He should next be led to see the relations among the interest, the sum loaned (the principal), and the sum called the amount. Suppose the sum loaned to be $100, the time to be six months, and the rate 6 per cent per annum. Take the line AB to represent six months, A the beginning of the time, and B the end of the time. Principal: $100 A $3 interest. B At the end of the time the sum $103 has to be paid to the loaner-that is, the $100 has to be restored, and $3 paid as interest. The sum, $103, is called the amount. It is plain then that (2) The principal = 100 of the interest; The use of a line to represent time will assist the pupil greatly, and after examining a few examples similar to the foregoing, he will know all the relations among principal, interest, and amount, and will see how to write them down when the rate and the time are given. When these relations are understood, the whole subject of interest is understood, the only care required on the part of the teacher consisting in making a careful gradation of problems. One of the most striking applications of interest is to problems relating to the so-called true discount-a term which should fall into disuse. There is but one dis count, the discount of actual business life; it is an application of percentage, and on account of its being calculated in the same way as interest it is erroneously spoken of as interest, and a confusion arises in the mind. of the pupil. Accordingly, the problem, Find what sum would pay now a debt of $150 due at the end of six months, the rate of interest being 6 per cent per annum, is a definite problem in interest. To solve it we have recourse to the line illustration given above. It is plain that if one had $100 now, and put it out at interest at the rate given, it would come back at the end of the time as $103. Thus, $100 now is the equivalent of $103 at the end of six months-that is, the sum now, equivalent to a certain sum due at the end of six months, is 188 of that sum. Therefore, in the case in question, the sum is 188 of $150. It is true that there is here an allowance off, a discount, so to speak, but until the pupil understands the whole question of interest and discount the term should not be used in this connection. We shall suppose, then, that the student has mastered simple interest, and shall turn to compound interest. Compound Interest.—The teacher should explain that the value of money-as the pupil has seen-depends, in some measure, on where it is placed in time; men in business always suppose interest to be paid when it is due, or if an agreement is made that its payment be deferred, they regard this interest in its turn a source of interest. An example worked out in detail will help the pupil to see just what is done. Suppose a sum of $1,000 loaned for three years, at 5 per cent per annum, interest to be paid at the end of the three years, and the interest at the end of each year to become a source of interest for the ensuing year or years: The pupil will work several such examples, and will find not a little pleasure in determining just how much interest has been paid as interest on interest. He is then ready to make a more general study of compound interest. Suppose a sum loaned at compound interest for three years at 5 per cent per annum. What is the interest on any sum for one year at 5 per cent? Plainly of the sum. What is the amount? 18% of the sum. What, then, is the amount of any sum for one year? 185 of that sum. What sum bears interest for the second year? 188 of the original sum. What will the amount of this be? 188 of itself, and therefore 188 of 185 of the original sum. Accordingly, the amount of the sum for two years is (185) of the original sum. What for three years? Plainly 185 of (188)' of original sum, and therefore (185) of original sum. This is found to be 1187838 of original sum. How much more 1576 have we than the original sum? 1000000 157625 of original sum; therefore, interest = 157625 of sum = 157825 of amount, etc. 1000000 The pupil should be told that in all transactions involving a time longer than one year (or it may be by agreement six months or three months) compound interest is alone employed where the interest is thought of as all being paid at the end of the time. From what has been said he will know at once how to solve the following problem of interest: Find what sum paid now will discharge a debt of $1,000, due at the end of three years, the rate of interest being 6 per cent. He should acquire a facility in thus transferring money from one time to another. Annuities.-Afew words may be said on the subject of annuities. If A gives B $100 to keep for all time, and the rate of interest be 6 per cent, B would be undertaking an equivalent if he would agree (for himself and his heirs) to pay to B (and his heirs) $6 at the end of each year, for all time. This $6 supposed paid at the end of each year is called an annuity; as it runs for all time, it is called a perpetual annuity, and is said to begin now, though the first payment is made at the end of the first year. The $100 is very properly called its cash value, and the relation of the $100 to the annuity of $6 is plainly that of principal to interest. Thus, it will be easy to find the cash value of any given perpetual annuity, or to find the perpetual annuity that could be purchased with a given sum. To illustrate this we should need a line extending beyond all limits: $100 $6 $6 $6 $6 $6 $6 $6 $6 (The divisions of the line represent each one year.) Next we may suppose an annuity to begin at the end of, say, three years, so that the first payment would be made at the end of the fourth year. Taking the annuity to be $6 and the rate 6 per cent per annum, we see that the value of this annuity at the beginning of the fourth year (represented by the point in the illustration below) is $100. A B $100 $6 $6 $6 $6 But that $100 is placed at the end of three years from now, and is therefore equivalent to (188)3 of $100 now. We have thus the cash value of an annuity deferred three years. When the pupil knows how to deal with the two cases discussed he can easily be led to find the cash value of an annuity beginning now and running for a definite number of years. When asked to compare the two perpetual annuities represented below, he will see that the first exceeds the second by three payments-$6 at the end of the first year, $6 at the end of the second year, and $6 at the end of the third year, and these constitute an annuity for three years beginning now. |