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Assume any principal at pleasure, and find the amount for the time and rate per cent. Then, as the amount found is to the amount or debt given, so is the principal assumed to the required principal, or present worth.
2. Suppose a debt of D810, were to be paid three months hence, allowing 5 per cent., what is its worth in cash ?
then, 81 :
3 months 1)4.00
3. Purchased goods to the amount of D750, on a credit of 9 months, at 5 per cent., but wishing to make immediate payment, it is required to know what sum in ready money would discharge the debt.
Ans. D722.39.1+ 4. What is the discount on D280, due in 6 months, at 7 per cent. ?
Ans. D9.46.9+ 5. What is the present worth of D840, due in 1 year, 6 months, at 6 per cent. ?
Ans. D770.64.2. 6. What is the present worth of D954, due in 3 years, at 4.5 per cent. ?
Ans. D840.52.8. 7. If you purchased goods to the amount of D796.49, on a credit of 4 months, at 3 per cent., what sum in ready money would discharge the debt ?
Ans. D788.60.3. 8. What difference is there between the interest of D1200 at 5 per cent. per annum, for 12 years, and the discount of the same sum at the same rate and time ?
9. What suun in ready money must be received for a bill of 900D. due 73 days hence, discount at 6 per cent. per annum ?
Ans. D889.32.8. 10. What sum will discharge a debt of D615.75, due in 7 months at 41 per cent. per annum ?
Ans. D600. 11. B. has D2000 due him from 4., of which D500 are payable in 6 months, D800 in 1 year, and the remainder at the exa piration of 3 years at 6 per cent.; but if A. should make present payment, how much would he have to pay ?
Ans. D1833.37.4+ Note.When payments are to be made at different times, find the present value of the several sums separately, and their sum will be the present value of the note or debt.
12. What sum will discharge a debt of D1000, whereof D600 is payable in one year, and the remainder in 6 months, at 4 per cent. ?
Ans. D969.08. 13. What is the present worth of D600 due in 5 years at 7 per cent. ?
Ans. 1444.44.4+ 14. What is the difference between the interest and discount of D1000 for 1 year at 6 per cent. ?
Ans. D3.39.6+ 15. What is the present value of a note for D3500, on which D300 are to be paid in 6 months, D900 in 1 year, D1300 in 18 months, and the remainder at the expiration of 2 years, the rato of interest being at 6 per cent. per annum?
Ans. D3225.83+ 16. What is the present value of D2880, one half payable in 3 months, one third in 6 months, and the remainder in 9 months, at 6 per cent. per annum ?
Ans. D2810.08+ 17. Bought goods to the amount of D1854 for which I gave my note for 8 months at 6 per cent. ; but being desirous of taking it up at the expiration of 2 months, what sum does justice require me to pay ?
Ans. D1800 18. What is the present worth of D515 due 6 months hence ? D500—duc 1 year hence ? D485.84,9—due 15 months hence ? D479.06.9—due 20 months hence? D468.18.1—due 4 years hence ? D415.32.2—at 6 per cent. ? Ans. D2348.42.1+
19. What is the present worth of D1350, due 5 years 10 months hence, at 6 per cent. ?
Ans. D1000. 20. What is the discount of D460, due 2 years 6 months hence at 6 per cent. ?
What is discount? What is present worth ? How will you first proceed to find the present worth ? After having found the interest of D100 at the given time and rate per cent. what is next to be rone ? After having added the interest so found to D100, by what rule do you work to find the discount? Repeat the rule. Is it correct to take the interest for the discount? What is the difference ? What is the face of a note ? When payments are to be made at different times, how do find the
present value ?
21. What is the difference between the discount of D227.66 for 2 years 3 months and 20 days, and the interest of the same sum for the same time, at 6 per cent. ?
Ans. D3.81.2. 22. What is the present worth of D1500 for 90 days at 7 per cent. per annum ?
1. Divide the given sum or debt by the amount of Dl for the given time. 2. The quotient will represent the present worth, which taken from the debt will leave the discount. Thus to find the present worth of D133.20 payable 1 year and 10 months hence, and discount. The amount of D1 for 1 year 10 months, is D1.11; then D133.20-D1.11=D120 present worth ; and D133.20-D120=D13.20 discount.
EQUATION. EQUATION is a rule used to find the mean or equated time of several payments which are due at different times, so ihat no loss shall be sustained by either parly.
