11.2740 10.4772 Y’rs./4 per cent. j5 per cent. (6 per cent. 17 per cent. 1 .9615 .9523 .9433 .9345 2 1.0860 1.8594 1.8333 1.8080 3 2.7750 2.7232 2.6730 2.6243 4 3.6298 3.5459 3.4651 3.3872 5 4.4513 4.3294 4.2123 4.1001 6 5.2421 5.0756 4.9173 4.7665 7 6.0020 5.7863 5.5823 5.3992 8 6.7327 6.4632 6.2097 5.9712 9 7.4353 7 1078 6.8016 6.5152 10 8.1109 7.7217 7.3600 7.0235 11 8.7605 8.3064 7.8868 7.4986 12 9.3850 8.3632 8.3838 7.9426 13 9.9856 9.3935 8.8526 8.3576 14 10.5631 9.8986 9.2949 8.7454 15 11.1184 10.3796 9.7122 9.1079 11.5523 10.8377 10.1058 9.4466 17 12.1656 9.7632 18 12.6593 11.6895 10.8276 10.059 19 13.1339 12.5883 11.1581 10.335 20 13.5903 12.4622 11.4699 10.594 21 14.0291 12.8211 11.7640 10.835 14.4511 13.1630 12.0415 11.061 14.8568 13.4885 12.3033 11.272 24 15.2469 13.7986 12.5503 11.469 25 15.6220 14.0939 12.7833 11.653 26 15.9827 14.3751 13.0031 11.825 27 16.3295 14.6430 13.2105 11.986 28 16.6630 14.8981 13.4061 12.137 29 16.9337 15.1410 13.5907 12.277 30 17.2920 15.3724 13.7648 12.409 16 22 23 10. What is the present worth of an annuity of $21.54, for 7 years; interest being 6 per cent. ? 11. What is the present worth of an annuity of $ 936, for 20 years, at 5 12. What is the present worth of an annuity of $ 258, for 17.years, at 4 per cent ? per cent. ? 13. Find the present worth of an annuity of $796.50, tò continue 28 years; interest being 7 per cent. ? 14: A young man purchases a farın for $924; and agrees to pay for it in the course of 7 years, paying 1 part of the price at the end of each year. Allowing interest to be 6 per cent., how much cash in advance will pay the debt ? 15. Allowing interest to be 5 per cent., which will be in my favor, to pay $15 a year for 10 years, or, to pay $ 160 in advance ?—by how much? When an annuity does not commence until a given time has elapsed, or some particular event has taken place, it is called a REVERSION. PROBLEM III. To find the present worth of an annuity in reversion. RULE. Find, (by Problem 2nd.), the present value of the annuity from the present time till the end of the period of its continuance: find, also, its value for the time before it is to commence: the difference of these two results will be the present worth. 16. What is the present worth of an annuity of $ 200, to be continued 7 years, but not to commence till 2 years hence; interest being 6 per cent. ? 17. Find the present worth of a reversion of $ 152 a year, to commence in 6 years, and to continue 18 years interest being 4 per cent. 18. What is the present worth of a reversion of $ 75 a year, to commence in 5 years, and to continue 24 years; interest being 6 per cent.? 19. What must be paid for the purchase of a reversion of $450 a year, to commence in 5 years, and to continue 13 years; interest being 5 per cent. ? 20. Find the present worth of a reversion of $ 942.30 a year, to commence in 2 years, and to continue 11 years; interest being 7 per cent. 21. A father leaves to his son, a rent of $310 per annum, for 8 years, and, the reversion of the same rent to his daughter for 14 years thereafter. What is the present worth of the legacy of each, at 6 per cent.? 22. What is the present worth of a reversion of $100 a year, to commence in 4 years, and to continue for ever; interest being 6 per cent? This annuity continuing for ever, will, when it commences, be worth that sum of money which would yield $100 a year, at 6 per cent. interest. Therefore, after finding the principal, whose interest is $100 per annun, deduct from it a compound discount for 4 years the re• mainder will be the present worth. 23. What is jhe present worth of a reversion of $824 a year to commence in 7 years, and to continue for ever; interest being 5 per cent. ? 24. What is the present worth of a reversion of $530 a year, to commence in 22. years, and to continue for ever ; interest 7 per cent. ? 25. How much must be paid, at present, for a share in a fund, which, after the lapse of 20 years, will yield an incorne of $100 a year; interest 6 per cent. ? 