[ .ums, each, from 1001. and multitinually together for a divisor, and division, will give the policy to covje round.* EXAMPLE. $1500 from Boston to Philadelphia, at 3 Guadaloupe, at 4, from thence to Nantz, at me at 6 per cent.; For what sum must he over his adventure the vogage round, supposqual out and home, and tantamount to the sev 100 X 100 X 100X1500 3X100—4× 100-5X 100-6 CASE VI. $1803-835, Ans. iven sum is adventured several voyages round, as in the last either at the same, or different risks, from port to port, and premium for the voyage round is required, iantamount to the eral given rates per cent. It is evident that the policy to be taken out for the first voyage becomes the a for which a policy is to be taken out for the second voyage, and so on. nce the examples of this case are to be solved by the rule for Case II. making he sum in the policy for the first voyage, the sum for which a policy is to be tak- out for the second voyage. Therefore the operation on the given example would be as follows. 100X1500 100-3: 100 :: 1500: policy for 1st voyage=100-3 the sum to be insured on the second voyage, we have, 100X1500 100-3 100X100X1500 Now as 100 x 1500 100-3 100X100X1500 100-4: 100 :: : 2nd policy= 100-3×100-4 100X100X100X1500 And 100-5: 100 :: : 3d policy== 100-3X100-4 × 100-5 × 100-6 100-3X 100-4 X 100-5 × 100-6 which is the Rule. The same may be shown by the Double Rule of Three, thus, 1004 X 1500 100-3: 100 :: 1500: 100-4: 100 :: 100-5: 100 :: 100-6: 100 :: It is plain that however numerous the voyages, the power of 100 must be equa to their number, and that the divisor must always be the continued product o* the differences between 100 and the several rates of insurance. If the rate of in1004 × 1500 surance had been the same on each of the voyages, then the policyif the rate had been 6 per cent. 100-6 RULE.* 1. Find the sum for which the policy must be taken, by the last case. 2. Multiply the sum adventured by 100, and divide that product by the policy. 3. Take the quotient from 100, and the remainder will be the premium per cent. on the policy, tantamount to the several premiums given in the question. EXAMPLE. A merchant adventured $1500 from Boston to Philadelphia, at 3 per cent. from thence to Guadaloupe, at 4; thence to Nantz, at 5; and thence home, at 6 per cent. : What will be the premium, tantamount to those given in the question, on a poliey for covering the first adventure, the whole voyage, supposing the risks out and home equal? In Case V. we found the policy, which would cover the adventure 1500×100 the voyage round, to be $1803-835. Then 100— 1803.835 16.844 the premium per cent. on the policy the voyage round, and tantamount to the several given premiums. CASE VII. If a policy be taken out for a given sum, to cover a certain adventure, from one port to another, on to several ports, at equal premiums from one place to the other, to find what that equal premium is. RULE.† 1. Involve 100 to that power denoted by the number of risks, and multiply this power by the sum adventured, (or covered.) *When the policy is found by Case V. the operation becomes the same as that directed by the Rule, Case III. which has been proved. The operations may be shortened in many cases, by keeping the terms separate in the t part of the process. Thus-by Case V. the policy in this example= 1004 × 1500 By the last remark in the demonstration of the Rule Case V. when the insurance is the same on each of several voyages, the policy is equal to the product of the sum to be insured and 100 raised to a power whose index is the umber of voyages, divided by the difference between 100 and the rate of inurance raised to the same power. Hence this product divided by the policy est give a quotient equal to the difference between 100 and the rate of insur 2. Divide the last product by the policy. 3. Extract that root of the quotient denoted by the number of risks. 4. Take this root from 100, and the remainder will be the equal premium from one port to the other. EXAMPLE. A merchant adventured $1500 from Boston to Philadelphia, thence to Guadaloupe, thence to Nantz, and thence home; to cover which all round he took out a policy for $1803-835; and the premium was equal from one place to the other: what was the premium per cent.? 100 100x100x100 × 100 × 1500, =4.507 per cent. Answer. 1803.835 CASE VIII. When an adventure is insured out and home at one risk, at a given rate per cent. and the voyage terminates short of what was at first intended: To find what the underwriter must receive per cent. RULE. 1. If just half the voyage is performed, it must be considered as two equal risks: If one third, then, as three equal risks; if but one fourth, then, as four risks, and so on; and by Case 2d must be found the amount which will cover the adventure the voyage round. 2. Involve 100 to that power denoted by the number of risks, and multiply this power by the sum adventured. 3. Divide this product by the aforesaid amount. 4. Extract that root of the quotient denoted by the number of risks. 5. Take this root from 100, and the remainder will be the sum per cent. which the underwriter must receive. EXAMPLE. A merchant covers $200 at 6 per cent. from Newburyport to the West Indies and home again; but the voyage terminating in the West Indies, what must the insurer receive per cent.? 100 6 94: 100: 200: 212-765957 amount to cover $200 voyage round. 2000000 212-765957 100X100×200=2000000 and -9400, and 100-9400 =3.0465 to be paid the insurer per cent. upon the above amount. ance raised to a power whose index is the number of years. If that root of the quotient, indicated by the number of years, be extracted you will have the difference between 100 and the rate per cent. and this difference taken from 100 gives the rate. COMPOUND INTEREST IS that which arises from the interest being added to the priacipal, and (continuing in the hands of the borrower) becoming part of the principal, at the end of each stated time of payment. METHOD I. RULE. Find the amount of the given principal, for the time of the first payment, by Simple Interest: next, find the interest of that sum, or principal, and add it as before, and thus proceed for any number of years, still accounting the last amount as the principal for the next payment. The given principal being subtracted from the last amount, the remainder will be the compound interest. In federal money, multiply the principal by the rate for the first time of payment, setting the product two places more to the right than the multiplicand, and the decimal point in the product under that in the multiplicand; then find the amount, and proceed as above. Note. It is not usually necessary to carry the work beyond mills; therefore, when the figure next beyond mills, at the right, exceeds 5, increase the number of mills 1; when it does not exceed 5, it may be omitted. The result will be exact enough for common purposes. EXAMPLES. 1. What will £480 amount to in 5 years, at 6 per cent. per annum? £ Principal 480 Rate of interest 6 Principal for the 1st year 480 0 It may be observed that all the computations, relating to Compound Interest, are founded upon a series of terms, increasing in Geometrical Progression, wherein the number of years assigns the index of the last and highest term: Therefore, as one pound is to the amount of one pound, for any given time, so is any proposed principal, or sum, to its amount for the same time. 1/24 £ S. d. s. d. Prin. for the 4th year 571 13 83 Prin. for the 4th year 571 13 8 Interest for ditto 34 6 01 Amount for 5 years 642 6 11 Subtract the first principal 480 0 0 Compound interest for 5 years 162 6 11 In federal money, thus: The principal being $1600 for five years. Principal for the 1st year $1600. Rate of interest 6 principal 5th year 2019-963136 |