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4. In what time will any sum double itself at 5 per cent . compound interest? Ans. 14y. 2mo. 13d.
5. In what time will any sum double itself at 6 per cent . compound interest? Ans. lly. lOmo. 20-)-d.
6. A gentleman has deposited $ 450, for the benefit of his son, in a savings bank, at compound interest at a semiannual rate of 3£ per cent. He is to receive the amount as soon as it becomes $ 1781.66^. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in possession of the money? Ans. 21 years.
DISCOUNT AND PEESENT WORTH.
381 i Discount is an allowance or deduction made for paying money before it is due.
382i The Present Worth is the amount of ready money that will satisfy a debt before it is due. It is equivalent to a principal which, being put at interest, will amount to the debt at the time of its becoming payable. Thus, $ 100 is the present worth of $ 106 due one year hence at 6 per cent.; for $ 100 at 6 per cent, will amount to $ 106 in that time; and $ 6 is the discount.
383i In discount, the rate per cent., the time, and the sum on which the discount is made, are given, to find the present worth, which corresponds precisely to the rate per cent., the time and the amount being given, in either simple or compound interest, to find the principal.
384. The interest or percentage of any sum cannot properly be taken for the true discount; for we have seen (Art. 382) that the interest for one year is the fractional part of the sum at interest, denoted by the rate for the numerator, and 100 for the denominator; and the discount for one year is the fractional part of the sum on which discount is to be made, denoted by the rate for the numerator, and 100 plus the rate for the denominator. Thus, if the rate of interest is 6 per cent., the interest lor one year is of the sum at interest; but if the rate per cent, of discount is 6, the discount for one year is t$b of the sum on which discount is made.
385. Business men, however, often deduct, or "take off," from the face of a bill or note due at some future time, a greater percentage than the interest would be for the given time at the given rate. Therefore, the true present worth and discount are not obtained by that method, but only a Nominal Present Worth and a Nominal Discount. The true discount is equal to the interest on the true present worth of the debt, while the nominal discount is equal to the interest on the face of the debt.
386i To find the true present worth of any sum, and the discount, for any time, at any rate per cent.
Ex. 1. What is the present worth of $ 12.72, due one year hence, discounting at 6 per cent.? What is the discount?
Ans. $12 present worth; $0.72 discount.
Amount of $ 1, 1.0 6 ) $ 1 2.7 2 ( $ 1 2, Present worth.
1 0 6
2 12 $ 1 2.7 2, Given sum.
$ 0.7 2, Discount.
Since $ 1 is the present worth of $ 1.06 due one year hence, at 6 per cent., it is evident that the present worth of $ 12 72 must be as many dollars as $ 1.06 is contained times in $ 12.72. We thus find the present worth to be $ 12, which, subtracted from the given sum, gives S 0.72 as the discount.
Rule. — Divide the given sum by the amount of $ 1 for the given time and rate, and the quotient will be the Present Worth.
From the given sum subtract the present worth, and the remainder will be the Discount.
2. What is the discount on $ 802.50, at 7 per cent., due one year hence? Ans. $ 52.50.
3. What is the present worth of $117.6O, due one year hence, at 12 per cent.?
4. What is the present worth of $ 769.60, due 3 years and 5 months hence? Ans. $ 638.672.
5. What is the present worth of $ 678.75, due 3 years 7 months hence, at lh per cent.? Ans. $534,975.
6. What is the discount on $ 600, due 5 years hence, at 5 per cent.?
7. A merchant has given two notes; the first for $79.87, to be paid January 21, 1856; the second for $87.75, to be paid December 17, 1856. How much ready money will discharge both notes February 10. 1855? Ans. $154,545.
8. C. Gardner owes Samuel Hall as follows: $ 365.87, to be paid December 19, 1855; $161.15, to be paid July 16, 1856; $ 112.50, to be paid June 23, 1854; $ 96.81, to be paid April 19, 1858. What should Hall receive as an equivalent, January 1, 1854? Ans. $ 653.40.
