21. A bankrupt owes $150000, to pay which he has the following property : real estate worth $17500, merchan. dise worth $3750, personal property worth $20000, and sums due from various individuals amounting to $100000. If he collects all his dues, how much can he pay on every dollar of his debts ? 22. A., B. and C. purchase a house for $10000, A. contributing $4000, B. $3600, and C. $2400. The house rents for $800, and the taxes and repairs amount to $50 a year. What income does each of the owners derive from the estate, and what per cent. of his investment? 23. Divide 7500 into 5 parts, in the proportions of l, 3, 4, 5, and a 24. In the distribution of a bankrupt's property, $763. 50 was divided among four of his creditors. Ai's bill was $250, B's. $300, C.'s $325, and D.'s $362.50. How much did each receive ? 25. Four men were joint owners of a farm, their shares being in the proportion of 6, 2, 5, and 3, respectively. It is required to divide the annual rent, which is $1500, equitably among them. CHAPTER XIII. ALLIGATION. When the values of a variety of ingredients are given, and it is desired to make a mixture of any fixed value, the quantity of each ingredient to be employed is determined by ALLIGATION. When there is no limitation in the quantity of either ingredient, we may make the desired mixture by the fol. lowing RULE. Having written the values of the ingredients in a perpendicular column, connect by a line each value that is less than the required average with one or more that is greater, and each value that is greater with one or more that is less. Write the difference between each value and the average, op are as the numbers 2, 3 and 4, respectively. What was euch man's stock and gain ? 13. A privateer captured a prize valued at $15000, which the crew agree to divide in proportion to their pay, and the time each has been on board ; the officers and midshipmen have been on board 4 months, and the sailors 3 months; the pay of the officers is $20 a month, of the midshipmen $15, and of the sailors $12, and there are 3 officers, 6 midshipmen, and 40 sailors. How much does each receive ? 14. Three men bire a pasture for $40. A. put in 4 horses for 3 months and 6 for 2 months, B. put in 5 for 4 months and 10 for 1 month, and C. put in 7 for 4 months and 3 for 2 months. How much ought each to pay ? 15. A man failing in trade owed $75000, to meet which he had property valued at $14500. How much can he pay A., who is a creditor for $10000, B., who is a creditor for $3750, and C., who is a creditor for $12362.50 ? 16. A. commenced business with a capital of $2500. Five months afterwards he received B. as a partner, who added $4000 to the stock. Six months after B. was admitted, C. joined the firm with a capital of $6000. Two years from the commencement, they had gained $8000. What was each man's share of the gain ? 17. It is required to divide $36000 among 4 persons, in such manner that the second may have twice as much as the first, the third as much as the first and second, and the fourth three times as much as the third ? 18. A legacy of $40000 was left to four heirs, in the proportion of k to the first, z to the second, to the third, and j to the fourth. What was the share of each? 19. A levy of 1200 horses is to be distributed among three regiments in the proportion of 1, and 7, respectively. How many will each regiment receive ? 20. Five men hire a pasture for $294. A. put in 10 cows for 3 months; B. put in 5 cows for 2 months ; C. put in 8 cows for 11 months ; D. put in 16 cows for 24 months; and E. put in 20 cows for 4 months. How much ought each to pay ? 21. A bankrupt owes $150000, to pay which he has the following property : real estate worth $17500, merchandise worth $3750, personal property worth $20000, and sums due from various individuals amounting to $100000. If he collects all his dues, how much can he pay on every dollar of his debts ? 