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Note When the payments are half-yearly, then 4a-dut must be divided; but when they are quarterly, then 8a-Sut must be divided as before.

5. If an annuity of 701 per ann. payable half yearly, being forborne 5 years, amounts to 3891 78 6d. I demand the rate per cent? Ans. 5l per cent.,

6. If an annuity of '70l per annum payable quarterly, amounts to 39 11 118 3d in 5 years; 1 demand the rate per eent? Answer, 51

CASE 4. Q. When U, A, and R, are given to find T, how is it discovered ? A. Thus; First 2

13

per cent.

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EXAMPLES:

Ans. 7 years.

1. In what time will 70l per annum amount to 3851 for: borne at 5 per cent? Answer 5 years.

2. In what tine will a pension of 561 per ann. amount to 4621 11s 2d 1.6qr. at 6 per cent?

3. If a house he let upon lease, for a certain time, for 501 per annum, and the amount be 3921 at 4 per cent; I demand the time that it was let for? Ans. 7 years.

4. If a salary of 100l per annum being forborne a certain time, amounts to 7941-10s at 45 per cent; I demand the time of forbearance? Answer, 7 years.

Note. If the payments were half-yearly, then T will be equal to the number of half years, or payments; but if they were to be made qudr. terly; then T' will be equal to the number of quarterly payments.

5. If an annuity of 70l per ann. payable half-yearly, being forborne, amounts to 3891 78 6d at 5 per cent; I demand the time and payment forborne ? Answer, 10 payments=5 years.

6. If an annuity of 70% per annum payable quarterly, being forborne, amounts to 3911 118 3d at 5 per cent; I demand the time and payments forborne ? Answer, 20 pay. ments=5 years. Of the Present Worth of Annuities, or Pensions, etc. Note. Prepresents the present worth--U, T and R, as in the last.

CASE 1. Q. When U, T and R, are given to find P, how is it discovered

Thus; rt-rt X2t

1+1=. 2rt+2

EXAMPLES. 1. What is the present worth of 50l per ann. to conţinue 6 years at 5 per cent? Ans. 2590 12s 3d 2.4+qrs.

2. What is 80l yearly rent, to continue 5 years, worth in ready money, at 6 per cent? Ans. 2441 13s 3d 5+2qrs.

3. What is a salary of 40l per ann. to continue 7 years, worth in ready money, at 4 per

cent. Ans. 2451. 4. What is a pension of 301 per ann. for 5 years worth in ready money, at 45 per cent ?. Ans. 1381 98 4d 2.6+qrs.

Note. Observe the same node here, which is given in Case 1, in An. nuities and Pensions in arrears, concerning half-yearly and quarterly payments.

5. What is the present worth of 50l per ann. payable balf-yearly for 6 years, at 5 per cent? Ans. 2621 10s.

6. What is the present worth of 50l per anu. payable quarterly for 6 years, at 5 per cent. Ans. 2681 188 903.6qrs.

Note. By comparing these two examples with the first, it may be observed, that the present worth of half-yearly payments i& more advaritageous than yearly payments, and the present worth of quarterly than half-yearly payments.

Q. When P, T, and R, are given to find U, how is it discovered? 1. Thus; rt+1

-: psu. rt-rt+2t 1. There is a lease of a house

years to come; I demand the yearly rent, when the present worth at 5 per cent is 2591 12s 3d 2.4qrs? Answer, 501

2. What yearly rent is that, the present worth of which for 5 years is 344l 128 2d 2qrs. at 6 per cent? Answer,

801

per annum. 3. What salary is that which for 7 years continuance at 4 per cent produces 245l for the present worth? Ans. 401.

4. If the present worth of a pension to continue 5 years at 4's per cent be 1331.98 40 , I demand the pension ? A.801.

Note. When the payments to be made are half-yearly, you must mulu tiply by 4p; but when they are quarterly, then multiply by 8p to find u.

5. There is a lease of a house, payable half yearly, for six years to come, I demand the yearly rent, when the presept wurth at 5 per cent is 2621 10s. Ans. 501.

6. There is a lease of a house, payable quarterly, for 6 years to come, I demand the yearly rent, when the present worth at 5 per cent is 5631 13s 9d 3.6qrs. Ans. 50%

EXAMPLES.

per ann.

CASE 3. Q. When U, P, and T, are given to find R, how is it discovered ? 2ut-2p

A. Thus:

per cent?

2pmut thout

EXAMPLES 1 I demand at what rate per cent will the yearly rent of 501 to continue 6 years, produce the present worth of 2591 128 3 2 qrs.

Ang. 5l per cent. 2 If the yearly rent of 80l per annum to continue 5 years, being 3442 125- 3d. 2 qrs. present worth, what is the rate

Ans. 6l per cent. 3. If a salary of 401 per ana. to continuo 7 years, produce 2452 for the present worth, what is the rate per cent? Ans. 71 per cent.