9. What is the present worth of $ 67.25 due 3 years hence?
10. What is the present worth of $80.095, due 3 years hence at compound interest? Ans. $ 67.25.
11. What is the discount on $ 110.364 due 5 years hence, at 7 per cent, compound interest? Ans. $31,677.
387. To find a nominal present worth of any sum due at some future time, and the discount on the same at a given rate, reckon the interest on the face of the debt for the gii-en time and rate, and the same will he the nominal discount; and this discount subtracted from the face of the note will give the nominal present worth.
1. I have bought of Paine and Woodard a bill of goods amounting to $960, on six months, but for ready money they take off from the face of the bill, for the time, 5 per cent. What was the amount paid? Ans. $ 912.
2. How much more is the nominal than the true discount on $ 5000 due one year hence, at 7 per cent.?
3. When money is worth 6 per cent. a year, how much may be gained by hiring money to pay $4440 due 6 months hence, allowing the present worth of this debt to be reckoned by deducting the nominal discount? Ans. $3,996.
388. Banking is the general business transacted at banks. A bank is a joint-stock company, established for the purpose
of receiving deposits, loaning money, dealing in exchange, or issuing bank-notes or bills, as a circulating medium, redeemable in specie at its place of business.
The capital of a bank is the money paid iu by its stockholders, as the basis of business.
The affairs of a bank are usually managed by a board o f directors chosen by the stockholders, and the principal officers are a president, a cashier, and one or more tellers.
Note. — The president and cashier sign the bills issued; the cashier superintends the bank accounts; and the tellers receive and pay out money. A bank check is an order drawn on the cashier for money.
389. Bank discount is the simple interest of a note, draft, or bill of exchange, deducted from it in advance, or before it becomes due.
The interest is computed, not only for the specified time, but for three days additional, called days of grace.
The legal rate of discount is usually the same as the legal rate of interest; and the difference between bank discount and true discount is the same as tlie difference bertveen interest and true discount (Art. 384).
390. A note is said to be discounted at a bank, when it is received as security for the money that is paid for it, after deducting the interest for the time until it becomes due.
The money paid for a note is called its avails, proceeds, or present worth.
391. A note, though nominally due at the time specified in it, is not legally due till the days of grace have been counted.
The time a note has to run is counted in days from after the day of its being discounted to the day of its becoming legally due.
392. To find the bank discount and the avails or proceeds of a note or bill for any time or rate per cent.
Ex. 1. What is the discount on a note for $ 1000 having 63 days to run, discounted at a bank at 6 per cent.? How much are the proceeds?
Ans. Discount, $ 10.50; proceeds, $ 989.50.
Operation. We find the interest
Sum discounted, $ 1 0 0 0.0 0 on the sum discounted
, c • j. r or, j i A n n as in Art. 354, and this
rlv of sum = int. for 60d. 1 0.0 0 interest js the bank
^5 of int. for 6Od. = int. for 3d. .5 0 discount (Art. 389),
Bank discount, $l"oTo which subtracted from
_ the sum discounted
Proceeds, or present worth, $ 9 8 9.5 0 gives the proceeds or
present worth. Hence,
Find the interest on the note, or sum discounted, for the given rate and time, including Three days of grace, and this interest will be the discount.
Subtract the discount from the face of the note, or sum discounted, and the remainder will be the proceeds, or present worth.
2. What is the bank discount on a note for $ 7800 on 90 days' time? Ans. $ 120.90.
3. What is the bank discount on a note for $ 1200, payable in 60 days, at 7 per cent.? Ans. $ 14.70.
4. What is the bank discount on $ 8000, payable in 60 days? What are the proceeds? ,
* Ans. Discount, $ 84; proceeds, $ 7916.
5. How much money should be received on a note for $ 760, payable in 5 months, discounted at a bank?
6. A merchant sold a cargo of hemp for $ 7860, for which he received a note payable in 6 months. How much money will he receive at a bank for this note? Ans. $ 7620.27.
7. If the following note was discounted April 3, 1857, how long had it to run, and what were the proceeds?
$ 160TV,5. Boston, December 3, 1856.
Six months after date, for value received, we promise to pay Robert S. Davis Co., or order, one hundred and sixty -j*0°0 dollars, at the Merchants' Bank.
Hai.lett, Osgood, & Co. Ans. 63 days; proceeds, $158.72.