ALGEBRA. Logarithmic Equations. [CH. XXIX 40. Ex. 1. Solve the equation log (x − 9) + log √(2 x − 1) = 1. log√(x-9)+ log√(2x-1)= log 10, ✓[(x − 9)(2 x − 1)] = 10, 2 x2 19x+9= 100. Therefore or The roots of this equation are 13 and or or Ex. 2. Solve the equation log(x+12) log x = 0.84510 + log(6 − 6 x). By the principles of logarithms, log x + 12 Consequently Ex. 3. Solve the equation x logz = 100 x. (log x)2= log 100+ log x, (log x)2 - log x = 2. Solving this equation as a quadratic in log x, we obtain 7. log(x + √x) + log (x − √/x)= log 4 + log x2 - log x. Compound Interest and Annuities. 41. To find the compound interest, I, and the amount, A, of a given principal, P, for n years at per cent. If the interest is payable annually, the amount of $1 at the end of one year will be 1+r dollars, and the amount of P dollars will be P(1+r) dollars. This amount, P(1+r), becomes the principal at the beginning the second year. Therefore, at the end of the second year the amount will be P(1+r) × (1+r), = P(1+r)2 dollars, and so on. Therefore, at the end of n years the amount will be P(1 + r)" dollars, or A = P(1+r)". 42. This formula can be used not only to find A, but also to find P, r, or n, when the three other quantities are given. Thus, 43. An Annuity is a fixed sum of money, payable yearly, or at other fixed intervals, as half-yearly, once in two years, etc. 44. To find the present value, P, of an annuity of A dollars, payable yearly for n years, at r per cent. The present worth of the first payment is A 1 + r dollars, of dollars, and, in general, of the (1+r)2 dollars. (1+r)" Multiplying numerator and denominator by 1+r, we have ALGEBRA. [CH. XXIX Ex. 1. Find the amount of $500 for 8 years at 5% compound interest. A = P(1+r)" = 500 × 1.058. log A= log 500 + 8 log 1.05. Ex. 2. Find the present value of an annuity of $1000 for 6 years, if the current rate of interest is 5%. P=5076. Therefore the present value of the annuity is $5076. EXERCISES IX. Find the amount at compound interest: 1. Of $3600 for 5 years at 41% 2. Of $1875.50 for 8 years at 5%. 3. Of $12,350 for 6 years at 31%. 4. Of $21,580 for 7 years 4 months at 4%. Find the principal that will amount to: 5. $7913 in 5 years at 5% compound interest. 6. $14,770 in 10 years at 41% compound interest. 7. $11,290 in 8 years at 4% compound interest. 8. $11,090 in 6 years 6 months at 3% compound interest. 9. In what time, at 4%, will $8010 amount to $11,400 at compound interest? 10. In what time, at 41%, will $3530 amount to $6023.50, if the interest is compounded semi-annually? Find the rate of compound interest: 11. If $1110 amounts to $1640 in 8 years. 12. If $3750 amounts to $6070 in 14. years. Find the present value of an annuity: 13. Of $1000 for 10 years, if the current rate of interest is 4%. 14. Of $1250 for 8 years, if the current rate of interest is 41%. 15. Of $2500 for 10 years, if the current rate of interest is 5%. 16. Of $3000 for 12 years, if the current rate of interest is 6%. CHAPTER XXX. PERMUTATIONS AND COMBINATIONS. § 1. DEFINITIONS. 1. The following examples will illustrate the character of an important class of problems. Pr. 1. Write the numbers of two figures each which can be formed from the three figures, 4, 5, 6. We have 45, 54, 46, 64, 56, 65. Pr. 2. What committees of two persons each can be appointed from the three persons, A, B, C? The committees may consist of A, B; A, C; or B, C. These problems make clear the difference between groups of things, selected from a given number of things, in which the order is taken into account, as in Pr. 1, and in which the order is not taken into account, as in Pr. 2. 2. We are thus naturally led to the following definitions: A Permutation of any number of things is a group of some or all of them, arranged in a definite order. A Combination of any number of things is a group of some or all of them, without reference to order. 3. It follows from these definitions that two permutations are different when some or all of the things in them are different, or when their order of arrangement is different; and that two combinations are different only when at least one thing in one is not contained in the other. Thus, ab and ba are different permutations, but the same combination. |