LESSONS IN AlilTHMETIC. 



nine months hence, and of 4416, due twelve months honoo, at 4 

 per cent. 



1 . Reckoning True Discount. 



Present value of 309, due 9 months hence, is . . . . 900 

 Present value of i'Uti, duo 12 months hence, is 



t. of 300 for U months is the wiiao as that of 2,71*) i<> 



1 mouth. 



> to rout of 400 for 13 months is the same as that of 4,800 foi 

 1 mouth. 



Hence tho interest on 4700 (the sum of the present values 

 for tho equated time must be equal to the interest on 

 42,700 + 44,800, or 47,500, for one month. Now the time 

 in whi.-li tho samo interest will bo produced by two differou 

 sums will be inversely proportional to tho sums ; hence tho 

 equated timo will be 'flg, or 10 months ; 



we see the truth of the following 



Rule for finding tJie Equated Time of .two or more Debts due at 

 ,;t times, at a given rate per cent., True Discount being 



Find the true present value of each debt, multiply it by its 

 corresponding timo, and add tho products. Divide the sum ol 

 tho products by the sum of tho present values. 



-. Reckoning Mercantile Discount. 



Interest of 309 for 9 months is the same as that of 9 x 309 or 

 2,781, for 1 month. 



That on 416 for 12 months is the same as that of 12 x 416, or 

 4,992, for 1 mouth. 



Hence the interest on 4309 + 4416, or 4725, for tho equated 

 time must be tho same as that on 42,781 + 44,992, or 47,773, 

 for one month. 



Hence the equated time is '^, or 10'72 months nearly. 



We get, then, the following 



Rule for finding tlie Equated Time of two or more Debts due at 

 different times, Mercantile Discount being reckoned : 



Multiply the amount of each debt by its corresponding time, 

 and add the products. Divide the sum of the products by the 

 sum of tho debts. 



N.B. When mercantile discount is reckoned (as is the case 

 in practice), the rate per cent, does not affect the calculation. 

 Tho times of both debts must of course be expressed in 

 tho same denomination, and the result will appear in that 

 denomination. 



EXERCISE 58. EQUATION OF PAYMENTS. 



1. Find the equated time of 800, payable in 3 ye/irs, and of 1,200, 

 payable in 4 years, at 5 per cent, simple interest, by reckoning (1) 

 true, (2) mercantile discount 



2. Find the equated time of payment of 201 5s., due 6 months 

 hence, and of 209, due 18 mouths hence, at 4i per cent., reckoning (1) 

 true, (2) mercantile discount. 



3. Find the equated time of 692, payable in 60 days, and 254, 

 payable in 96 days. 



4. I owe 500, due 50 days hence, and 750, due 100 days hence ; 

 when should I liquidate the debt equitably by paying down 1,500, 

 interest being reckoned at 4 per cent, per annum ? 



STOCKS, SHARES, BROKERAGE, INSURANCE, ETC. 



19. Suppose that I lend a sum of money to the Government 

 or to a company, on the understanding that I am to receive a 

 certain fixed annual per-centage upon it (say 3 per cent.), and 

 that at any timo after this transaction it is found that more 

 than 3 per cent, can be commonly got for money ; it is clear 

 that if I sell then my claim upon the Government or company 

 to another person, ho will not give me so much as I gave. The 

 name given to money so lent is Stock, and the price given at 

 any time for 4100 of this stock is the price of stock at that timo. 



The Funds are properly tho money raised by the Government, 

 by taxes, etc., to pay the interest of the debt, but the term 

 is often applied to the debt itself. Thus, when wo hear that 

 the Funds are at 90>- it means that 490 5s. must bo paid for 

 4100 worth of stock, this entitling the purchaser to receive 

 from the Government the sum of 43 (in the Three per Cents.) 

 agreed to be paid upon the 4100 originally lent. 



Diffnrent names are given to different descriptions of stock, 

 according to the original conditions of the formation of the 

 debt. For instance, the Three per Cent. Consols i.e., the Throo 

 per Cent. Consolidated Annuities, etc. 



