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56. The principal use of the arithmetical complement, is in working proportions by logarithms; where some of the terms are to be added, and one or more to be subtracted. In the Rule of Three or simple proportion, two terms are to be added, and from the sum, the first term is to be subtracted. But if, instead of the logarithm of the first term, we substitute its arithmetical complement, this may be added to the sum of the other two, or more simply, all three may be added together, by one operation. After the index is diminished by 10, the result will be the same as by the common method. For subtracting a number is the same, as adding its arithmetical complement, and then rejecting 10, 100, or 1000, from the sum. (Art. 53.)
It will generally be expedient, to place the terms in the same order, in which they are arranged in the statement of the proportion. 1. As 6273 a. c. 6.20252 2. As 253 a. c. 7.59688 Is to 769.4 2.88615
Is to 672.5 2.82769 So is 37.61 1.57530
So is 497 2.69636
57. In compound, as in single proportion, the term required may be found by logarithms, if we substitute addition for multiplication, and subtraction for division.
Ex. 1. If the interest of $365, for 3 years and 9 months, be $82,13; what will be the interest of $8940, for 2 years and 6 months ?
In common arithmetic, the statement of the question is made in this manner,
: 82.13 dollars ::
And the method of calculation is, to divide the product of the third, fourth, and fifth terms, by the product of the two first.* This, if logarithms are used, will be to subtract the sum of the logarithms of the two first terms, from the sum of the logarithms of the other three.
365 log. 2.56229 Two first terms
Sum of the logarithms
Third term Fourth and fifth terms
1.91450 3.95134 0.39794
Sum of the logs. of the 3d, 4th, and 5th 6.26378
1st and 2d 3.13632
58. The calculation will be more simple, if, instead of subtracting the logarithms of the two first terms, we add their arithmetical complements. But it must be observed, that each arithmetical complement increases the index of the logarithm by 10. If the arithmetical complement be introduced into two of the terms, the index of the sum of the logarithms will be 20 too great; if it be in three terms, the index will be 30 too great, &c.
365 a. c. 7.43771 Two first terms
3.75 a. c. 9.42597 Third term
8940 3.95134 Fourth and fifth terms
The result is the same as before, except that the index of the logarithm is 20 too great.
* See Arithmetic.
Ex. 2. If the wages of 53 men for 42 days be 2200 dollars; what will be the wages of 87 men for 34 days?
Two first terms
53 a. c. 8.27572
42 a. c. 8.37675
59. In the same manner, if the product of any number of quantities, is to be divided, by the product of several others; we may add together the logarithms of the quantities to be divided, and the arithmetical complements of the logarithms of the divisors.
Ex. If 29.67 X 346.2 be divided by 69.24 x 7.862 X 497 ; what will be the quotient?
In this way, the calculations in Conjoined Proportion may be expeditiously performed.
6C. In calculating compound interest, the amount for the first year, is made the principal for the second year; the amount for the second year, the principal for the third year, &c. Now the amount at the end of each year, must be
proportioned to the principal at the beginning of the year. If
the principal for the first year be 1 dollar, and if the amount of 1 dollar for 1 year=a; then, (Alg. 377.) a : a’=the amount for the 2d year, or the princi
pal for the 3d ; a? ; q3=the amount for the third 1:a::
year, or the prin cipal for the 4th ; as : a*=the amount for the 4th year, or the prin
cipal for the 5th. That is, the amount of 1 dollar for any number of years is obtained, by finding the amount for 1 year, and involving this to a power whose index is equal to the number of years. And the amount of any other principal, for the given time, is found, by multiplying the amount of 1 dollar, into the number of dollars, or the fractional part of a dollar,
If logarithms are used, the multiplication required here may be performed by addition ; and the involution, by mul. tiplication. (Art. 45.) Hence,
61. To calculate Compound Interest, Find the amount of 1 dollar for 1 year ; multiply its logarithm by the number of years; and to the product, add the logarithm of the principal. The sum will be the logarithm of the amount for the given time. From the amount subtract the principal, and the remainder will be the interest,
If the interest becomes due half yearly or quarterly ; find the amount of one dollar, for the half year or quarter, and multiply the logarithm, by the number of half years or quarters in the given time. If P=the principal, a=the amount of 1 dollar for 1
year, n=any number of years, and A=the amount of the given principal for n years ; then
Taking the logarithms of both sides of the equation, and reducing it, so as to give the value of each of the four quantities, in terms of the others, we have
Any three of these quantities being given, the fourth may be found.
Ex. 1. What is the amount of 20 dollars, at 6 per cent. compound interest, for 100 years?
2. What is the amount of 1 cent, at 6 per cent. compound interest, in 500 years ? Amount of 1 dollar for 1 year 1.06 log. 0.0253059 Multiplying by
More exact answers may be obtained, by using logarithms of a greater number of decimal places.
3. What is the amount of 1000 dollars, at 6 per cent. compound interest, for 10 years ?
4. What principal, at 4 per cent. interest, will amount to 1643 dollars in 21 years ?