2u x then use X = t. t. 7 years. Note-If the paye halir kenarly {\"\u, will give 4 t. Note-If the pay-Shalf-yearly then use S4a, ir, and 2 t. ments are made { quarterly 28a, 1 r, and 4 t. 2 a III. Given, u, a, and t, to findr.-RULE. u XtXt-uxt EXAMPLE, 20. If an annuity of $250 per annum, amounts to S2065 in 7 years : Required the rate per cent ? Ans. 6 per cent. 4a-4X}ux 2t Note-If the pay- Shalf-yearly ments are made {quarterly jux2x2– ux2t 8a-8Xtu X 4t 4ux40 X 4t-$4X4t IV. Given, u, a, and r, to find t. 2 2 a xxx и Хr EXAMPLE. 21. If a pension of 8250 per annum, being unpaid a certain time, amounts to $2065, at 6 per cent: What time has the payment been delayed ? Ans. Sir, 2 t made quarterly PRESENT WORTH OF ANNUITIES. Here p, represents the present worth ; u, t, and r, as before. I. Given, u, t, andr, to find p.-RU. -txr+2t :Xu=p. 2t Xr+2 EXAMPLE. 22. What is a pension of $250 per annum worth in ready money, at 6 per cent. for 7 years? Ans. $1454.225. Note-Respecting half-yearly and quarterly the same as I. II. Given, p, t, and r, to find u. txr-L1 RULE. .: X2p=u. txtxr-tXr+2t EXAMPLE. 23. What annuity is that which for 7 years continuance, at 6 per cent. produces $1454.225, present worth? Ans. $250. Note-If the pay- Shalf-yearly so Str, 2t, and x by 4 p. ments are made quarterly, 1 {r, 4t, and X by 8 p. III. Given, u, p, and t, to findr.-RULE. 24 Xt--PX2 2pXtti Xt-utx Xt EXAMPLE. 24. If an annuity of 8250 per annum, to continue 7 years, produce $1454.225 for the present worth, what is the rate per cent ? Ans. 6 per cent. Note-If the pay. Shalf-yearly ? S 4, 27, will give }.r. ments are made quarterly 40,4t, will give fr. r u IV. Given, u, p, and r, to find t. 2 2p RULE. xXx 2p + uXr EXAMPLE. 25. Required the time that $250 per annum, may be purchased for $1454.225 at 6 per cent. Ans. 7 years. If the --yearly , 2t. a ANNUITIES, &c. TAKEN IN REVERSION. I. To find the present worth of an annuity, &c. taken in reversion. RULE 1.–Find the present worth of the yearly sum at the Thus, tXtXr--tXr+2t:Xu=p. given rate, and for the time of its continuance, 2tXr+2 2. Change pinto a, and find what principal being put to interest will amount to (Thus, a, at the same rate, and for the time to tXr+1 come before the annuity, &c. commences, EXAMPLE. 26. What is the present worth of 351. per ann. to continue 12 years ; but is not to commerce till the end of 5 years, al. lowing 10 per cent. to the purchaser ? Ans. 1971. 58. 5d. 1.792 qr. II. To find the yearly income of an annuity, &c. in reversion. RULE 1.--Find the amount of the present worth at the given rate, and Thus, pXtXr+p=d. for the time before the reversion, 2.--Change a into p, and find what annuity being sold Xr+1 will produce p, at the same Thus, :X2p=4. rate, and for the time of its tXtXr--tXr+2t continuance, EXAMPLE. 27. A person having an annuity left him for 12 years, which does not commence till the end of 5 years, sold it for 1971. 5s. 5d. 1.7924r. allowing 10 per cent. to the purchaser : What was the yearly income? Ans. 351. REBATE OR DISCOUNT. Here s, represents the sum to be discounted : Do the pres. ent wortii, t, and r, as before. I. Given, s, t, and r, to find p. -Rule. eXr+1 EXAMPLES. 28. What is the present worth of $600 due 3 years hence, at 5 per cent per annum? Ans. $508.4745. 29. What is the present worth of $357.50 to be paid 9 months hence, at 5 per cent per annum ? Ans. $344.5783. II. Given, p, t, and r, to find s. -RULE. PXtXr+p=s. EXAMPLES. 30. If the present worth of a sum of money due 9 months hence, allowing 6 per cent, be $508.4745, what was the sum first due ? Ans. $600. 31. A person paid $344.5783 for a debt due 9 months hence he being allowed 5 per cent for the discount, how much was the debt? Ans. $357.50. per cent. III. Given s, p, and t, to find r. -RULE. txe EXAMPLES. 32. At what rate per cent, will $600 payable 3 years hence, produce $508.4745 for present payment ? Ans. 6 per cent. 33. At what rate per cent, will $357{ payable 9 months hence, produce the present payment of $344.5783. Ans. 5 sop IV. Given s, p, and r, to find t. -RULE. rix p EXAMPLES. 