64. If an annuity of £250 per annum, to continue 7 years, produce £1454...4...6, for the present worth, what is the rate per cent. ? Ans. 6 per cent.

65. If a pension of £28 per annum, to continue eight years, produce £188 for the present worth, what is the rate per cent. ? Ans. 5 per cent. Note. When the annuities, or rents, &c. are to be paid half-yearly, or quarterly, then,

For half-yearly payments, take half of the annuity, &c. and twice the number of years, the quotient will be the ratio of half the rate per cent.—and,

For quarterly payments take a fourth part of the annuity, &c, and four times the number of years, the quotient will be the ratio of the fourth part of the rate per cent.

66. If an annuity of £150 per annum, payable half-yearly, having 5 years to come, is sold for £667...10...0, what is the rate per cent.? Ans. 5,per cent. 67. If an annuity of £150 per annum, payable quarterly, having years to come, is sold for £671...5...0, what is the rate per cent. ? Ans. 5 per cent

IV. When U, P, R, are given, to find T.

68. If an annuity of £150 per annum, produce £660 for the present worth, at 5 per cent. what is the time of its continuance? Ans. 5 years.

660 X 2

69. For what time may a salary of £60 be purchased for £291...6...3, 15 at 4 per cent. ?

499

Ans. 5 years. 70. For how long time may £250 per annum, be purchased for £1454...4...6.,, at 6 per cent.?

Ans. years.

71. What time may a pension of £28 per annum be purchased for £188, at 5 per cent.

Ans. 8 years Note. When the payments are half-yearly, then U will be equal to the half annuity, &c. R half the ratio, and T the number of payments ; and

When the payments ure quarterly, U will ve equal to a fourth part of the annuity, &c. R the fourth of the ratio, and T the number of payments.

72. If an annuity of £150 per annum, payable half-yearly, is sold for £667...10...0, at 5 per cent. I desire to know the number of payments and the time to come?

Ans. 10 payments, 5 years. 73. An annuity of £150 per annum, payable quarterly is sold for £671...5...0, at 5 per cent. what is the number of payments and time to come? Ans. 20 payments, 5 years.

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 tir-tr+20° yearly sum at the given rate and for the time of its continuance, thus :

2tr +2

—P.

2. Change P into A, and find what prinipal being put to interest will amount to a A at the same rate, and for the time to come before the annuity, &c. commence, tr+1 thus:

EXAMPLES.

74. What is the present worth of an annuity of £150 per annum, to continue 5 years, but not to commence till the end of 4years, allowing 5 per cent. to the purchaser ?

5×5,05-5,05+2X5

5×,05×2+2

-=4,4×150=

660

Ans. £550.

-=500.

4x,05+1

75. What is the present worth of a lease of £50 per annum. to continue 4 years, but is not to commence till the end of 5 years, allowing 4 per cent. to the purchaser ?

Ans. £152...5...11. 76. A person having the promise of a pension of £20 per unnum, for 8 years, but not to commence till the end of 4 years, is willing to dispose of the samne at 5 per cent. what will be the present worth? Ans. £111...18...1. 77. A legacy of £40 being left for 6 years to a person 15 years of age, but is not to commence till he is 21; he, wanting money, is desirous of selling the same at 4 per cent. what is the present worth ¿ Ans. £171...13...117

To find the Yearly Income of an Annuity, &c. in Reversion.

RULE 1 Find the amount of the pre

sent worth at the given rate, and for the ptr+pŹA. time before the reversion, thus,

78. A person having an annuity left him for 5 years, which does not commence till the end of 4 years, disposed of it for £550 allowing 5 per cent. to the purchaser, what was the yearly income? Ans. £150.

550×4,05+550=660, then

5X,5+1

5×3×,05—5,05+5×2

=,113636 × 660 × 2=£150.

79. There is a lease of a house taken for 4 years, but not to commence till the end of 5 years, the lessee would sell the same for £152...5...11. 25, present payment, allowing 4 per cent. to the purchaser, what is the yearly rent? Ans. £50.

