whole numbers than 365, for convenience in reckoning it would have been better to assume 360 days for the nominal year in fixing the rate, rather than 365. The time in expressing the rate is arbitrary, and as neither 360, 365, nor 366 is the exact number of days in all years, either civil or astronomical, would not the increased facility in computation, and the perfect accuracy in the result, warrant the change?

The division of the year into twelfths, called months, is purely imaginary; for no month, either lunar or calendar, was ever known which occupied just one-twelfth of a year. Manifestly, if we assume a year of 365 days as the standard for reference in expressing the rate, we never can introduce the denominations of months in any form whatsoever without inaccuracy, unless we involve in the calculation fractional parts of days, which would be as absurd as it would be difficult.

365

If, however, we assume a year of 360 days, we may have assumed months of 30 days. Then 6 per cent. per annum of 360 days would be 1 per cent. for 60 days, and all time being reduced to days or months of 30 days each, or years of 360 days each, the computation would be simple, rapid, and perfectly accurate. As it is, the law having accurately determined when a paper matures, however the time may be expressed in the paper, the only accurate rule for computing interest is to ascertain the actual number of days, and make each day's interest of the annual interest. Some banks are restricted by their charters in their discounts to "6% per annum," but are allowed to compute by Rowlett's Tables. But Rowlett's rule "To find bank interest," makes all time reducible to days, and the interest for each of the year's interest, so that when the time in the note to be discounted reads "two months," the interest for of the year should never be taken except when February 29th of a leap year is included in the term, for in that case only will the "two months" contain just 60 days and no more. In all other cases, the interest should be 59, 61, or 62-360ths of the year's interest, according to the actual number of days contained in the time of the note. In Massachusetts and some other States interest computed on the supposition that 360 days make the

이

year is regarded valid. But in New York each day's interest must be only of the year's interest.

ART. 96. RULES FOR COMPUTING THE DIFFERENCE OF TIME BETWEEN DATES.-Besides counting the exact number of days as referred to above, two rules are in common use.

RULE I.—By compound subtraction, reckoning 30 days for a month.

RULE II.—By finding the number of entire calendar months from the first date, and counting the actual number of days left.

Note.-By "calendar month" is meant the time from any day of one month to the corresponding day of the next month. If the days of the first month is a higher number than the greatest number of days in the last month, the calendar month ends with the last day. Thus from Oct. 31 to Nov. 30 is a calendar month.

From Aug. 20, 1854, to March 10, 1857, would be, according to the 1st Rule, 2 yrs. 6 mo. 20 d. ;

2d Rule, 2 yrs. 6 mo. 18 d. From Aug. 31, 1854, to March 10, 1857, would be, according to the 1st Rule, 2 yrs. 6 mo. 9 d.

.6

66 2d Rule, 2 yrs. 6 mo. 10 d.

It will be observed that in these particular examples, though the actual difference of time in the two cases is 11 days, the result by the second rule shows only 8 days. A discrepancy of 2 days may also arise in the use of the first rule, for by it the time from Feb. 28, 1857, to March 2, 1857, would be 4 days, while the actual time is only 2 days. The first rule also shows no difference of time between March 31 and April 1. Each rule will give a result sometimes too large and sometimes too small.

The examples in this work, except those in Bank Discount, and those otherwise restricted, may be wrought by the second rule.

ART. 97. PROBLEMS IN WHICH THE INTEREST IS KNOWN.— Of the four quantities, the principal, time, rate per cent., and interest, to find either one of the first three, the remaining three being given, we have the following

GENERAL RULE.

Find the interest by the given conditions, assuming one dollar for the principal, one per cent. for the rate, or one year for the time, in place of the unknown quantity, as the case may be, by which divide the given interest, and multiply the assumed amount by the quotient.

Unity is assumed for convenience only in multiplication. Note. When the amount is given instead of the interest, to find the latter subtract the principal from the amount.

Examples.

1. What is the rate of interest if I receive $20.96 for the use of $126.75 for 2 yrs. 24 d. ?

Solution.-At 1% I would have received $2.62, and since the given interest is eight times this, the rate should be eight times 1%.

2. What sum invested at 10% per annum will secure an income of $1000 semi-annually?

Solution. One dollar thus invested would yield an income of 5 cents semi-annually, and since $1000 is 20,000 times 5 cents, the sum loaned should be 20,000 times one dollar.

3. In what time will $512.60 amount to $538.31 at 7% per annum ?

Solution. The interest of $512.60 in one year would amount to $35.88, and since the given interest is only $25.71, the required time would be 3 of 1 year, which by reduction will be found to be 8 mo. 18 d.

25.71

35.88

Fractional days in the result may of course be neglected. ART. 98. The same result may be obtained by making the statement in the form of a proportion, though it is better to work by analysis.

The above examples would be thus stated:

As $2.62 int. at 1% is to $20.96 given int., so is 1% the supposed rate to 8% the required rate.

4. In what time will any sum double itself by simple interest at 5 per cent. ?

Solution. The required interest must be 100% of the principal, and as there is a gain of only 5% in one year, it will take as many years as 5 is contained times in 100.

Note. To treble itself, the required interest must be 200% of the principal.

PRESENT WORTH.

ART. 99. Simple interest varies directly as the principal, time, and rate per cent. Either two of the latter terms remaining the same, interest varies as the other. The principal being given or fixed, the amount, consisting of the sum of principal and interest, or of a constant and variable quantity, can not vary as the time and rate per cent. But if the time and rate per cent. are constant quantities, the interest varies as the principal, and the amount being in this case the sum of two equally varying quantities, varies also as the principal. From this we see the truth of the following

PROPOSITION. For the same time and rate per cent., whether the interest be simple or compound, the amount due varies as the principal.

The Present Worth of any debt is the sum or principal which at the current rate of interest will amount to that debt when it becomes due.

For example, $100 at 10% will amount in one year to $110. The Present Worth then of $110 due one year hence is $100.

The amount, rate, and time being given, to find the principal or Present Worth, we have the following

RULE.

Assuming any principal, determine the amount for the given rate and time, by which divide the given amount, and multiply the assumed principal by the quotient.

Note. To render the multiplication easy, assume $1 or $100.

Remark.-The difference between the Present Worth and the Amount of the debt is called the Discount; and is really the interest on the Present Worth. For Bank Discount, see Art.

Examples.

1. What is the present worth and discount of a debt of $1000 due in 1 yr. 6 mo., the current rate of interest being 6 per cent. ? Ans. Pres. Worth, $917.431; Dis., $82.569.

2. What sum must I put at interest at 10 per cent. to liquidate a debt of $3000 due 3 years hence ?

3. A man can sell his farm for $5000 cash, or for $6000 payable in 2 years; if he accept the last offer, and receive instead its present worth at 8% interest, how much better would it be than the first offer? If he accept the first offer, and loan the $5000 at 8% interest, how much less would he receive at the end of the 2 years than if he accept the last? What is the present worth of that difference?

ANNUAL INTEREST.

ART. 100. If a note reads "with interest payable annually," or "with annual interest," the interest may be collected at the close of each year; but, if not paid, the interest due draws only simple interest to the time of maturity, or until paid.

It is a principle in law, that money due or on interest always draws simple interest, unless a condition to the contrary is expressly stated. The condition, "with interest payable annually, applies to the interest which accrues on the principal or face of the note, and not to the interest on the annual interest.