To express this formula in words: Add 1 to the product of the time multiplied by the rate, and divide the amount by the sum. The quotient will be the principal. This is a general rule for calculating Discount. When a man pays a sum of money before it has become due, he is evidently entitled to some reduction from the debt. Equity requires that he should pay such a sum as would amount to the sum due, if put at interest during the time for which it is advanced, at any rate per cent. agreed upon by the parties. The difference between such a sum, which is called the present worth of the debt, and the debt itself, is the discount. 69. What is the present worth of 392 dollars, due in 2 years, the discount being reckoned at 6 per cent.? It is here required to find what sum will amount to 392 dollars, in 2 years, at 6 per cent. In other words, the amount, time and rate are given, to find the principal. 70. A gentleman hired a sum of money at 5 per cent.; and at the end of 3 years, he paid 8234 dol lars, for principal and interest. What was the sum hired? 71. A merchant sold an invoice of goods, amounting to 1961 dollars, on a year's credit. What dis count should he make for present payment, allowing money to be worth 6 per cent.? 72. Required the present worth of 713 dollars, discounted for 4 years at 6 per cent. 73. What sum will amount to $667 in 3 years, at 5 per cent. ? 74. Given the amount, time and principal, to find the rate. T p = the interest for 1 year. That is, From the amount subtract the principal; and divide the remainder by the product of the principal multiplied by the time. The quotient will be the rate. 75. A man lent $420; and, in 5 years, he received in payment $546. At what rate per cent. was the money lent? 1 76. At what rate per cent. will $380 amount to $513, in 7 years? 77. Given the interest, time and rate, to find the principal. Expressed in words: Multiply the time by the rate, and divide the interest by the product. The quotient will be the principal. 78. A paid B $126 for the use of a certain sum of money 3 years; the interest being reckoned at 5 per cent. What was the sum lent? 79. In the course of 4 years, a man paid interest to the amount of $288, which was reckoned at 6 per cent. What was the debt? 80. The amount, principal and rate being given, to find the time. That is, From the amount subtract the principal; and divide the remainder by the product of the rate and principal. The quotient will be the time. 81. A man lent $460 at 5 per cent. and received, for principal and interest, $529. How long was the money kept? 82. In what time will $780 amount to $1014, interest being reckoned at 6 per cent.? These examples show the manner in which general results, or formulas, are obtained; and also how they may be used in solving particular questions. Let the learner now turn back to Chapter VIII., and generalize the questions marked with a star (*); and then solve the same questions, numerically, by their respective formulas. He will thus be prepared to generalize some of the more difficult questions, in the same chapter, which are not marked, CHAPTER X. EVOLUTION. SECTION I. Introduction. WHEN a quantity is multiplied by itself one or more times, the product is called a Power of that quantity. Thus, a2, being the product of a Xa, is the second power or square of a; and 63, that is, b xbx b, is the third power or cube of b. [See Chap. VII. Sec. I.] On the contrary, the quantity which is multiplied by itself to produce any power, is said to be the Root of that power. Thus, a is the second or square root of a2; and b is the third or cube root of b3. Powers and Roots are, therefore, correlative terms; and Evolution and Involution are the reverse of each other. Involution is the method of raising a given root to a proposed power; but Evolution is the method of finding the roots of given powers. Involution is more perfect, however, than Evolution; for if any proposed power of a given quantity be required, it can be exactly obtained; but there are many quantities whose exact roots cannot be found. It is evident, for instance, that the square root of a cannot be determined; for there is no quantity, which, being multiplied by itself, will produce a. The roots and powers of numbers have the same relation to each other as those of literal quantities. Thus, the second powers of 2 and 3 are 4 and 9; and the square roots of 4 and 9 are 2 and 3. The exact roots of the intermediate numbers, 5, 6, 7 and 8, cannot be found. 5th powers. 1 32 243 1024 3125 7776 16807 32768 59049 100000 The roots of quantities are indicated either by means of the radical sign ✔✅✅, or by a fractional index. Thus, a, ora, is the square root of a, 2 3 a is the cube root of a. a3 is the 4th root of a3. 64 is the square root of 64, which is 8. 3a+x is the cube root of a + x. If the quantity affected by the radical sign be not a complete power, that is, if its root cannot be exactly found, it is called a Surd, or Irrational Quantity. Thus, 35, x2, 5/a3, &c., are surd quantities. 3 Express the roots of the following quantities by means of the radical sign: |