(50) The sum of 7431. 4s. 9d. is payable in 6 years' time, and the present value of that sum is 5871. 7s. 03d. I demand at what rate per cent. the rebate must be made.

IS to find the present worth of an annuity, &c, to continue for ever.

When U and R are given, to find P.

(51) Suppose a freehold estate of 500l. per annum were to be sold; what is the worth, allowing 5 per cent. to the buyer?

(52) What is an estate of 251. per annum, to continue for ever, worth in present money, allowing 44 per cent, to the buyer?

When P and U are given, to find R.

(53) Suppose one gave 10,000l. for a freehold estate of 5001. per annum, what rate per cent. has the purchaser for his money?

(54) If an estate of 257. per annum be bought for 5551. 11s. 1d. what is the rate per cent,?

When P and R are given, to find U.

THEOREM 22. P+r−1=U.

EXAMPLES.

(55) Suppose a person would lay out 10000l. on a freehold estate, and so as to be allowed 5 per cent. for his money, what must be the annual rent of such an estate?

(56) If a freehold estate be bought for 5557. 11s. 1 d. and the allowance of 4 per cent. be made to the buyer, what is the yearly rent?

LXXIII. PURCHASING FREEHOLD ESTATES in

REVERSION.

To find the worth of a freehold estate in reversion..

(57) What is an estate of 500l. per annum worth in ready money, to continue for ever, but not to commence till the end of 4 years, allowing 5 per cent. to the purchaser?

To find the yearly rent of an estate taken in reversion.

(58) A freehold estate is sold for 82271. 1s. 4d. which does not commence till the end of 4 years, the buyer being allowed 5 per cent. for his money: I desire to know the yearly income.

QUESTIONS for Exercise.

(59) Held of a college 4861. 10s. a-year, on a reserved rent of 941. Money being at 5 per cent. interest; what

fine ought severally to be paid on a 7, a 14, and a 21 years' lease?

(60) Suppose I would add five years to a running lease of 15 years to come, the improved rent being 1861. 7s. 6d. per annum; what ought I to pay down in ready money for this favour, discounting 4 per cent. ?

(61) A. has a term of 7 years in an estate of 501. per annum ; B. has a term of 14 years in the same estate; and C. has a farther term of 10 years after B. in the same estate. What is the present value of the several interests in the said estate?

(62) For a lease of certain profits for 7 years, A. offers to pay 1501. gratuity, and 3007. per annum; B. offers 4007. gratuity, and 250l. per annum; C. bids 6501. gratuity, and 2007. per annum; and D. offers 18007. for the whole purchase, without any yearly rent. Query, which is the best offer? and what difference, computing at 4 per cent.?

(65) Value the lease of a house in tolerable repair, the rent 541. 17s. a-year, the ground rent 7 guineas; 3 years of it only to come; the rent payable every 6 months; discount per compound interest on this kind of purchase at 101. per cent.

(64) A fine for a lease of a tenement is settled at 1537. under a reserved rent of 16. a-year. Now the tenant cannot conveniently pay more than 501.; but for the 6 years to come of the term, is willing rather to pay an adequate rent, computing 10l. per cent. per compound interest. What ought that rent to be?

(65) Another lease for 7 years is agreed for at 250l. Fine, on the old rent 447. a-year; but considering the contractor desires to reduce the rent to 201. a-year, and pay a proper fine, computing as before, after the rate of 101. a-year; to what must the fine be advanced?

(66) A son, previcus to his marriage, is inclined to have 50l. a-year freehold settled on his family; and to have immediate possession of it, offers his father, in lieu, an annuity for his life, valued at 12 years'

purchase, discounting at 4 per cent. thereon: whereas he is content the estate should be valued at a discount of 3 per cent. and consequently will be worth 33 years' purchase. Pray what had the father for his life?

(67) A gentleman took a college lease of 2371. a-year, for 21 years, and paid the full fine: the rent reserved was 101. a-year; but when 4 years were elapsed, against his marriage, he renewed the lease, and filled up the 21 years. In 14 years after that, his wife dying, he again renewed it in favour of his daughter, then 7 years of age; and by the time she was 19, it was a third time renewed in order to her settlement. The question is, what money the society must have received from this family from first to last, allowing 51. a-year discount on the fines.

GEOMETRICAL DEFINITIONS.

GEOMETRY contains the nature and properties of lines, angles, surfaces, and solids.

A point is that which has no parts nor magnitude.

A line is length conceived without breadth,

An angle is the mutual inclination of two lines which

meet.

When a straight line, as CD (Fig. 4.), standing upon another, A B, makes the angles, ADC, and CDB, on each side equal to one another, each of these equal angles is called a right angle, and the dotted line, CD, is said to be perpendicular to the line A B.

An angle is commonly expressed by three letters; that placed at the angular point being always wrote in the middle, as ADC (Fig. 4.) denotes the angle b.

An obtuse angle is that which is greater than a right angle, as CAB (Fig. 3.).

An acute angle is that which is less than a right angle, as DCB (Fig. 4.).

Parallel lines are those of which every point of the one is at the same distance from the other, as the lines A B and CD (Fig. 2.).