became due to the time of payment; while if it be paid before it becomes due, the debtor loses the use of the sum paid from the time of payment to the time when it would justly have been due. The use of any sum of money is regarded as worth its interest for the time it is used. (c.) The application of the foregoing principles is illustrated in the following problems and solutions: 1. Mr. Lincoln owes Mr. Wood $400, due in 5 months, $600, due in 9 months, and $200, due in 12 months. When can the whole be paid without gain or loss to either party? Solution. By the conditions of the question, Mr. Lincoln is entitled to the use or Interest of $400 for 5 mo. = $10.00 Or to use $1200 till its int. = $49.00 The interest of $1200 being $6 per month, he is entitled to keep it as many months as there are times $6 in $49, which are 8 times. Therefore he is entitled to keep it 8 months, or 8 months and 5 days. First Proof. By paying the whole at the equated time, Mr. Lincoln gains the use of the first debt from the time it was due to the equated time, and loses that of the second and third from the equated time to the time when they would otherwise have been due. That is, he gains interest of $400.00 for 3 mo. 5 da. Sum of losses = $6.33 which shows the work to be correct. = = $6.33 = = $2.50 = the gain, Second Proof. If each debt should be paid when it becomes due, Mr. Wood will, when the last debt is paid, have had the use of $400 for 7 mo. and of $600 for 3 mo., which at 6 per cent is equivalent to $14 + $9 $23 interest. If, however, the sum of the debts should be paid at the equated time, Mr. Wood would, at the end of 12 months, when the last debt would otherwise have been paid, have had the use of $1200 for 3 mo. 25 da., which, at 6 per cent, is worth $23 interest. This shows that he would have the same interest in one case as in the other, and thus proves the first result correct. Second Solution. Another solution similar in character to the last can be obtained by ascertaining how much would be gained or lost by paying the entire debt at any assumed time, and from that getting the equated time. For instance, suppose that the entire debt had been paid at the end of 9 months. Then Mr. Lincoln would have gained interest on $400 for 4 months = $8.00 and lost interest on $200 for 3 months $3.00 equivalent to a gain of $5.00 which shows that 9 months is as many days longer than the true time as it will take for $1200 to gain $5 at interest. We find (by 191) that it will take $1200, at 6 per cent interest, 25 days to gain $5. Therefore the equated time = 9 mo. 25 da. = 8 mo. 5 da. Third Solution. - When the numbers are convenient, as in this ex ample, a method like the following can be used to advantage : The sum of the debts is $1200, of which the first debt is, the second, and the third. But the use of of a sum 5 mo. is worth as much as the use of the whole of it for of 5 mo., or 1 mo.; the use of of a sum for 9 mo. is worth as much as the use of the whole of it for of 9 mo., or 4 mo.; and the use of of a sum for 12 mo. is worth as much as the use of the whole of it for Therefore Mr. Lincoln is entitled to the use of the sum of the debts for 1 mo. +42mo. + 2 mo. = 8 mo. = 8 mo. 5 da. of 12 mo., or 2 mo. Fourth Solution. - The following method is much used, but we think the method by interest will ordinarily be found preferable : The use of $400 for 5 mo. is worth as much as the use of $1 for 400 times 5 mo., or 2000 mo. The use of $600 for 9 mo. is worth as much as the use of $1 for 600 times 9 mo., or 5400 mo. The use of $200 for 12 mo. is worth as much as the use of $1 for 200 times 12 mo., or 2400 mo. Therefore Mr. Lincoln is entitled to the use of the entire debt for such time as will be equivalent to the use of $1 for 2000 mo. + 5400 mo. 2400 mo., or 9800 mo. But as the use of $1 for 9800 mo. is equivalent to the use of $1200 for 12ʊʊ of 9800 mo., or 8 mo., he can keep the entire debt 8 mo., or 8 mo. 5 da. NOTES. First. As the equated time will be the same, whatever be the rate of interest, the rate may be considered to be that which is most easily calculated. Second. The equated time will frequently contain a fraction of a day; but if the fraction be less than 2, it may be disregarded, or if it be more than 1, 1 may be added to the number of days. 2. A owes B $250, due in 3 mo., $400, due in 6 mo., and $350, due in 8 mo. What is the equated time of payment? 