3. Find the average price per pound of the following mixture: 40 pounds of tea at 60 cents per pound, 60 pounds at 80 cents per pound, and 60 pounds at $1.20 per pound? 4. A liquor dealer mixed 30 gallons of whiskey of 60% proof, 50 gallons of 70% proof, and 20 gallons of 90% proof. What was the % of proof of the mixture? 5. A goldsmith mixed 14 ounces of gold, 23 carats fine; 6 ounces 16 carats fine; 6 ounces, 18 carats fine; and 6 ounces, 19 carats fine. How many carats fine was the mixture? 6. A liquor dealer mixed 2 gallons of water with 9 gallons of whiskey costing $2 per gallon and 14 gallons of whiskey costing $3 per gallon, and sold the mixture at a profit of 25%. What was his selling price per gallon? To find the proportional quantity of each ingredient ́required in a mixture of a given average value. INDUCTIVE EXERCISE. If a gallon of one kind of wine worth $3 be mixed with a gallon of another kind worth $5, and the mixture be sold at $4 per gallon, what will be the gain on the cheaper wine? The loss on the higher priced wine? How do the gain on one kind and the loss on the other compare? If the mixture had been sold at $4.50 per gallon, prove by comparing the price of each ingredient with $4.50 that there would have been a gain, and how much? If the mixture had been sold at $3.50 per gallon, that there would have been a loss, and how much? [b] A gain in the price of one ingredient must be balanced by a corresponding loss in the price of another ingredient, to produce a mixture of a given average price. 7. A grocer wishes to mix 4 kinds of coffee worth respectively 17, 18, 24, and 25 cents per pound, in such proportions that the mixture may be worth 20 cents per pound. How many pounds of each kind must the mixture contain? 8. How many pounds of tea, worth respectively 60, 70, 85, and 90 cents per pound, must be taken to form a mixture worth 75 cents per pound? 9. To form a mixture worth 8 cents per pound, how many pounds of sugar worth 8, 9, and 10 cents per pound should be combined? 10. A dealer combined a quantity of alcohol 80% proof from 4 different barrels, respectively 60%, 70%, 75%, and 90% proof. How many gallons were drawn from each barrel ? 11. How many barrels of flour at $6, $63, $7, and $8 per barrel, must be sold to realize an average price of $6 per barrel? 12. In what proportion can vinegar worth 20, 25, and 40 cents per gallon be mixed to form a compound worth 30 cents per gallon? 13. What proportion of water, whose respective temperatures are 50°, 70°, 80°, and 100° Fahrenheit, must be combined to obtain water of 95° Fahrenheit? 14. A liquor merchant compounded a kind of whiskey worth $1.20 per gallon by mixing water with 3 kinds of whiskey worth respectively $1.10, $1.40, and $1.50 per gallon. What proportional amount of each did he take? To find the proportional quantity of each ingredient when the average price, the price of each ingredient, and the relative quantity of two or more of the ingredients are given. INDUCTIVE EXERCISE. 1. If by selling 3 pounds of one ingredient of a mixture produces a loss of 1 cent, and by selling 2 pounds of another ingredient of that mixture produces a gain of 1 cent, how many pounds of each ingredient must be sold to produce a counterbalancing gain and loss of 2 cents? Of 3 cents? Of 5 cents? 2. If 12 pounds of the ingredient sold at a loss be used in the mixture, how many pounds of the ingredient sold at a gain must the mixture contain to offset that loss? [c] To multiply or divide вOTH proportionals of a comparison by the same number preserves their counterbalancing gain and loss, and produces any desired relation of those proportionals to other quantities. 15. A man bought Rio coffee at 14 cents per pound, Laguayra at 16 cents, Maracaibo at 18 cents, and Java at 24 cents. His purchase averaged 20 cents per pound, and the Rio was to the Laguayra as 2 to 5. What number of pounds of each did he purchase? 16. A grocer combined teas worth 50, 60, 75, and 80 cents a pound in such proportions as to produce a mixture worth 70 cents a pound. The quantity of tea worth 75 cents was to that worth 60 cents as 4 to 1. What proportion of each kind of tea did he take? 17. A dairyman averaged 32 cents per pound upon 5 kinds of butter which he sold at 20, 25, 30, 35, and 40 cents per pound. He sold as much butter at 30 cents as he sold at 35 cents. What proportional quantity of each kind did he sell? 18. A jeweler combined three kinds of silver, respectively pure, pure, and pure, in such proportions that the mixture was pure. What proportion of each did he take, if the mixture contained 5 times as mcuh silver pure as of silver pure? 19. A liquor dealer mixed a quantity of whiskey 65% proof by combining whiskeys which were respectively 50%, 55%, 60%, 70%, and 75% proof. What proportion of each did he take, if the 50% proof in the mixture was to the 75% proof as 3 to 2? 