4 If a pension of 30l per annum to continue 5 years, produce 1331 98 4d 2 qrs. for the present worth, what is the rate per cent?

Ans. 4 per cent. Note. When the Annuities or Rents are to be paiel half-yearly capa quarterly, then,

For half-yearly payments take half of the Annuity or yearly Rent, and twice the Number of Years, that is, reduce the Years into half years, and then the Quotient of the upper part divided by the lower, will be the ratio of half the Rate per Cent. But,

For quarterly payments, take a fourth part of the Annuity or yearly Rent, and four times the Number of Years; that is, reduce the Years into quarters; and then the quotient of the upper Part divided by the . lower will be the Rutio of a 4th part of the Rate per C'ent.

5 If a lease of a house of 50l. per ann. payable halfyearly, having 6 years to come, is sold for 2621. 10s. I demand the rate per cent?

Ans. 6. A lease of a house of 50l per ann. payable quarterly having 6 years to come, iş sold for 2631 18s 9d 3 qrs. "I demand the rate per cent?

Ans. 51

per cent. OASE 4. Q. When U, P, and R, are given to find T, how is it discovered ? ? 2p A. Thus: first

lex

per cent.

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ance

ments.

EXAMPLES. 1 If 501 yearly rent produce the present worth of 2591 12s 3d 2 qrs. at 5 per cent what is the time of its continue

Answer 6 years. 2 I demand how long sol per annum may be purchased for 3441 12s 3d 2 qrs at 6 per cent? Answer 5 years.

3 How long must a salary of 401 per annum be enjoyed for 2451 at 4 per cent? Ans. 7 years,

4 What time may a pension of 301 per ann. he bought for 1331 Os 4d 2 qrs. at 4 per cent? Auswer 5 years.

Note. 1 If the payments are to be half yearly, then U will be half of the given lease, pension, &c. and R will be half of the Ratio of the given rate, and T which is required, will be=the number of payments or half years.

2 If ihe payments are to be quarterly, then U will be=a fourth part of the given Lease, Pension, &c. and R will be a fourth part of the ratio of the given rate, and T will be=the number of quarterly pag.

5 A lease of an house of 50l per annum payable halfyearly is sold for 2621 10s at

per cent. I demand the number of payments and the time to come. Ans. 12 payments=-6 years.

6 A lease of an house of 50l per annum payable quarterly, is sold for 2631 188 9d 3qrs. at 5 per cent. I demand the number of payments, and the time to come! Ans. 24 payments,=6 years.

Of Annuities, Leases, &c. token in Reversion. Q. How do you find the present worth of an annuity &c. in reversion ?

A. Thus; first find the present worth of the yearly sum at the given rate, and for the time of its continuance: to do which, there are given U, T, and R, to find P, which is thus discovered:

rttrt+2t

tu=pa

2rt+2 Secondly, find what principal being put to interest will amount to P, at the same rate, and for the time to come before the annuit;, &c. commences; and that will be the present worth of the annuity, &e. in Reversion: therefore let P be changed into A=the amount, and then there will be given A, R. and T, to find P, or the principal which is :has discovered:

fr for

EXAMPLES 1 What is the present worth of a lease of 30? per ann. to continue 3 years, but is not to commence till the end of 2 years, allowing 4 per cent to the purchaser?. Ans. 771 78 7.2d.

2 I have the promise of a pension of 171 per ann. for 7 years, but it does not commence till the end of 4 years, and I am willing to dispose of the same for the present payment at the rate of 5 per cent-I demand the present worth? Ans. 841 9s 6d.

3 There is a legacy of 20l per ann, for 8 years, left to a person of 16 years of age; the time of payment is to commence at the year of perfection, i. e. at 21 years; but he wanting a sum of money, is minded to sell the same at 4 per cent-1.demand the present worth ? Ans. 1151 38 od 1.44 qr.

4 A good natured gentleman being minded to bestow a favour upon an unthankful wretch, settled upon him an income of 35l per ann. for 12 years, to commence five years after such settlement; but he wanting money to follow his extravagances, sold it at the rate of 19 per cent.Idemand how much he received forthe present worth? Ans. 1971 5s 5d 1.696 qr.

Q. How do you find the yearly income of an annuity, &c. in Reversion ?

A. Thus: First, find the amount of the present worth of the yearly sum, at the given rate, and for the time before the reversion: to do which, there are given P, T, and R, to find A, which is thus discovered :

ptr+p=a Secondly, Find what yearly rent being sold will produce A, for the present worth, at the same rate and for the time of its continuance: and that will be the yearly sum required : therefore change A into P, and then there will be given P, R, and T, to find U, or the yearly sum, thus :

rt+1

.: x 2p=üo:. rtt-rt+ 2t.

CASE 5.

EXAMPLES.

1 There is a lease of a house taken for 3 years, but. commences not till the end of 2 years : and the lessee would sell the time for 761 78 7.2d present payment, al-. lowing 4 per cent to the purchaser: 1 demand the yearly rent? Ans. 30l per ann.

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