3. I owe $700, payable as follows: $150 in 3 mo., $184 in 7 mo., and the rest in 11 mo. without gain or loss? When can I pay the whole 4. I owe $960, payable as follows: $180 in 4 mo. 20 da., $348 in 6 mo. 15 da., $234 in 8 mo. 5 da., and the rest in 10 mo. 13 da. Required the equated time of payment. 5. A trader bought $1800 worth of goods, agreeing to pay of the money down, & of it in 5 mo., § of it in 6 mo., ₫ of it in 9 mo., and the rest in 12 mo. At what time may the whole be paid? 6. Bought a lot of goods, for which I agreed to pay $437.75 in 3 mo., $394.25 in 6 mo., and $628.19 in 8 mo. When may the whole be paid without gain or loss? 7. A owes B $800, payable in 10 mo. ; but to accommodate B, he pays $250 down. When ought the remainder to be paid? Solution. - After paying $250, he will owe $800 $250 = $550, which he ought to keep till its interest shall equal the interest of $800 for 10 months. But the interest of $800 for 10 mo., equals the interest of one dollar for 800 times 10 mo., or 8000 mo. equals the interest of $550 for of 8000 mo., which is 14 mo. Hence it ought to be paid in 14 mo. 6 8. I owe $1000, payable in 9 mo.; but to accommodate my creditor, I pay $300 down, and agree to pay $300 more in 2 mo. How long ought I, in justice, to keep the remainder? 9. I owe $600, payable in 8 mo. 15 da., and $400, payable in 12 mo.; but afterwards agree to pay $400 down, and $300 in 2 mo. 20 da., on condition that I may keep the remainder enough longer to compensate for my loss. When will the remainder become due? 10. A owes B $480, due in 1 yr., and B owes A $720, due in 1 yr. 6 mo. If A should pay his debt at once, when ought B to pay his? 193. To find Date of Equated Time. (a.) The best method of solving such examples as the following is to see how much interest will be gained or lost by paying the sum of the debts at any assumed time. (b.) It will be well as a general thing, to select for the assumed time a date on which one of the debts becomes due, as by that means we shall avoid the necessity of reckoning interest on that debt. Reference should also be had to the probable equated time. (c.) The time is reckoned by counting the days between the dates considered, as in the English method of computing interest. 1. James Brown owes William Greene the following debts, viz.: $534.83, due Jan. 7, 1855; $285, due April 4, 1855; $327.38, due July 3, 1855; and $438,75, due Aug. 17, 1855. When may the whole be paid without gain or loss? Solution. Suppose that April 4, 1855, be selected as the assumed time. Then Mr. Brown would gain interest on Showing that Mr. Brown is entitled to keep $1585.96, the entire debt due, as many days after April 4 as it will take it to gain $6.94 interest. This, found by 191, is 26 days, plus a fraction less than . Therefore the equated time is 26 days after April 4, which is April 30. NOTE. The above shows that on April 4th Mr. Brown could justly have settled the account by paying $1585.96 $1579.02. $6.94 = Again. Suppose that July 3 be selected as the assumed time. Then Showing that Mr. Brown ought to pay $1585.96, the entire debt due, as many days before July 3 as it will take it to gain $16.84 interest. This, found as before, is 64 days nearly. Therefore the equated time is 64 days before July 3, which is April 30, as before. - NOTE. The above shows that if the account should not be settled till July 3, Mr. Brown ought justly to pay $1585.96 + $16.84 $1602.80. = Proof. By paying the debt on April 30, Mr. Brown will gain interest on $327.38 from April 30 to July 3, 64 da. = $3.49 $11.39 = $7.97 $11.46 $00.07 which, being less than the interest of $1585.96 for a half day, shows that April 30 is the correct equated time. 2. I owe $387.53, due Nov. 7, 1851; $467.81, due Dec. 21, 1851; $256.19, due Feb. 11, 1852; $136.43, due March 1, 1852; and $387.59, due May 3, 1852. What is the equated time of payment? 3. I owe $2867, due April 15, 1850; $1642, due July 27, 1850; $4371, due Oct. 8, 1850; and $5940, due Jan. 1, 1851. What is the equated time of payment? 4. I owe $628.13, due Dec. 17, 1852; $427.19, due Dec. 23, 1852; $371.16, due Dec. 30, 1852; $587.83, due Jan. 3, 1853; $987.62, due Jan. 7, 1853; and $843.28, due Jan. 14, 1853. What is the equated time of payment? How much is due on the above Jan. 1, 1853 ? 5. I owe $543.28, due April 24, 1855; $723.13, due May 11, 1855; $484, due Sept. 3, 1855; $426.18, due Oct. 10, 1855; $236, due Nov. 10, 1855. What is due on the above Sept. 1, 1855, interest being reckoned at 5 per cent? 6. What is the equated time for paying the following debts: $600, due March 7, 1850; $400, due June 11, 1850; |