31. The present worth of $600 due for a certain time to some, is $508.4745 at 6 per cent, in what time should the sum have peen paid without any rebate ? Ans. 35. I have received $344.5783 for a debt of $3574 allowing the person 5 per cent for prompt payment, I desire to know when the debt would have been payable without the discount? Ans. 9 months. EQUATION OF PAYMENTS. I. 1 o find the equated time for the payment of a sum of money due at several times. 3 years. of} Thus, Rule 1 Find the present worth of each payment for its respective time tXr+1=p. 2. Add all the present worths together, and call that sum ps then will s-p=d the rebate. d 3. Andze, the true equated time, EXAMPLES. 36 M. owes N. $200, whereof $40 is to be paid at 3 months, 860 at 6 months, and $100 at 9 months ; at what time may the whole debt be paid together, rebate being made at 5 per cent ? Ans. .57315 years=6 months, 26 days. 37. P. owes Q. $800, whereof $200 is to be paid in 3 months 8200 at 4 months, and $400 at 6 months : but they agreeing to make but one payment of the whole, at the rate of 5 per cent discount: the true equated time is demanded ? Ans. 4 months 22 days. 38. R. owes S. 81200, which is to be paid as follows: $200 down, $500 at the end of 10 months, and the rest at the end of 20 months ; but they agreeing to have one payment of the whole, rebate at 3 per cent; the true equated time is de. manded! Ans. 1 year 11 days. COMPOUND INTEREST. Compound Interest is that which arises from a principal increased by its interest, as the interest becomes due. The letters here made use of, are, a, the amount. p, the principal, hence, amp, the interest. t, the time. r, the ratio, or amount of 81, or £. for 1 year at any given rate, which is thus found : $100 : 105 :: 1 : 1.05=r, at 5 per cent. As 100 : 106 : :1: 1.06=r, at 6 per cent. 100 : 107 :: 1 : 1.07=r, at 7 per cent. I. Given, p, t, and r, to find a. Rule. pXr? EXAMPLES. 1. What will $200 amount to in 4 years, at per cent per Ans. $243.10125. 2. What will 8480 amount to in 5 years, at 6 per cent per annum ? Ans. $642.348288. II. Given, a, r, and t, to find p. -RULE, =p. annum ? EXAMPLES. 3. What principal being put to interest will amount to $243.10125 in 4 years, at 5 per cent ? Ans. $200. 4. What principal being put to interest will amount to $642.348288 in 5 years at 6 per cent ? Ans. $480. III. Given, p, a, and r, to find t. ROLE. «En which P being continually divided by r, till nothing remains, the num. ber of those divisions will be =t. EXAMPLES. 5. In what time will 8200 amount to $243. 10125 at 5 per cent? 6. In what time will $480 amount to $642.348288 at 6 per cent? Ans. 4 years. Ans. 5 years. a IV. Given, p, a, and t, to find r.-Rule. -=r*: then * vrt=r. P EXAMPLES 7. At what rate per cent will $200 amount to $243.10125 in 4 years ? Ans. 5 per cent. 8. At what rate per cent will $480 amount to $642.348288 in 5 years ? Ans 6 per cent. ANNUITIES, OR PENSIONS, IN ARREARS. Here u represents the annuity, pension, &c, a, r, t, as before uXr I. Given, u, t, and r, to find a.- -Rule. -1 EXAMPLES. 9. What will an annuity of $50 per annum, payable yearly, amount to in 4 years, at 5 per cent ? Ans. $215.50625. 10. What will an annuity of $75 per annum, payable yearly, amount to in 6 years, at 6 per cent ? Ans. $523.14885. aXra II. Given, a, r, and t, to find u. -Rule. EXAMPLES. 11. What annuity being forbomne 4 years, will amount to $215.50625 at 5 per cent? Ans. $50. 12. What salary being omitted to be paid 6 years, will amount to $523.14885 at 6 per cent? Ans. $75. aXr tikama III. Given ú, a, and r, to find t. -Rule. which being continually divided by r, till nothing remains, the number of those divisions will be=t EXAMPLES. 13. In what time will $50 per annum amount to $215.50625 at 5 per cent ? Ans. 4 years. 14. In what time will 875 per annum amount to $523.14885 allowing 6 per cent for forbearance of payment? Ans. 6 years. PRESENT WORTH OF ANNUITIES, PENSIONS, &c. I. Given, u, t, and r, to find p.Rule. 4 u u |