EQUATION OF PAYMENTS. Q. What is EQUATION OF PAYMENTS? A. Equation of Payments teaches how to find one mean or equated time for the payment of several debts, due at different times, so that no loss shall be sustained by either party. RULE. Q. How do you state and work the questions in Equation of Payments? 4. Each payment must be multiplied by its time, and the several products must be added together; ther, the sum of the products must be divided by the whole debt, and the quotient will be the answer, or equated time for the payment of the whole. EXAMPLES. 1. A owes B $1600, of which $800 are to be paid in 4mo., and $800 in 8mo. ; but they agree that the whole shall be paid at one time; what is the equated time for the payment of the whole? Ans. 6mo. EXPLANATIONS. $ mo. In this example, you multiply the 800 by 4, and again by 8, and add the two products for a dividend, and the two payments, making 1600 1600) for a divisor. The principle of this operation is perfectly 9600 (6mo. Ans. plain; for, if B shall extend the payment of one half of his debt two months after it is due, he should, unquestionably, receive the other half two months before it is due. 2. Cowes D $2000, of which $400 are now due, $800 to be paid in 5mo., and $800 at 15mo., and they agree to make one payment of the whole; at what time must it be paid.? Ans. 8mo. 3. A merchant purchased goods, amounting to $4000, of which $800 are to be paid present, $1600 at 5mo., and the balance, $1600, at 10mo.; but they agree to make one payment of the whole; what is the equated time? Ans. 6mo. ANNUITIES. Q. What is an ANNUITY? A. An Annuity is a sum of money payable every year, for a certain number of years, or for ever. EXPLANATIONS. As was stated on page 168, this rule is merely a particular application of the Rule of Three Direct. Amount is the sum of the annuities for the time it has been forborne, with the interest due on each payment or annuity. Present worth of an annuity is such a sum as being now put to interest, would exactly pay the annuity when it becomes due; and, it is such a sum as must be given for the annuity, if it be paid at the commencement Contingent annuity is when the annuity depends on some contingency, as the life or death of a person Reversion is when the annuity does not commence until a number of years has elapsed. Arrear is when the debtor keeps the annuity beyond the time of payment. RULE. Q. How do you find the amount of an annuity at simple interest? A. First find the interest of the given annuity for one year; and then for 2, 3, 4, &c., up to the given number of years, less 1: then multiply the annuity by the given number of years, and add the product to the whole interest, and the sum will be the amount sought. EXAMPLES. 1. What is the amount of an annuity of $100 for 5 years, simple interest, computed at 7 per cent? Ans. $570. EXPLANATIONS. The interest of $100, at 7 per cent. for 1 year, is for 2 years, $7. 14. 21. 28. $100×5-500 for 3 years, Five years' annuity, at $100 a year 2. What is the amount of an annuity simple interest, computed at 7 per cent? is Ans. $570 of $800, for 5 years, Ans. $4560. 3. A man let a house upon a lease for 8 years, at $200 per annum, and the rent being in arrear for the whole term; what sum must he demand at the end of the term, simple interest being allowed at 6 per cent? Ans. $1936. RULE. Q. How do you find the present worth of an annuity at Simple Interest? A. First find the present worth of each year by itself, discounting from the time it becomes due; then the sum of all these will be the answer or present worth required. EXAMPLES. 1. What is the present worth of $200 per annum, to continue 4 years, at 7 per cent.? Ans. $683,89,1m. Ans. $683,89,1m. present worth required. 2. What is the present worth of $800 per annum, to continue 4 years, at 6 per cent.? Ans. $2792,12c. 3. What is the present worth of $200 per annum, to con→ tinue 3 years, at 4 per cent.? Ans. $556,06,3m. INVOLUTION. Q. What is INVOLUTION? A. Involution teaches how to find the powers of numbers, by multiplying any given number into itself continually a given number of times: and the several products which arise are called powers. EXPLANATIONS. The number denoting the height of the power, is called the index or exponent of that power; thus, the number itself is called the first power, or root. If the first power be multiplied by itself, the product is called the second power or square; and if the square be multiplied by the first power, the product is called the third power, or cube, &c.; thus, 4 is the root or first power of 4; 16 is the 2d power, or square of 4, produced thus, 4×4 = 16; 61 is the 3d power or cube of 4, produced thus, 4X4X4 = 64, and so on. Thus, you will readily perceive, that to find the square of any given number, you multiply once; and to find the cube you multiply twice. EXAMPLES. 1. What is the square, or 2d power of 5? Ans. 25. EXPLANATIONS. In this example, you merely multiply the 5 by itself, 5 and the product is the answer. 5 1 25 Ans. 2. What is the square, or 2d power of 7? Ans. 49. 3. What is the cube of 3? Ans. 27. Ans. 1024. 4. What is the 5th power of 4? 9. What is the square of? Ans. 292,41. Ans. §. 10. What is the the cube of? Ans. 27. NOTE. A decimal fraction is raised to any power, the same as a whole number, and the same rules are observed in pointing off as in multiplication of decimals. A vulgar fraction is raised to any power by multiplying the numerator of the fraction by itself, and the denominator by itself, until, as in whole numbers, the number of multiplications be one less than the index, or exponent of the power to be found. TABLE of the powers of the 9 digits, from the 1st to the 5th. Roots, 12 31 41 51 61 7 81 9 quares, 1 4 9 16 25 36 491 6i4| 81 Cubes, || 8|27|61| |25| 216||343 512 72|| Biquadrates [116] 81| 256] 625|1296| 2401| 4096| 6561 Sursolids, |32|243) 10243125|7776|16307|3276859049, EVOLUTION. Q. What is EVOLUTION? A. Evolution is the extracting or finding the root of any given power or number. EXPLANATIONS. Evolution is, as you will at once perceive, the reverse of Involution. The root, you have seen, is that number which, being multiplied into itself continually, will produce the given power. The Square Root of any given number is a number which, being multiplied into itself, will produce that number. The Cube Root is a number, which, being cubed, or involved to the third power, will produce the same given num ber. The power of any given number, or root, may be found exactly by multiplying the number continually into itself; yet, there are numbers, a proposed root of which can never be exactly found; but, by means of decimals, you may approximate or come near to the root, to any degree of exactness. Those numbers whose exact roots can not be obtained are called surd numbers; and those whose roots can be exactly found, are called rational numbers. This character v placed before any number, expresses the square root of that number; thus, V25 expresses the square root of 25. The same character is made to express any other root, by placing the index of the root above it. Thus, 27 expresses the cube root of 27; and 625 expresses the fourth root of 625. &c. Thus, you can always tell how many figures there will be in the SQUARE ROOT of any Humber, by pointing it off from unit's place, into periods of two figures each. You can likewise ascertain how many figures there will be in the CUBE ROOT of any number, by pointing it off from unit's place, into periods of three figures each. EXTRACTION OF THE SQUARE ROOT. Q. What is extraction of the Square Root? A. Extraction of the Square Root is to find a number, which, being multiplied into itself, will produce the given number. RULE. Q. How do you extract the square root of any given number? A. 1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another |