EQUATION OF PAYMENTS. ART. 278. EQUATION OF PAYMENTS is the method of find ing the mean or average time of paying debts due at different times, without loss either to the debtor or to the creditor. The date at which the several debts may be paid is called the EQUATED TIME. Ex. 1. A miller owes a farmer $90, of which $45 is to be paid in 4 months; $30 is due in 7 months; and the remain ing $15 in 8 months; at what time may these debts be paid in one single payment, so that no loss may be sustained by the miller or the farmer? Analysis. If the miller can keep $45 for 4 months, he can keep $1 for 45 times as many months 180 months. Again, if he can keep $30 for 7 months, he can keep $1 for 30 times as many months 210 months; and if he can keep $15 for 8 months, he can keep $1 for 15 times as many months = 120 months. By adding these several sums, we see that the miller can keep $1 for 510 months; and if he can delay the payment of $1 for 510 months, he can delay the payment of $90 (the whole debt) only as many months 50 months 5 months, or 5mo. 20da. 510 months Ans. 5mo. 20da. From the Analysis we derive the following RULE. Multiply each payment by the time before it is due, and divide the sum of these products by the whole debt; the quotient will be the equated time. EXAMPLES. 2. A debt of $975 is due as follows: viz., $150 in 5 months, $375 in 4 months, and $450 in 8 months; at what time should the whole sum be paid? Ans. 6mo. 3. A merchant of Savannah bought a bill of goods in Boston amounting to $3600, of which $1500 was to be paid in 3mo., $1200 in 4mo., and $900 in 5 months; what is the average time for the payment of the whole? Ans. 3mo. 25da. 4. A gentleman owes the following sums, due as follows: viz, $175 in 20 days, $150 in 1mo. 15da., and $200 in 60 days; at what time should the whole debt be paid? Ans. 42da. 5. A drover bought 80 horses and 110 mules, giving $120 apiece for the horses and $100 for the mules. He agreed to pay of the money in 4 months, in 6 months, and the balance in 10 months; how long a credit should he have on the whole sum if he pays it at one time? Ans. 5mo. 12da. 6. R owes S $750; of which $200 is to be paid down, $250 in 3 months, and the balance in 4 months; if he agrees to pay the whole at one payment, what is the equated time? Ans. 2mo. 18da, NOTE. In examples like the last the amount paid down must be regarded as a payment due in 0 days or months; and while the product of this first payment by 0 is nothing, yet, in dividing the sum of the products by the whole debt, the cash payment must be reckoned as a part of the divisor. 7. A farmer purchases $300 worth of fertilizers, agreeing to pay down, in 2 months, in 4 months, and the balance in 6 months; what credit should be allowed to pay the whole at one time? Ans. 3 months. Observation. The foregoing Rule, though sanctioned by custom, is not perfectly accurate. It proceeds upon the sup position that a sum of money paid a certain length of time before it becomes due balances an equal sum paid the same length of time after it becomes due; or, in other words, that the discount on a sum of money is the same as the interest on the same sum for an equal time; which is not true. For short periods and for small sums of money the difference 18 so slight that it is practically disregarded. ART. 279. The TRUE EQUATED TIME can be found when desired. We will illustrate by an example: 8. I cwes B $1339, of which $663 is due in 4 months, and $676 in 8 months; what is the true equated time of paying the whole at one payment? Analysis. If money is worth 6%, the present worth of $663 due in 4mo. is $650 (Art. 258); and the present worth of $676 due in 8mo. is also $650. The sum of the two present worths is $1300. If H pays B $1300 down, exact justice will be done. But the sum to be paid at some time is not $1300, but $1339. The question then is: "In how long a time will $1300 amount to $1339 at 6 per cent. ?"-i. e., how long a time will it require for $1300 principal to gain $39 interest? PROBLEM II., Art. 250, answers this question. We say the interest of $1300 for 1 year at 6% is $78. interest requires 1 year, $1 will require will require 39 times as long or 6 months. The true time is therefore 6 months. If we find the average time according to the Rule, it will be found to give 6.019 months of a : Then, if $78 year, and $39 of a year .5 of a year, 6mo. Oda. 13.68hr., making a differ ence of a little more than half a day. What we have said in regard to finding the true equated time may be summed up in the following RULE. Find the present worth of each payment due; then find the CIME in which the sum of all these present worths will amount to the sum of all the payments. EXAMPLES. 9. P owes W $785, of which $255 is due in 3mo., and $530 in 9mo.; if money is worth 8%, what would be the true equated time to cancel the whole debt by one payment? Ans. 7mo. 10. A farmer owes a commission merchant the following sums: viz., $306 payable in 4mo., $412 in 6mo., and $530 in a year; what would be the true equated time of payment, money being worth 6% ? Ans. 8mo. RATIO. ARTICLE 280. When we compare two quantities of the same kind, we frequently say that one is twice as great, or three times or four times as great, as another. Thus, we say 8 pounds is twice as great as 4 pounds, because 4 is contained twice in 8; 12 shillings is three times as great as 4 shillings, because 4 is contained 3 times in 12. This is expressed concisely by saying that the RATIO of 8 to 4 is 2; of 12 to 4 is 3, etc. Hence, ART. 281. RATIO is the relation of one quantity to another of the same kind. The RATIO of two numbers is indicated by two dots, placed one above the other, between the two numbers; thus 93 expresses the RATIO of 9 to 3. The two quantities compared are the terms of the ratio, the first term being called the antecedent,* the second the conse quent, and the two terms taken together are called a couplet ART. 282. RATIO is also expressed in the form of a fraction by making the antecedent the numerator, and the consequent the denominator; thus * Antecedent, from the Latin antecedo, signifying to go before. + Consequent, from the Latin consequor, to follow after. Analysis. The question simply means: "What part of 54 is 18?" We say 1 unit is part of 54 units, and 18 units will be 18 times as much, which will give 9. What part of 27 is 15? Ans. 5. 10. Express 24 as a fractional part of 56. Ans. 4. ART. 283. To find the ratio of two quantities, they must be of the same denomination. Thus, if we were required to find the ratio of 5 gallons to 7 quarts, we could not get it directly, but by reducing the 5 gallons to 20 quarts, the ratio would be that of 20 quarts to 7 quarts, or 29. 11. What is the ratio of 2ft. 3in. to 10in.? 12. What is the ratio of 3s. 6d. to 7d.? Ans. 7. Ans. 6. 13. What is the ratio of 3pk. 6qt. to 2bu. 2pk. 4qt.? Ans. 14. What is the ratio of 528yd. to 1mi.? Ans. 15. What is the ratio of 6hr. 24min. to 2da.? Ans. 1; Ans. 45 = 75 Ans. 2. 16. What is the ratio of 45ct. to $3? 17. What is the ratio of 75ct. to $5? 18. What is the ratio of $7 to 85ct.? 19. What is the ratio of $2.25 to $9? 20. What is the ratio of $3.75 to $11.25? 21. What is the ratio of Analysis. It is obvious that hence the ratio of to is 3. 10 to ? Ans. is 3 times as much as; Ans. 3. |