16. If 4% of the taxes of a certain town are spent in collection, find the tax levy so that $166,032 may be available for public purposes. 17. In a certain city the tax for school purposes is 3 mills on $1. If this tax amounts to $24,210, find the assessed valuation of taxable property. 18. If $18,477.90 is raised by taxation at the rate of 40 cents on $100, find the valuation of taxable property. 19. Find the rate of taxation when $29.14 is paid on property assessed at $7285. United States Customs Duties 20. The cost of 10 opera glasses imported from England after paying a duty of 45% ad valorem is $84.39. (a) Find the invoice price in United States currency. (b) Find the invoice price in British currency (£1= $4.85). 21. The duty on treble ingrain carpet is 22¢ per square yard and 40% ad valorem. Suppose the invoice price is $1 per square yard, and the cost of a rug after the payment of duty is $32.40. How many square yards are in the rug? 22. The duty on razors is $1.75 per dozen and 20% ad valorem. The cost to an importer of 6 dozen Sheffield razors is $94.31. (a) Find the invoice price in United States currency. (b) Find the invoice price in British currency (£1=$4.85). (c) Find the selling price of a razor if the importer makes a profit of 20%. (d) If the razors are sold at $2 apiece, what is the rate per cent of profit? 23. A jeweler imports 5 microscopes which cost him after paying a duty of 45% ad valorem $290. (a) Find the amount of the invoice. (b) How should the microscopes be marked so as to make a profit of 40% after allowing a discount of 10% on the marked price? 24. The duty on lead pencils is 45 per gross and 25% ad valorem. (a) Find the invoice price per gross of imported pencils which cost the importer $2.45 per gross. (b) If these pencils are sold at 3 apiece, find the gain per cent. Insurance 25. The annual premium on a life insurance policy of $2500 is $49.80. Find the rate on $1000 insurance. 26. If the annual premium on a life insurance policy at the rate of $27.10 per $1000 is $130.08, find the amount of the policy. 27. A residence is insured at the rate of $1.10 per $100. If the premium is $37.95, find the amount of the policy. 28. A house is insured at the rate of $1.30 per $100. If the premium is $109.59 and the amount of the policy is of the value of the house, find the value of the house. 29. A man has a life insurance policy at the rate of $29.40 per $1000, and a policy for the same amount at $28.30 per $1000. He pays for both $201.95. Find the amount of each policy. CHAPTER X INVOLUTION. SQUARE ROOT 111. The process of raising a quantity to a power is called involution. 112.. Find the third power of a2. (a2)3 = a2. a2. a2 = a2+2+2 = αβ. Hence, To find any power of a letter affected with an exponent, write the letter, and for its exponent take the product of the exponent of the power by the exponent of the letter. 113. Find the fourth power of abc. (abe)1 = abc · abc · abc · abe = aaaa · bbbb • cccc = a4b4c4. Associative Law. Hence, A power of a monomial expression is obtained by raising the factors of the expression to the required power and taking their product. Hence, A power of a fraction equals the power of the numerator divided by the power of the denominator. Even powers of negative quantities are positive. 116. In involution a quantity is given and a power of the quantity is sought. In evolution a power of a quantity is given and the quantity is sought. Evolution is the process of finding a root of a quantity. The square root of a quantity is one of its two equal factors. The square root of 16 is 4, since 164 x 4. Similarly, the cube root of a quantity is one of its three Thus, the cube root of 125 is 5, since equal factors. 125 = 5 x 5 x 5. What is the square root of 225? This problem may be solved by resolving 225 into its prime factors. 2253.3.5.5=(3.5)(3·5). √225 = 3 x 5 = 15. Hence, Since and (+15)(+15)= + 225, (-15)(-15)=+225. Therefore, the square root of 225 is ±15. The double sign is read plus or minus. 117. Since an even power of a positive or negative quantity is positive, hence, inversely, An even root of a positive quantity is ±. Since an odd power of a positive quantity is positive, hence, An odd root of a positive quantity is positive. Also, since an odd power of a negative quantity is negative, therefore, An odd root of a negative quantity is negative. 118. An even root of a negative quantity gives rise to a new kind of number known as an imaginary. For example, √ — 4. This is neither +2 nor - 2. 119. Since therefore, (3x2y1)5 = 243 x10 y20, √243 x10 y20 = 3x2y*. |