EXERCISES 1. If $50 be deposited annually in a building and loan association paying 6%, compounded annually, what will the savings amount to in 10 years? 2. How much must a man save annually, and deposit in a savings and loan company paying 5%, compounded annually, in order to pay off a mortgage for $2000 after 5 years? 3. How long will it take to accumulate $2000 if $75 are deposited annually in a savings and loan company paying 6 %, compounded annually? 4. A father buys a bond for $1000, due after 18 years, which bears interest at 5% payable semi-annually, for the college expenses of his infant boy. The interest payments are deposited in a bank paying 4%, compounded semi-annually. How much money will be available 18 years later? 5. A man buys a house and lot, paying $1000 down, and agreeing to pay $1000 annually for 4 years. What is the equivalent cash price if money is worth 6% per annum? Note that the first payment is not part of the annuity, since the first payment of an annuity is due at the END of the first period. 6. A piano is sold for $100 cash and $25 quarterly for 2 years. What is the equivalent cash price, if money is worth 4% compounded quarterly? 7. A man buys a house, giving back a mortgage for $5000 with interest payable annually at 6%. If the mortgage is to be paid off by five equal annual payments, covering principal and interest, what should be the amount of the annual payment? Hint: The annual payments constitute an annuity whose present value is $5000. 8. Which is the better offer for a house from the seller's standpoint, $5000 down and $1000 annually for 4 years, or $5000 down and $2000 annually for 2 years, if money is worth 4 %? 9. Six months before a boy enters college, his father wishes to deposit a sum in a savings bank paying 4%, interest compounded semi-annually, which will enable the boy to draw $300 every 6 months during his college How much should he deposit? course. 10. It is estimated that a certain mine will be exhausted in 10 years. If the mine yields a net annual income of $2000, what would be a fair purchase price, money being worth 5%? 11. A man holds a mortgage for $6000, due after 3 years, interest 6% payable quarterly. If he disposed of it how much might he expect to receive, if money is worth 4% compounded quarterly. Hint: The present value of the mortgage is the sum of the present value of $6000 due after 3 years and of a quarterly annuity of $90 (one fourth of a year's interest). 12. A bond for $100 is to be redeemed at the end of 10 years, and bears interest at 5 per cent, payable semi-annually. At what price for the bond will the purchaser realize 4 per cent on his investment? Hint: The present value of the bond consists of the present value of $100 due 10 years hence and the present value of the annuity consisting of the semi-annual interest payments. 13. What is the purchase price of the bond in the preceding exercise if the buyer is to realize 6 per cent? 14. An insurance company desires to offer, at the time a policy matures, an option between a 20-year bond for $1000, bearing interest at 5 per cent, payable semi-annually, and an equivalent cash payment. If the computations of the company are on a 3 per cent basis, what should the cash payment be? 15. At the maturity of an insurance policy, the company offers a cash payment of $1000 or the option of five equal annual payments, the first to be made one year after the policy matures. If the company assumes that money is worth 3 per cent, what should be the amount of the annual payment? 16. Three months before a boy enters college, his father deposits on the boy's account $1200 in a bank paying 4 per cent interest, compounded quarterly. The boy wishes to withdraw the money quarterly, in equal amounts, during the four years of his course. How much should he draw at a time? 17. A city issues 20-year bonds to the amount of $100,000 in order to raise money for the improvement of its water supply. A sinking fund to provide for the extinction of the debt can be accumulated at 4 per cent, interest compounded annually. How much must be deposited in the sinking fund at the end of each year? 18. A man buys an automobile for $1000, and estimates that he will be allowed $400 for it in purchasing a new car three years later. How much should he save every three months for the purchase of the new car, if he deposits his savings in a bank paying 4 per cent interest, compounded quarterly? 19. A furnace costing $450 must be installed in a house every 10 years. How much should the landlord save each year for this purpose, if the savings can be accumulated at 5 per cent, interest compounded annually? 