20. Given the price of Stock, to find the actual Rate per Cent, 

 received. 



EXAMPLB. If the Three per Cento, are at 93J, find the rat* 

 per cent, of interest received. 



For 93 5s. paid, 3 is received yearly. 



Henoe. as (8* : 3 : ! 100 rate per cent. received. 



t 



Therefore = 3 217, the rate per cent. 



21. Given the sum invested, and the price of the Stock, tojha 

 the Income. 



EXAMPLE. If 41,200 be invested in the Three per Cento 

 when they are at 88), find tho income. 



Since for every 88} paid, 100 worth of stock is received, 

 88} paid will produce an annual income of 3. 



Hence, as 881 : 3 : : 1,300 : required income. 



Therefore tho required income = -^ 



47 



40 17s. OiW. 



22. Given tJie income received from an investment of a given 

 Sum of Money, to find the price of the Stock. 



EXAMPLE. If .1,200 invested in the Three per Cent*, pro- 

 duces a yearly income of 440 17s. OJJd., find the price of the 

 stock. 



40 17s. OJ|d. i J V- of a pound. 



Hence } x J$2 i a the number of times 43 is contained in the 

 income ; that is, it is the number of times 4100 is contained in 

 the amount of stock bought. 



Hence, 41,200 divided by this will give the actual money paid 

 for each 4100 of stock. 



Therefore the required price = - - x 1,200 : 

 1990 



88J. 



23. When Government stock is purchased, the transaction is 

 effected through the agency of a broker, who charges Jth per 

 cent, upon tho stock bought i.e., 2s. 6d. upon every 4100 of 

 stock purchased. 



Thus, if 4500 worth of stock be purchased when the funds 

 are at 92, the actual price paid will be (5 x 492) + (5 x 4), 

 or 4460 12s. 6d. And, similarly, the seller of stock pays his 

 broker ith per cent, upon the amount of stock sold for him. 

 This charge is called Brokerage, or Commission. In the examples 

 we give, however, it need not be reckoned unless it is expressly 

 mentioned. 



24. Exactly the same principles hold with reference to Shares 

 of any kind. Originally they are fixed at a certain price, and 

 then, according to the success or failure of the company, and 

 the probable amount of dividend it will pay, etc., the value of 

 tho shares fluctuates. 



When a share, or 4100 of stock, will sell for the original 

 price which was paid for it, then the shares are said to bo at 

 par. When the price is less by a certain amount than the 

 original price, they are said to bo at so much discount ; and 

 when tho price is more by a certain amount, they are said to be 

 at so much premium. 



25. EXAMPLE. The income derived from investing a sum in 

 the Three per Cents, at 90 differs by 41 from that derived from 

 an equal sum invested in railway shares at 140, paying 5 per 

 cent, upon the original shares. Find tho sum. 



In tho first investment, 



90 produces annually 3 ; 

 Therefore 1 , or &. 



In the second investment, 



140 produces annually 5 ; 

 Therefore 1 rle, or A. 



Hence tho difference of income produced by 1 is -,' f j'j, or K 

 Hence tho difference of income produced by 420 ia JJ/or 1.* 

 The answer is therefore 430. 



INSURANCE. 



2C. By the yearly payment of a certain sum called a Premium 

 x> an insurance company, a person can secure at his death the 

 >ayment of a certain larger sum. The document by which the 

 company binds itself to pay over tho money at the death of the 

 insurer is called tho Policy of Insurance. Thus, a man of 30, in 

 ordinary health, by paying about 425 a year to a company, ia 

 able to " insure his life " for 41,000. 



Tho principles which determine tho amount of premium to be 



tid depend upon carefully prepared tables of statistics, show- 



ng tho average rate of mortality at different ages, and also upon 



;he doctrine of chances and annuities, but they are too compli- 



ated to be introduced here. 



There are various other kinds of insurance, as, for instance 



