865200 398.934000) = B's share 443260 = £398 : 18 : 81 and C's share. 452.125200) = A share = £452 : 2:6) IV. TO DETERMINE THE LIABILITY OF EACH PARTNER IN A CONCERN, ACCORDING TO THE NUMBER OF SHARES HELD BY EACH, AND THE TIME FOR WHICH HE HAS HELD THEM. Rule. Multiply the number of shares held by each, by the number of units in the time, (all the times being expressed in terms of the same unit;) add the products, and divide the whole liability by the sum; multiply the quotient by each product; the results will be the shares of each partner. EXAMPLES. 1. Three persons rented a field for £68 : 10:0. A puts in 140 sheep for 14 days; B 160 sheep for 12 days; C 200 sheep for 8 days; what should each pay? £. 140 X 14 = 1960 548,0)68.50.0125 160 X 12 = 1920 548 200 x 8 = 1600 1370 5480 1096 2740 TO FIND THE EQUITABLE TIME AT WHICH SEVERAL PAYMENTS DUE AT DIFFERENT TIMES MAY BE ALL PAID AT ONCE. Rule. Multiply each sum due (expressed in the same unit) by the number of units in the respective times (all expressed in the same unit), add the products, and divide by the sum of all the debts. The result will be the number of units in the equated time, the unit being that in which the mes were expressed. EXAMPLES. 1. Find the equated time at which the following debts may be paid: viz. £250 due in 7 months; £500 in 9 months; £250 in 12 months. 250 x 7 = 1750 £250 Sum of products .. =9250 £1000 Sum of debts. 9250 =91. .. Equated time = 94 months. 1000 2. Find the equated time at which the following sums may be paid: viz. £347 : 11 : 6 due now, and £72: 5:6 to be paid every month for the next 12 months. £347 : 11:6 = £347.575 £72:5:6 = £72.275 12 X £72: 5:6 = £867.300 Sum of products = (1+2+3 + &c. to 12 terms) X 72.275 = 78 x 72.275 = 5637.450. 4859 500 Def. 1. Any income payable at stated periods is called an annuity. Def. 2. Annuities to continue a fixed number of years are called annuities certain. Def. 3. Annuities depending on the life or lives of individuals are called contingent or life annuities. Def. 4. Annuities, to commence after the lapse of a certain time, are called deferred or reversionary annuities. Def. 5. If the payment of an annuity is forborne for a certain number of years, the sum of the amounts of the payments at compound interest is the amount of the annuity. Def. 6. The present value of an annuity is that sum, which put out at compound interest, would be sufficient to pay the annuity as it becomes due. TO FIND THE AMOUNT OF AN ANNUITY FOR BORNE FOR ANY NUMBER OF TERMS. Rule. Find the ratio of the Compound Interest on £l for the given time to the interest for 1 term, and by it multiply the given annuity. EXAMPLE. Find the amount of an annuity of £50 : 10 : 6 forborne for 4 years at 5 per cent. (1.05)4 – 1 Amount = x 50.525 £. .05 1.05 .05).2155063 = (Int. of £1.) 1.05 4.310126 525 52505 1050 21550630 1.1025000 215506 52011 8620 2155 £217.76911 = Amt. of annuity. £1.2155063 = Amt. of £1. Ans. £217.76911 = £217 : 15:44. II. TO DETERMINE THE PRESENT VALUE OF AN ANNUITY TO LAST A GIVEN NUMBER OF TERMS. Rule. Find the compound interest of £l for the given time; divide this by the product of the amount of £1 and the interest for 1 year; multiply the annuity by the quotient. EXAMPLE. (1.04)6 1 Pres. val. of Int. of £1. £l annuity. 1.04 .05061264).26531600(5.242086 £. 1.04 25306320 X 36£. Ill. TO FIND THE VALUE OF A PERPETUAL ANNUITY. Rule. Find the ratio of the annuity to the interest on £l for one year ; this will be the number of pounds in the value. EXAMPLE. Find the present value of £3 perpetual annuity, interest being at the rate of 24 per cent. 3 600 23 5 IV. TO FIND THE PRESENT VALUE OF A DEFERRED ANNUITY. Rule. Find the value of the annuity, supposed perpetual, and multiply this by the difference between the present values of £1, due at the time at which the annuity is to commence, and due at the time at which it is to cease. EXAMPLE. Find the present value of an annuity of £130 to commence 8 years hence, and to continue 15 years, at 45 per cent. Present value of perp. ann. of £130 = £130 ;- .045 = £2888.8. .3633501 23 Difference =.3398350 £2888.8888888 5389330 8666666666 866666 Ans. £982:0:81. £982.0344441 V. TO DETERMINE WHAT ANNUITY MAY BE PURCHASED WITH A GIVEN SUM, TO LAST A GIVEN TIME. Rule. Find the present value of an annuity of £1 for the given time; the ratio of the given sum to this will be the number of pounds in the annuity. EXAMPLE. If a person lay out £3000 in the purchase of an annuity, to continue 25 years, find his yearly income at 5 per cent. compound interest. Present value of an annuity of £1 for 25 years = 14.0939446. 14.0939446)3000.0000000(212.8573718 281878892 181211080 40271634 12083745 808590 704697 103893 5236 4228 1008 22 Ans. £212 : 17 : 2 nearly |