20. A man owes $2000 on which he pays 5% interest. If he pays the debt, principal and interest, in 6 equal annual installments, what is the amount of each payment? How much will he owe at the beginning of the second year? At the beginning of the third year? 21. Find the limit of the present value of an annuity as the number of payments increases indefinitely. Verify the result by finding the sum of the infinite geometric progression whose terms are the present values of the several payments. 22. A college graduate wishes to provide for a scholarship of $90 a year. Find the amount which he must present to the college, if collegiate funds can be invested at 4%, using the result obtained in the preceding exercise. By what other method can the amount be found? 23. A railroad spends $900 annually to provide a watchman for a grade crossing. If money is worth 5 per cent, how much can they reasonably be expected to spend for the elimination of the crossing? 88. Graph of the Exponential Function kb"x. The form of the graph of b was determined on page 216. From it the graph of kb may be found by multiplying ordinates by k (Theorem, page 89). Fig. 140 gives the graph of k2 for several values of k. Notice that yk is the intercept on the y-axis. k2 6 k-1 The form of the graph of the function bnx may be found by dividing by n the abscissas of several points on the graph of several values of n is given in Fig. 141. The combination of the two theorems quoted shows that the graph of kbn may be obtained as Ул follows: Plot the graph of be, divide the abscissas of points on it by n, which gives the graph of b1x, and multiply the ordinates of points on the new graph by k. y 7 The form of the graph of kb depends on the value of b, but for suitable values of k' and n' the same curve is the graph of k'cn'*. We shall use b = 10 as a standard value, and we shall now show that we can determine a constant m such that the graphs of kb and k10m2 are identical. The graphs will be identical if and only if This is essentially an exponential equation to be solved for m. Equating the logarithms of both sides of the equation, Hence it is always possible to use the base 10, and we shall do so in the future unless the contrary is indicated. = k10mx The most characteristic properties of the graph of y are that it does not cross the x-axis, that it does cross the y-axis, and that it always rises or always falls. 2. Plot the graph of y = e−x2. How can the graph of y = ke 2x2 be obtained from it? This graph is called the probability curve. 3. Find the inverse of the function k10mx. What are the most important characteristics of the graph of y c log nx? = 4. (a) Construct the graph of k log2 x for several values of k. (b) Construct the graph of log2 nx for several values of n. (c) Construct the graph of log2 3x. 89. The Logarithmic Scale. The ordinates of points on the graph of log10 x corresponding to integral values of x from 1 to 10 inclusive, are a set of segments on the y-axis whose lengths from the origin are equal to the logarithms of the corresponding abscissas. The line O'A' is an enlargement of the segment 04 on the y-axis, the numbers on the right being the logarithms of those on the left. Α' 10-9-1.00 9-0.95 800.90 0.85 60.78 5--0.70 -0.60 3--0.48 2-0 -0.30 1-4,0.00 FIG. 144. A scale is called a uniform scale if the distance of a number from the point marked 0 is equal to the number, the distance from 0 to 1 being the unit segment. A scale is called a logarithmic scale if the distance of a number from the point marked 1 is equal to the logarithm of the number, the distance from 1 to 10 being taken as unity. The logarithms are spaced uniformly along the line O'A', while the integers are spaced non-uniformly. The utility of a logarithmic scale lies in the fact that the addition and subtraction of the logarithms of numbers, and hence the multiplication and division of numbers, may be effected mechanically by the addition and subtraction of line segments on the scale. For instance, to find the product 3 × 2, we add the segment 12 to the segment 1-3. For this purpose, place a pair of dividers so that the points are on the extremities of the segment 12, and then with this opening place one end of the dividers on the point 3 and the other will touch the scale exterior to the segment 1 - 3 in the point 6, the required product. To find the quotient, subtract the segment 1 - 2 from the segment 1 - 3 in a similar manner by means of the dividers. The logarithmic scale is sometimes called Gunter's scale after Edmund Gunter, who first made use of it for purposes of calculation in 1620. The dividers may be dispensed with if two identical logarithmic scales are arranged to slide along one another as shown in Fig. 145, which shows the method of finding the product 3 x 2 and the quotient 3/2. For the product 4 x 5, the above method gives a point out |