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14. My note on 4 months, for $2000, was discounted at a bank, and I immediately put the proceeds on interest. When the note became due, I renewed for the samo term as before; and, when this note became due, I again renewed it for the same time. When the last note became due, I collected the amount of the money I had lent, and paid the note at the bank. How much did I lose by the transaction?
118. To find the Time, when the Principal, Interest, and
Rate are known.
(a.) To find the time, when the principal, interest, and rate are given, we first compute the interest for any convenient assumed time. Then the required time will be the same part of the assumed time, that the given interest is of the interest for the assumed time.
NOTE.--The selected or assumed time should be one for which the interest may be readily computed. When the rate is 6 per cent, the assumed time should be 6 days, 60 days = 2 months, 600 days = 20 months, or 6000 days = 200 months. When the rate is other than 6 per cent, the assumed time should be 1 yr. = 360 days, or such multiple or part of a year as will give for the interest 1 per cent, or some other convenient part of the principal.
1. How long will it take $749.76 to gain $21.868, at 6 per cent?
SOLUTION. — The interest of $749.76 for 6 da. is $.74976. If it takes 6 days to gain $.74946, it will take the same part of 6 days to gain $21.868, that $21.868 is of $.74976, which is 2186880 of 6 days = 175 da. = 5 mo. 25 da.
2. How long will it take $856 to gain $17.976, at 7 per cent ?
Solution.—The interest of $856 for 1 year at 7 per cent is $59.92. Now, if it takes 1 year or 360 days to gain $59.92, it will take the same part of 360 days to gain $17.976 that $17.976 is of $59.92, which is 17976 of 360 days = 108 days = 3 mo. 18 da. (b.) In how long time at interest will —
3. $975.73 gain $97.573 at 6 per cent?
(a.) When one man owes another sums of money payable at different times, it may be desirable to determine when the whole can be paid without gain or loss to either party. The process of doing this is called EQUATION OF PAYMENTS, and the time sought is called the EQUATED TIME.
(b.) It is obvious that if a debt be not paid till after it has become due, the debtor gains the use or interest of it from the time it became due to the time of payment; while, if it be paid before it becomes due, the debtor loses the use or interest of it from the time of payment to the time when it would justly have been due. The thing sought, then, is to ascertain when any sum must be paid, in order that the interest lost by paying one portion of it before it is due, shall be equal to the interest gained by keeping another portion after it has become due.
(c.) In solving problems in equation of payments, we first find to what interest a person is entitled on account of the various debts considered, and then find how long the sum of the debts, or the balance due, must be kept to gain this interest.
1. A owes B $150 due in 4 mo., $200 due in 5 mo., and $450 due in 7 mo. When can the whole be paid without gain or loss to either party?
1st Solution. — By the conditions of the question, A is entitled to the use, or
Interest of $150 for 4 mo. = $3.00
"o $200 for 5 mo. = 5.00
6 $450 for 7 mo. = 15.75 or to the use of $800 till its int. = $23.75, which, determined as in Art. 118, is 1783 days = 5 mo. 28} da., or, as it would be reckoned practically, 6 mo. 28 da.
. 5. I owe $250 due in 10 days, $300 due in 15 days, $450 due in 30 days, and $125 due in 36 days. What is the equated time of payment ?
6. What is the equated time for paying $100 due March 7, $75 due March 15, $350 due April 6, and $500 due April 29 ?
NOTE. - Find in how many days from any assumed time, as, for instance, March 7, each debt is due, and then proceed as before explained. The time is found to be 3628 days, or, practically, 37 days after March 7, which is April 13.
7. What is the equated time for paying $326.35 due July 8, $529.47 due Aug. 1, $823.14 due Aug. 19, and $500 due Aug. 27 ?
8. What is the equated time for paying $296.36 due March 4, $896.82 due April 9, $794.15 due May 16, and $823.10 due July 10?
9. A owes B $1000, payable in 6 mo.; but, to accommodate B, he pays half of it in 3 mo. When ought he to pay the remainder?
10. I owe $500 payable in 8 mo., $600 payable in 9 mo., and $900 payable in 10 mo. To accommodate my creditor, I pay $500 in 1 mo., $300 in 2 mo., and $200 in 3 mo. When ought I to pay the remainder ?
11. A owes B $600 due in 5 mo., and B owes A $900 due in 8 mo. At what time can both debts be paid without gain or loss to either party?
12. A owes B $500 due in 2 mo., and $400 due in 4 mo., while B owes A $200 due in 3 mo., and $600 due in 5 mo. A agrees to pay the whole of his debt in 1 mo., if B will pay his enough sooner to compensate for the loss. When ought B to pay his ?
13. I owe $225 due in 30 days, $325 due in 45 days, $130 due in 50 days, and $300 due in 60 days. If I should not pay anything till the end of 60 days, and then should pay the whole, how much ought I justly to pay, money being worth 6 per cent per year?
14. I owe $800 due in 2 mo. 15 da., $900 due in 3 mo. 10 da., $1000 due in 4 mo., $800 duo in 5 mo., and $600 due in 6 mo. 20 da.; but, to accommodate my creditor, I make a complete settlement in 4 mo. How much ought I to pay, money being worth 6 per cent?
15. I owe $450 due Sept. 3, $500 due Sept. 29, $375 due Oct. 10, and $750 due Nov. 1. If I should pay nothing till Oct. 20, and should then make a complete settlement, how much ought I justly to pay, money being worth 6 per cent? What is the equated time for paying the above debts ?
120. Equation of Accounts.
1. The account-books of A and B show that A owes B
and that B owes A $425.37 due Feb. 3, 1857. T. $219.86 due Jan. 27, 1857. $428.13 due March 7, 1857. | $1000. 'due March 30, 1857. $285.93 due April 23, 1857. $900. due May 1, 1857. $428.25 due May 5, 1857.
$2119.86 = sum due A. $1567.68 = sum due B.
$2119.86 — $1567.68 = $552.18 = balance due A. · When can this balance be paid without gain or loss to either party?
» SOLUTION. — The first step towards solving this problem is to find the interest which will be gained or lost by paying the balance at any assumed time, say Ipril 1, 1857. It is obvious that A should allow interest on all sums which he keeps after they have become due, and on all which he receives before they become due; and that interest should be allowed him on all sums which he pays before they are due, and on all which he fails to receive till after they become due. Hence, the interest account would be as follows:
$10.325 - $6.153 = $4.172 = Balance of interest due B, i. e. gained by A and lost by B.
The question now resolves itself into this : If, by B's paying A the balance, $552.18, April 1, he loses $4.17 interest, when can he pay it without any gain or loss of interest ?
The answer evidently is as many days after April 1 as it will take for $552.18 to gain $4.17 interest, which, found as explained in 118, is about 45 days, and 45 days after April 1 is May 16, which is, therefore, the equated time.
Note. Had the above account been settled April 1, 1857, B should have paid A $552.18 – $4.17 = $548.01. If the balance of interest had been due to A instead of B, B should have paid $552.18 + $4.17 = 556.35.
2. The account-books of Henry Clark and George Barton show that
Clark owes Barton ! And that Barton owes Clark $328.14 due Oct. 1, 1857. $148.16 due Sept. 28, 1857. $425.96 due Nov. 3, 1857. $452.19 due Oct. 17, 1857. $604.50 due Nov. 25, 1857. I $83.75 due Dec. 1, 1857. What is the equated time for payment?