Multiply each payment by its time, and divide the sum of the several products by the whole debt, and the quotient will be the equated time for the payment of the whole.
QUESTIONS. 1. A. owes B. D380 to be paid as follows, namely : D100 in 6 months ; D120 in 7 months; and D160 in 10 months ; what is the equated time for the payment of the whole debt ? Thus : 100 x 6
= 600 120 X7
= 840 160 x10=1600
380 )3040/8 mo. Ans. 2. I have D200 due me, of which D100 is to be paid in 6 months, and D100 in 12 months, but it is agreed to make one nayment; required the time.
Ans. 9 months.
3. A gentleman has due him D600 to be paid as follows, D400 in 10 months, and D200 in 6 months ; what is the equated time ?
Ans. 8 months. 4. A. has due him a certain sum of money to be paid, £ in 2 months, 1 in 3 months, and the remainder in 6 months; what is the equated time?
Ans. 4 months. 5. What is the equated time for paying D2000, of which D500 is due in 3 months, D360 in 5 months, D600 in 8 months, and the balance in 9 months ?
Ans. 6 months (nearly). 6. What is the equated time for paying D380; whereof D100 is payable in 180 days, D120 in 210 days, D160 in 300 days ?
Thus, 100 X 180=18000 : 120 x 210=25200 : 160 x 300= 48000=91200 dividend = 380=240 days, Ans.
RULE II. See by rule 1, at what time the first man mentioned, ought to pay in his whole money; then, as his money is to his time, so is the other's money to his time; inversely, which, when found, must be added to, or subtracted from, the time at which the second ought to have paid in his money, as the case may require, and the sum, or remainder, will be the true time of the second payment.
7. P. is indebted to Q. D150, to be paid, D50 at 4 months, and D100 at 8 months ; Q. owes P. D250, to be paid at 10 months ; it is agreed between them that P. make present payment of his whole debt, and that Q. shall pay his so much sooner, as to balance that favor ; I demand the time at which Q. must pay
the D250. Thus, 50 x4=200; 100 x 8=800=1000-150=6 months, P.'s equated time ; then, D150 : 6g months :: D250 : 4 months ; then 10 months —4=6 months, time of Q.'s payment. Ans.
Note.-Notwithstanding the general use of the rules of equation, they are manisestly incorrect. It is argued by those who defend the principle, that what is gained by keeping some of the debts after they are due, is lost by paying others before they are due ; but this can not be the case, for though by keeping a debt after it is due, there is gained the interest of it for that time, yet by paying a debt before it is due, the payer does not lose the interest for that time, but the discount only, which is less than the interest; consequently, the rule can not be strictly correct; although in most questions which occur in business the error is so trifling that it will always be made use of as the most eligible method. The same system of erroneous calculation in regard to interest and discount is exhibited in equation, by a misapplication of terms, and a false principle.
Note 2.-Suppose a sum of money be due immediately, and another at the expiration of a certain given time forward, and it is proposed to find a time, so that neither party shall sustain loss; now, it is plain that the equated time must fall between the two payments, and that what is gained by keeping the first debt after
is due, should be equal to what is lost by paying the second debt before it is due; but the gain arising from the keeping of a sum of money after it is due, is evidently equal to the interest of the debt for that time; and the loss which is sustained by the paying of a sum of money before it is due, is evidently the discount of the debt for that time; therefore it is obvious that the debtor must retain the sum immediately due, or the first payment, till its interest shall be equal to the discount of the second sum for the time it is paid before due; because in that case the gain and loss will be equal, and consequently neither party can be a loser.
For what purpose is equation used? What is the first rule ? The second ? What can you say in relation to the correctness of the principle of calculating equations of time of payment?
This is a rule by which merchants and others exchange one commodity for another, and by which they know how to make the exchange, or proportion the quantities without loss to either party; the rules in Barter, Profit and Loss, and Partnership, are only applications of the rules of proportion which have been explained and are easily understood.
Find the value of what you propose to exchange at the price at which you wish to exchange it, by any rule most convenient. Then, as the price of one of the articles which you receive, is to the whole quantity, so is the whole value of what you give in exchange, to the answer required.
QUESTIONS. 1. A. has 200 pounds of tea at D1.25 per pound, which he will let B. have in exchange for sugar at llc. per pound; how much sugar must A. receive for his tea ?