21. If a pension of $250 per annum, being unpaid a certain time, amounts to $2065, at 6 per cent: What time has the payment been delayed? Note-If the pay- Shalf-yearly ments are made quarterly Ans. 7 years. r, u, will give 2 t. r, ₫ И, will give 4 t. PRESENT WORTH OF ANNUITIES. Here p, represents the present worth; u, t, and r, as before. I. Given, u, t, and r, to find p.-RUDE.txtxtxr+2t 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. RULE. txr+1 : 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, Sr, 2t, and X by 4p. r, 4t, and X by 8p. Note-If the pay- Shalf-yearly ments are made quarterly, III. Given, u, p, and t, to find r.-RULE. EXAMPLE. uxt--px2 2pxtuXt-utXXt 24. If an annuity of $250 per annum, to continue 7 years, produce $1454.225 for the present worth, what is the rate per cent? Ans. 6 per cent. EXAMPLE. Ans. 7 years. 25. Required the time that $250 per annum, may be purchased for $1454.225 at 6 per cent. Note-If the pay- Shalf-yearly ments are made quarterly Su, r, will give 2t. tu, tr, will give 4t. ANNUITIES, &c. TAKEN IN REVERSION. 1. 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 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 commence till the end of 5 years, allowing 10 per cent. to the purchaser ? Ans. 1977. 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=a. for the time before the reversion, 2.-Change a into p, and find what annuity being sold will produce p, at the same rate, and for the time of its continuance, Thus,: tXr+1 : X2p=4. 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. 1.792qr. allowing 10 per cent. to the purchaser: What was the yearly income? REBATE OR DISCOUNT. Ans. 351. Heres, represents the sum to be discounted: p, the pres. ent worthi, t, and r, as before. I. Given, s, t, and r, to find p.-RULE. txr+1 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. -RULE. pxtxr+p=s. II. Given, p, t, and r, to find s.— 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. III. Given s, p, and t, to find 7. -RULE. EXAMPLES. s-p -=r. txp 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 per cent. IV. Given 3, p, and r, to find t.- -RULE. EXAMPLES. s-p rxp =t. 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. 3 years. 35. I have received $344.5783 for a debt of $357 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. 7o find the equated time for the payment of a sum of money due at several times. Rule 1-Find the present worth of 8 each payment for its respective time Thus, xr+1=p. 2. Add all the present worths together, and call that sum P, then will s-p=d the rebate. d 3. Ande, the true equated time pxr EXAMPLES. 36 M. owes N. $200, whereof $40 is to be paid at 3 months, $60 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 $200 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. $1200, 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 demanded? 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, a-p, the interest. t, the time. r, the ratio, or amount of $1, or £. for 1 year at any given rate, which is thus found: As 100 105 100 1: 1.05=r, at 5 per cent. 1: : 106 1.06r, at 6 per cent. 100 107: 1: 1.07r, at 7 per cent. I. Given, p, t, and r, to find a. RULE. pXrt: EXAMPLES. 1. What will $200 amount to in 4 years, at 5 per cent per Ans. $243.10125. annum ? 2. What will $480 amount to in 5 years, at 6 per cent per Ans. $642.348288. annum ? II. Given, a, r, and t, to find p.—RULE. EXAMPLES. a 3. What principal being put to interest will amount to $243.10125 in 4 years, at 5 per cent? 4. What principal being put to interest $642.348288 in 5 years at 6 per cent? III. Given, p, a, and r, to find t. Ans. $200. -RULE. t which being continually divided by r, till nothing remains, the num ber of those divisions will be =. EXAMPLES. 5. In what time will $200 amount to $243.10125 at 5 per cent ? Ans. 4 years. 6. In what time will $480 amount to $642.348288 at 6 per Ans. 5 years. cent? a IV. Given, p, a, and t, to find r.-Rule. -r: then *rt=r. 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 I. Given, u, t, and r, to find a.— -Rule. EXAMPLES. uXrt U 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? II. Given, a, r, and t, to find u.-Rule. EXAMPLES. Ans. $523.14885. aXra Ans. $50. 11. What annuity being forborne 4 years, will amount to $215.50625 at 5 per cent? 12. What salary being omitted to be paid 6 years, will amount to $523.14885 at 6 per cent? Ans. $75. III. Given u, a, and r, to find t. -Rule. aXru-a t น 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 $75 per annum amount to $523.14885 allowing 6 per cent for forbearance of payment? Ans. 6 years. PRESENT WORTH OF ANNUITIES, PENSIONS, &c. 1. Given, u, t, and r, to find p.-- Rule. น น- r-1=p. ..f |