nomination of the notes, and the common interval of time are required. 5567.83 ) 619001111 days, the mean or average time of maturity from the present of the obligations to be converted; and 5567.835 = $1113.57, the common denomination of the notes; and (5+1) = 37 days, the common difference of time, or 111(5+1) common interval. The special times to maturity, therefore, of the five notes of $1113.57 each, are 37, 74, 111, 148, 185 days from the present time. When one of the common substitutes is to be treated as cash, or is to be considered as due at the present time, and the common interval is to be measured from the present, 8 t— PROP. 47. — Several matters of indebtedness, amounting in the aggregate to $2175.44, and which will collectively mature, or become due by average, at the close of 68 days from the present time, are to be cancelled by the payment of one-fourth of their sum down, and by passing three notes, made for one-fourth of their sum each, and payable at the close of a common difference of time from the present and from one to another. The common sum and common difference are required. 2175.44-4 $543.86, the common sum; and 681 (4 − 1) = 68÷1.5—46 days, the common interval. The three notes, therefore, are to be made for $543.86 each, and are to be made payable, the first at 46, the second at 92, and the third at 138 days from the present time. When the common interval is to apply between all the common substitutes, and is to be measured from an assigned time for the maturity of one of them, S Sn, and t= PROP. 48. A debt of $4500, due 123 days hence, without interest, is to be substituted by 4 notes, made for one-fourth of the sum each, which are to run an equal interval of time from maturity to maturity, and one of them is to be made payable at the close of an interval of 30 days from the present time. The common sum and common interval are required. 45004 $1125, the common denomination of the notes; and (123-30)1.562 days, the common difference of time. The times to maturity of the substitutes, therefore, are 30, 92, 154, 216 days later than the present time. PROP. 49. The last preceding proposition (Prop. 47), except that the given debt of $4500 has but 75 days to maturity; and one of the notes is not to become due until the lapse of 105 days from the present time. (75 105)—1.5 = 20 days, the common interval. The times from the present to the maturities of the notes, therefore, are 45, 65, 85, 105 days. PROP. 50. It is proposed to relinquish obligations, amounting in the aggregate to $2554.72, and which will collectively become due by equation at the close of 40 days from the present time, and to receive in their place 3 notes, made for one-fourth of the sum each, and the balance in ready cash; the said notes to run equal · intervals of time from maturity to maturity, measured from the present, and one of them to run 85 days; the adjusting interest, or discount, to be at 7 per cent. (85—40) (4—1) = 30 days, the common interval. The three notes of $638.68 each, therefore, are to be made payable at 85, 55, 25 days; and the present worth of the cash payment of $638.68, due 30 255 days hence, is $638.07. To invest a given sum of money in parts, at unlike rates of interest, and the parts to gain like interest in equal intervals of time. P, p', p", &c.— the parts, or partial investments. S r, r', r'', &c. the sum of the investments. the given rates, or these in their relations to each other, expressed in any proportion preferred. the product of r, r', r", &c., as expressed. m PROP. 51. Twenty thousand dollars ($20,000) are to be placed at interest in three such parts that the interest on them, at their respective rates of 6, 7, and 8 per cent. a year, shall be equal for all like intervals of time. The parts, or special investments, are required. m 6 X 7 X 8 336; and 336 To invest a given sum of money in parts, at like rates of interest, and· for unequal intervals of time; and the parts to gain like interest at the close of their respective times. t, t', t", &c.—the given times, or these in their relations to each other, expressed in whatever proportion preferred. m=product of t, t, t", &c., as expressed. S, and p, p', p", &c., as in the last preceding proposition. P, p', p", &c., Sm Sm Sm Nt Nt Nt" &c., inversely. PROP. 52. It is proposed to place $25,000 at interest in four separate sums, one of them for 60, one for 80, one for 110, and one for 150 days' time; and that these sums shall be such, that, at like rates of interest, they will gain like interest at the close of their respective times. The special sums are required. 60 X 80 X 110 X 150, or 6 X 8 X 11 X 15, or To invest a given sum of money in parts, at unlike rates of interest, and for unequal intervals of time; and the parts to gain like interest at the close of their respective times. m product of the given times, or of their relations to each other, by any measure whatever, that is common to them; or of the given rates, if preferred. S, N, p, p', p'', &c., as in the preceding. P, p', p", &c. == Sm Sm Sm &c. PROP. 53. Ten thousand dollars ($10,000) are to be placed at interest in three separate sums, one of them at 4 per cent., for 240 days; one at 6 per cent., for 120 days; and one at 8 per cent., for 80 days; and these sums are to be such that they will gain like interest, one with another, at the close of their respective times. The special sums are required. 240×120×80, or 24×12×8, or 6×3×2 4, 6, 8, or 2, 3, 4, 36 txtxt"=m. =r, r', r'l 12, 9, 8, the products of tr, t'r', 8-4.5 11.5: N, therefore To invest a given sum of money in parts, at like rates of interest, and for unequal intervals of time; and the amount (principal and interest) of the parts to be equal at the close of their respective times. PROP. 54. It is proposed to place $16,000 at interest in four separate sums, each at 7 per cent. a year: one of them for 80, one for 100, one for 150, and one for 200 days' time; and that these sums shall be such that their amount shall be equal, one with another, at the close of their respective times. The special sums are required. 365 365+ rt 365 + 365+rt' 365 365+r Therefore, A SN $4101.33, the common amount. If the times, t, t', t", &c., be taken in years instead of days, then 1 ·1 + N- + 1 1+rt 1+rt' 1 + rt" &c.; and p $ ÷ N(1 + ri); p'=S÷N(1+r), &c. And, if the times be taken in months, N: 12 12 + 12+ rt 12+rt" 12SN(12+rt'), &c. &c.; and p=12S ÷ N(12+rt) ; p′ : To invest a given sum of money in parts, at unlike rates of interest, and for unequal intervals of time; and the amount of the parts to be equal, one with another, at the close of their respective times. PROP. 55. The last preceding proposition, except that the rates are to be 6 per cent. for the 80 days' term, 7 per cent. for the 100 days' term, 8 per cent. for the 150 days' term, and 9 per cent. for the 200 days' term, instead of 7 per cent. for each of the terms. 365 369.8 0.98702 |