26

80. A person having the promise of a pension for 8 years, which does not commence till the end of 4 years, has disposed of the same for £111...18...1, present money, allowing 5 per sent. to the purchaser, what was the pension? Ans. £20.

81. There is a certain legacy left to a person of 15 years of age, which is to be continued for 6 years, but not to commence till he arrives at the age of 21; he wanting a sum of money, sells it for £171...13...11.7, allowing 4 per cent. to the buyer, what was the annuity left him. Ans. £30.

REBATE OR DISCOUNT.

NOTE. S represents the sum to be discounted.

P the present worth.

T the time.

R the ratio.

1. What is the present worth of £357...10, to be paid 9 months hence, at 5 per cent.?

357,5

-=£344,5783

,76X,05+1

Ans. £344...11...64. 168.

2. What is the present worth of £275...10 due 7 months hence, at 5 per cent.? Ans. £267...13...10 247 3. What is the present worth of £875...5...6 due 5 months hence, at 4 per cent.? Ans. £859...3...3

4. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent.? Ans. £70...11...93

II. When P, T, R, are given, to find S.
RULE. ptr+p=S.

EXAMPLES.

5. If the present worth of a sum of money due 9 months hence, allowing 5 per cent. be £344...H...6. 3,168 qrs. what was the sum first due ? Ans. £357...10.

344,5783 X,75×05+344,5783=£357...10. 6. A person owing a certain sum, payable 7 months hence, agrees with his creditor to pay him down £267...13...103 allowing 5 per cent. for present payment, what is the debt?'

3

7. A person receives £859...3...33 5 months hence, allowing the debtor 4 payment, what was the sum due ?

Ans. £275...10...0, for a sum of money due per cent. for present Ans. £875...5...6. debt due 15 months discount, how much Ans. £75.

8. A person paid £70...11...9 for a hence, he being allowed 5 per cent. for the was the debt?

III. When S, P, T, are given, to find R.

9. At what rate per cent. will £357...10, payable 9 months hence, produce £344...11...6. 3,168 qrs. for present payment? Ans. 5 per cent.

38

10. At what rate per cent. will £275...10, payable 7 months hence, produce £267...13...103, for the present payment ? Ans. 5 per cent. 11. At what rate per cent. will £875...5...6, payable 5 months hence, produce the present payment of £859...3...3 is

3

Ans. 41 per cent. 12. At what rate per cent. will £75, payable 15 months hence, produce the present payment of £70...11...9 ¦, ? Ans. 5 per cent. IV. When S, P, R, are given, to find T.

RULE.

=T.

rp

EXAMPLES.

13. The present worth of £357...10, due for a certain time to come, is £344...11...6. 3,168 qrs. at 5 per cent. in what time should the sum have been paid without any rebate?

357,5-344,5783

344,5783X,05

38

-=,75=9 months.

Ans. 9 months.

14. The present worth of £275...10. due for a certain time to come, is £267...13...10 at 5 per cent. in what time should the sum have been paid without any rebate? Ans. 7 months.

2479

15. A person receives £859...3...3, for £875...5...6, due at a certain time to come, allowing 4 per cent. discount, I desire to know in what time the debt should have been discharged without any rebate? Ans. 5 months.

16. I have received £70...11...9.3 for a debt of £75, allowing the person 5 per cent. for prompt payment, I desire to know when the debt would have been payable without the rebate?

Ans. 15 months.

EQUATION OF PAYMENTS.

TO FIND THE EQUATED TIME FOR THE PAYMENT OF A SUM OF MONEY DUE AT SEVERAL TIMES.

RULE. Find the present worth of each payment for its respective time, thus,

tr+1

Add all the present worths together, then, s-p=D.

EXAMPLES.

d.

1. Dowes E £200, whereof £40 is to be at three 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. 6 months, 26 days.

194,4281X,05

,57315 6 months, 26 days.

Dowes F £800 whereof £200 is to be paid in 3 months