26. How much must be paid, at present, for the title to an annuity of $1000, to commence in 40 years; interest being 5 per cent. ? XXXV. ALLIGATION. ALLIGATION relates to finding the mean value of a mix. ture composed of several ingredients of different values, and is considered under two heads, viz. Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL. We rank under the head of Alligation medial, those questions, in which the several ingredients and their respective values are given, and the mean value of the compound is required. For example, a wine merchant bought several kinds of wine, as follows; 160 gallons at 40 cents per gallon; 75 gallons at 60 cents per gallon; 225 gallons at 48 cents per gallon; 40 gallons at 85 cents per gallon; and mixed them together. It is required to find the cost of a gallon of the mixture. Now, if we find the whole cost of the several kinds of wine, and divide it by the whole number of gallons, it is evident, that the quotient will be the cost of a single gallon of the mixture. 160 gallons, at 40 cents per gal., cost $ 64.00 75 gallons, at 60 cents per gal., cost $ 45.00 225 gallons, at 48 cents per gal., cost 109.00 40 gallons, at 85 cents per gal., cost $ 35.00 500 the whole number of gallons, cost $ 252.00 $ 252.00 – 500=.504, or 50 cents and 4 mills. Therefore, to find the mean value of a compound, composed of several ingredients, of different values, we give the following Find the value of each ingredient, add these values together, and divide their sum by the sum of the ingredients. The quotient is the mean value. RULE. 1. A farmer mixed together 5 bushels of rye worth 70 cents a bushel, and 10 bushels of corn worth 60 cents a bushel, and 5 bushels of wheat worth $1.10 a bushel. What is a bushel of the mixture worth? 2. A grocer mixed together 38lb. of tea at 50 cents a pound, 1541b. at 80 cents, 12;lb. at 60 cents, 8 lb. at 96 cents. 773lb. at 32 cents, and sold the mixture at a profit of 20 per cent. At what price per pound did he sell it ? 3. A goldsmith melts together 11 ounces of gold 23 carats fine, 8 ounces 21+ carats fine, 6 ounces of pure gold, and 2 ounces of alloy. How many carats fine is the mixture ? We remark, that a carat is a 24th part. Thus, 23 carats fine, means t of pure metal. Pure gold is it. Alloy is considered of no value. 4. On a certain day, the mercury in the thermometer was observed to stand 2 hours at 60 degrees, 3 hours 62°, 4 hours at 64°, 3 hours at 67°, 1 hour at 72°, and 1 hour at 75°. What was the mean femperature for that day? 5. A dealer bought 241 gallons of syrup at 34 cents a gallon, and 24į gallons at 38 cents a gallon, and mixed both quantities and 14 gallons of water together, and sold the mixture at a profit of 50 per rent. At what price per gallon did he sell it? 6. A goldsmith melts together 3 ounces of gold 18 carats fine, 2 ounceș 21 carats fine, and 1 ounce of pure gold. What is the fineness of the compound? ALLIGATION ALTERNATE. Under the head of Alligation Alternate are included those questions, in which the respective rates of the different ingredients are given, to compose a mixture of a fixed rate. It is the reverse of Alligation Medial, and may be proved by it. If we would find what quantities of two ingredients, different in value, would be required to make a compound of a fixed value, it is evident, that, when the value of the required compound exceeds that of one ingredient just as much as it falls short of the value of the other, we must take equal quantities of the ingredients to make the compound; because there is just as much lost on the one, as is gained on the other. If the value of the compound exceeds that of one ingredient twice as much as it falls short of the value of the other, we must take of the ingredients in the ratio of 1 to 1, or 1 to 2. For instance, if we would mix wines, at 4 dollars and 1 dollar a gallon, in such proportion that the mixture should be worth 2 dollars a gallon, we must take 1 gallon at 4 dollars to 2 gallons at 1 dollar; because there is just as much lost on i gallon at 4 dollars, as is gained on 2 gallons at 1 dollar. If we would mix wines, at 6 dollars and 2 dollars a gallon, in such proportion as would make the mixture worth 3 dollars a gallon, we should take of the two kinds |