20. A farmer sold turkeys at $1.25 each, ducks at 75 cents, chickens at 50 cents, and geese at 90 cents. He sold 3 times as many ducks as turkeys, and realized an average price of 80 cents. How many of each did he sell? To find the proportional quantity of each ingredient when the quantity of one or more of the ingredients 21. A dealer sold 10 barrels of flour at $6.25 per barrel. What is the least number of barrels which he must sell at $6.30, $6.75, and $7.00 per barrel to realize an average price of $6.50 per barrel? 22. A clothier bought 24 pairs of pantaloons at $12 per pair. How many must he buy at $5 and $7 per pair that the average price will be $8 per pair? 23. A merchant sold 15 yards of muslin at 18 cents per yard. What is the least number of yards at 14, 10, and 9 cents per yard which he can sell to realize an average price of 12 cents per yard? 24. A dealer so mixed 8 pounds of coffee at 18 cents per pound with other grades at 14, 17, 21, and 22 cents per pound, that the price of the mixture could be fixed at 19 cents per pound. What is the least number of pounds of each which he could take? 25. A broker purchased 90 shares of railroad stock at 70. How many additional shares at 90 must he purchase that the average price per share may be 75? 26. A farmer bought 20 calves at $15 each, and a sufficient number at $8 and $10 each to make the average price of the lot $12 each. How many calves at $8 and $10 did he buy? To find the proportional quantity of each ingredient, when the total quantity of the ingredients is limited. 27. A broker sold mining stock at 60, railroad stock at 65, steamboat stock at 80, and bank stock at 88. Altogether, he sold 243 shares, and received an average price of $75 per share. How many shares of each did he sell? 28. A miller mixed 240 bushels from 4 grades of wheat, costing $.90, $.95, $1.10, and $1.20 per bushel, so that the average cost of the mixture was $1 per bushel. How many bushels of each grade did he take? 29. What quantity of water must be added to 16 quarts of acid 90% • proof and 24 quarts 95% proof, that the mixture may be an acid 80% proof? 30 A produce merchant bought 130 barrels of 3 kinds of onions at an average price of $2 per barrel. If he had purchased each kind separately the prices would have been respectively $1.90, $2.25, and $2.50 per barrel. How many barrels of each kind did he buy? 31. A liquor dealer diluted a quantity of rum for which he paid $2.50 per gallon with a quantity of water, making 20 gallons in all. How many gallons of each did the mixture contain, if it was sold at $2 per gallon without loss to the dealer? 32. A husband, wife, and son received $27.50 for 22 days' labor; the husband receiving $2 per day, the wife $1.10, and the son $.75. How many days' labor did each perform? FIRE AND MARINE INSURANCE. [a] The sum insured is the base [multiplicand]. EXAMPLES. 1. A merchant insured his warehouse worth $10800 at What was paid for premium if the rate of insurance was ? of its value. Sum insured X rate of premium [usually called rate of insurance] = premium. the following rates: 2. The stock of a manufacturing company was insured for $75000 at of the stock at %, of the remainder at %, and What was the total premium paid? the remainder at 1%. 3. A stock of goods valued at $45000 was insured in the Eagle Fire Insurance Company for $10000, in the Manhattan F. I. Co. for $15000; and in the Globe F. I. Co. for $9000. If the stock was damaged by fire to the extent of $18000, what proportion of the loss should be paid by each company? 4. A store and contents were insured as follows: $10000 in the Home Insurance Co., $6000 in the Essex Insurance Co., and $8000 in the Howe Insurance Co. A partial loss of $6000 having been caused by fire, how much thereof should each company contribute? 5. A stock of goods was insured for one year for $12000 at %. Six months thereafter the policy was cancelled at the request of the insured. What was the return premium, the short rate for 6 months being %? What would have been the return premium if the policy had been cancelled at the request of the underwriters ? 6. A fire insurance company took a risk of $75000 at 1% premium, and reinsured of the risk in another company at §%, and of it in another at 1%. What did the company gain by reinsurance? 7. A ship worth $75000 was insured for $20000 in one insurance company at 3% premium, and for $35000 in another at 7% premium. She was afterwards damaged by a storm to the extent of $3000. What insurance did the owners of the ship receive from each company? 8. A vessel worth $30000 was insured for $5000 in each of 4 companies. What insurance would its owners receive from each company if the vessel sustain a partial loss of $8000? If the vessel be totally destroyed? 9. I paid $175 premium on a schooner worth $25000, which was insured at of its value. What was the rate % of insurance? What was the rate of insurance in cents on the $100? Premium + sum insured rate % of premium. (Premium >10000) + sum insured rate of premium in cents per $100. 10. An insurance company received $120 premium on property insured for $16000. If a block of buildings be insured for $40000 in another company at the same rate, what will be the premium on the latter? |