annuity, t=the time, r = interest of 1 dollar; 1+r, and I have found pR = S, 528. To find the value of an annuity to continue forever, at a given rate of compound interest. : and a= = pr; but in the first value of p. where t is infinite, R' is infinitely greater than 1, where R'—1— R', and p: as before. 529. At what rate of interest will $100 amount to $200 in 94 years, at compound interest? Put r = rate of $1; R=1+r, t=92; then by 525, I have 39 39 100R = =200 per question; .. R1 =2, or R3 — 16, and R= 39/16 = 1.0737 by logarithms; R-1 = r = .0737, and then 100 X .0737—7.37 the rate per cent. = 530. If a principal, x, be put out at compound interest, for x years, at x per cent., to find the time, x, in which it will gain x, I Log. ;) xX (1+ (1+100) M=.3010300, and by page 301-4, I have 531. Given the rate per cent. for 1 year $5, to find what the amount of any sum, $100, will be at the year's end, at compound interest, supposing it to arise from the principal and interest due every day, &c. Letr interest of $ for a year, n = 365, the parts of a year, -interest for one day; 1+ money due at one day's end; n and (1+;)"= n money due at the year's end, (by Log's, p. 301;) nxlog. 3. (1+2)=log. amount for a year=.0215694, and 1.0509 amount for a year; 105.09 = amount of $100; 105.09 = amount of $100; (1+1)"= 1+r+ n(n-1) n(n-1)(n-2) 2n2 2.3n3 r3, &c., the amount for a year. If the interest is supposed to gain interest every moment, by becoming part of the principal, then n is infinite, and =1+r+ + 2 2.3 2.3.4' &c., the amount at the year's end; but this series is the number belonging to the hyperbolic logarithm r, whence the number belonging to the log. .43429448r — amount of $1 for a year = 1.0513; and from 100 =105,13 to gain interest continually. If the interest for a day be required, so that it may amount to 1+r at the year's end, at compound interest; then the amount at 1 day's end will be (1+r); which is something less than 1+ r n 532. A man puts out a sum of money, at 6 per cent., to continue forty years, and then both principal and interest is to sink; what is that per cent to continue forever? Or, if $100 be paid for annuity of $6 a year for forty years, what is that per cent.? Put a= Log. R ; Log. R = t Log.6-Log. (6-100r) (Suppose R=1.05; then r = .05, and 40 which is too little. Let R Log.r.019454; whence R= 1.046, 1.053; then r=. :.053, and Log. R -.023324, and R= 1.055, too large; then, by p. 272, art. 111, I find R = = 1.052, and the rate = 5.2 per cent. 533. If $200 be due 3 years hence; $80, 5 years hence; in what time must both be paid together, at 5 per cent.? (1.05)6 + (1.05)5 =(172.76 + 62.68) = $235.44 = the whole present worth of both sums; and then Log. 280-Log. 235.44 t Log. 1.05 =3.5527 years, Ans. 534. What must I pay for an annuity of $70, to begin 6 years hence, and then to continue 21 years, at 5 per cent. ? Let a = R-1 7 S 70, t21, R= 1.05, x = 6; then = 1— TR* a = $669.704; a the present worth of the annuity years hence =s, and the present worth of S, 7 years hence, 669.704, the present worth of the annuity in reversion. = 535. The discount of a bill came to $112. Had the rate per cent. been $1 more, it would have cost $126, and if the rate per cent. had been $1 less, only $96. Required the value of the bill, time when due, and the rate of interest. Let p = principal, n = time, and r = interest of $100 in one rate of interest, and a = 112, b = 126, and c=96; |