Given the above table, the calculation of elementary tables of the same kind for the other rates of this section is not laborious, provided a table of reciprocals is at hand. The exercise is all the more valuable because the results can be compared with the published tables. EXAMPLES. Ex. 1.-What is the equivalent of compound interest at 2 per cent. per quarter, payable each quarter, in terms of per cent. per annum, payable each year? Now, 1+£ at end of 1st 40 1+£ at end of 3rd 1+£ at end of 4th 40 quarter = £ at beginning, (1 +)1£ at end of year = £ at beginning; 40 (1+1)4-1£ interest = £ principal by year, payable yearly; 100 {(1 + 1) − 1}£ interest = 100£ principal by year. 1 40 1 1 1 1 (1 + 2)2 = 1 + 436 +6 (40)2 + 4 (40)3 + (40)4; 9761 The first approximation to the value is 10, the second 10%, the third 10%, and the full value is 100. The first approximation is equivalent to reckoning simple interest. Ex. 2. Find the amount at compound interest, payable yearly, at the end of four years, of 2,345£, the rate of interest for the first year being 4 per cent., for the second 5 per cent., for the third 6 per cent., and for the fourth 7 per cent. hence, 1+8£ at end of 3rd = 1£ at end of 2nd, 2345 1.04 2345 The calculation is as follows: Three places of decimals in the answer are sufficient, for a farthing is very nearly the thousandth of a 9380 pound. Hence, when the decimals amount to 4, we begin to apply contracted multiplication. Put the 2438.80 highest figure of the multiplier under the lowest figure 1.05 to be retained, reverse the order of the figures of the 1219400 multiplier, and begin the multiplication by a figure at 243880 the figure of the multiplicand below which it falls. 2560-7400 By this means we are able to cut off the unnecessary 601 figures. 25607400 1536444 Ex. 3.-Find the number of years in which 1,000£ will amount to 2,400£ at 5 per cent. per annum, compound interest, payable yearly. Given log 3=47712, 701 log 569897, and log 784510. 2714.3844 27143844 1900069 2904.3913 By decimalising we get 2,9047. 78. 10d. Suppose in n years, then (1+1)"£ at end = £ at beginning, 1000£ at beginning, 1000 (1+)"£ at end. But the amount at end is given to be 2400£; ... n{log 3 + log 7 - log 5 - 2 log 2} = log 3 + 2 log 2 – log 5, n = log 3 + 2 log 2 - log 5 log 3+ log 7 - log 5 – 2 log 2° All the logs are given directly, excepting log 2. log 3+ log 7+ log 5-2 Ex. 4.-A tenant whose annual rent is 400£, due at the end of each year, wishes to pay the whole of his rent in advance at the beginning of a nineteen years' lease; what sum ought he to pay, interest being reckoned at 4 per cent. per annum ? (1 + 2)19 - 1 400£ per annum, 25 £ at end of 19 years = £ paid per annum, 1 25 1£ at beginning of 19 years=(1+)19£ at end; ... 400 x 25(1+)19-1}(1+)-19£ at beginning. 10000{1-(3)19}£ at beginning. i.e., 1. Find the simple and the compound interest on 3781. for three years at 5 per cent. per annum. 2. Find to a penny the compound interest on 377 571. for 3 years at 3 per cent. 3. Which is greater, and by how much-the simple interest on 201. for 3 years at 5 per cent., or the compound interest for the same time at 4 per cent. ? 4. Find the amount, at compound interest, of 2001. for 3 years at 5 per cent. per annum. 5. A sum of money amounted to 465l. 2s. in 2 years at 5 per cent. compound interest; what was the sum? 6. Find the amount, at compound interest, at the end of 2 years, of 1237. at 4 per cent. per annum, payable half-yearly. 7. In how many years will a sum of money double itself at 5 per cent. compound interest? 8. A noble Scotch family have retained in their possession since the death of the Regent Murray in 1570, 5007. in gold coins. Find to what this sum would have amounted by the present year (1884) if it had been invested at 5 per cent. compound interest. 9. Find to the nearest shilling the present value of 2731., payable after 3 years, the rate of interest being 3 per cent. per annum. 10. Find the compound interest and true discount on 8001. for 3 years at 5 per cent. 11. The present value of a certain debt due 3 years henge is 1127. 10s. In 2 years' time, if it is not paid, its value will be 1201. 2s. 6d. What is the debt? 12. A gentleman leaves his property, worth 40007., to be divided between his sons, who are aged respectively 15 and 18 years at the date of his death. What sums ought his executors to set apart for the sons in order that they may receive the same amount on coming of age, taking the price of money at 5 per cent. per year? 13. A ship is valued at 14,7201. What sum should be insured at 8 per cent. by a person who owns one sixteenth of the ship, so that in case of loss he may recover both his share of the vessel and his insurance? 14. A gentleman insured his life for 2507. at a premium of 51. per annum; he died after n years, and the insurance office neither gained nor lost in the transaction. Find n, reckoning compound interest at the rate of 5 per cent. per annum. 15. Find the present value of an annuity of 50l. payable for 12 years, first, when the annuity begins to be paid at the end of 1 year hence; and, second, when it begins to be paid at the end of 10 years hence. Money at 3 per cent. per annum. 16. Find the amount accumulated at the end of 3 years by a person who invests 500l. now, and does the same at the beginning of each succeeding year, at 8 per cent. compound interest. 17. A county borrows 150,000 dols., to be paid off, principal and interest, in 20 equal annual instalments. Find the annual payment, interest at 6 per cent. 18. A cargo of goods was insured at 33 per cent. on the price they were expected to fetch abroad, which was 20 per cent. over the price paid at home. The insurance came to 1177. 10s. Find the cost price of the goods. 19. How many years will it take 1007. to accumulate to 500l., at 4 per cent. compound interest? 20. What is the rate of interest corresponding to 16 years' purchase? 21. The reversion of a freehold estate worth 2001. per annum to comn.ence 4 years hence is to be sold. Ascertain its value at 5 per cent. compound interest. 22. Required the commutation for a perpetual pension of 4,000l. per annum, taking money at 4 per cent. 23. What is the number of years' purchase when 3 per cent. consols are bought at 96? SECTION VII.-SHARES AND STOCKS. ART. 39.-Shares. By Shares is meant the equal sums of money by which the capital of a public company is at first brought together. The original or nominal value of a share is the amount of money subscribed to the undertaking by a person for each share he receives; it is expressed in the form m£ nominal per share. Generally a portion of the value of a share is paid on application therefor, another portion on allotment, and the remainder may be called up at intervals as required by the state of the company. Hence besides the nominal value of a share we may have its paid-up value. ART. 40.-Premium, Par, Discount. A person who has invested money in the shares of a company, may afterwards wish to exchange his shares for cash; the price received for a share will depend on the degree of prosperity of the company. Suppose that he sells at n£ per share. If n is greater than m, the shares are said to sell at a premium, the rate of premium being (nm)£ premium per share. If n is less than m, the shares are said to sell at a discount, the rate of discount being (m-n)£ discount per share. If n is equal to m, the shares are said to sell at par. ART. 41.-Stock. So long as the shares are not fully paid up, the capital of the company is sold only in shares; but when that has been done the shares are blended into one stock, and any integral number of pounds of the stock can be sold. The price of the capital of the company may then be no longer quoted at so much per share, but at so many £ cash for 100£ stock. The portion of the profit accruing in the course of a year or a. half year which it is thought safe to distribute is called the yearly or half-yearly dividend. It is divided among the shareholders in |