22. A., B. and C. purchase a house for $10000, A. contributing $4000, B. $3600, and C. $2400. The house rents for $800, and the taxes and repairs amount to $50 a year. What income does each of the owners derive from the estate, and what per cent. of his investment? 23. Divide 7500 into 5 parts, in the proportions of 1, 3, 4, , and 24. In the distribution of a bankrupt's property, $763. 50 was divided among four of his creditors. A.'s bill was $250, B's. $300, C.'s $325, and D.'s $362.50. How much did each receive ? 25. Four men were joint owners of a farm, their shares being in the proportion of 6, 2, 5, and 3, respectively. It is required to divide the annual rent, which is $1500, equitably among them. 7 CHAPTER XIII. ALLIGATION. When the values of a variety of ingredients are given, and it is desired to make a mixture of any fixed value, the quantity of each ingredient to be employed is determined by ALLIGATION. When there is no limitation in the quantity of either ingredient, we may make the desired mixture by the fol. lowing RULE. Having written the values of the ingredients in a perpendicular column, connect by a line each value that is less than the required average with one or more that is greater, and each value that is greater with one or more that is less. Write the difference between each value and the average, op posite the ingredient with which that value is connected, and the difference (or the sum of the differences, if there be more than one) opposite each ingredient will be the quantity of that ingredient required. EXAMPLE FOR THE BOARD. 12 1st Ans. 41b. at 5, 1 lb. at 7, 41b. 2d Ans. 3lb. at 5, 3lb. at 7, lib. 30 Ans. Ilb. at 5, 316. at 7, 3lb. at 8, at 8, 71b, at 10, 51b. at 12. at 8, lib. at 10, 61b. at 12. How much sugar, at 5cts., 7cts., 8cts., 10cts., and 12cts., must be mixed together, that the mixture may be worth Octs. a pound ? 5 1+3 5- 3 5 1 7 3 9 8- 1+3 9 8 1 9 8 3 10. 4 12 4+1 4+2 12 12+1 1 4lb. at 10, 31b, at 12. We may obtain as many answers as there are different ways of connecting the numbers above, with those below the average. To prove the rule correct, let us examine the second of the above answers. If we were mixing sugars at 5 and 12cts. to sell the mixture at 9cts., we should gain 4cts. on every pound of the former, and lose 3cts. on every pound of the latter. Then, on 3lb. of the former we should gain 12cts. and on 4lb. of the latter we should lose 12cts. ; therefore, if we mix these quantities, we shall neither gain nor lose by selling the mixture at 9cts. In the same way it may be shown that 31. at 7cts. and 2/b. at 12cts., 1lb. at 8cts. and ilb. at 10cts. may be sold at the average of 9cts., and the same reasoning will prove the truth of each of the other answers. 1. How much tea at 50cts. 75cts. 90cts, and $1.00, must be taken to form a mixture worth 80cts. 2. A jeweller has gold of 16, 17, 18, 20, 22, and 24 carats fine. What proportion of each must he take to make a mixture 21 carats fine? Pure gold is 24 carats fine, or 31. 3. How much grain, at 50cts. 75cts. $1.00, and $1.10 per bushel, will make a mixture worth 90cts. a bushel ? 4. How much water must be mixed with wine at $1.50 and $2.00 a gallon, to make the whole worth $1.00 per gallon ? 5. What quantity of raisins, at 10cts. 18cts. and 20cts. per lb. must be mixed together, to fill a cask containing 1501b, and to be worth 19cts. a lb.? [After obtaining the proportions by Alligation, find the exact quantities by Fellowship.] 6. It is required to mix sugar at 7cts. 8cts. 10cts. and 12cts. per lb. in such manner as to form a mixture of 2cwt. 3qr., worth 11cts. per lb. 7. Mix tobacco at 8cts. 10cts. 12cts. and 16cts, so as to make 100 pounds worth 11cts. a pound. EXAMPLE FOR THE BOARD. { A farmer wishes to mix 10 bushels of barley at 50cts., 4 bushels of oats at 45cts., and 16 bushels of rye at 75cts. with wheat at $1.25 and corn at 90cts. a bushel, so that the mixture may be worth $1.00 per bushel. We may regard the limited quantities as a single ingredient of 30 bushels, worth 62cts. a bushel. Proceeding in the usual way, 62 25 we find that 25 bushels at 62cts., 25 at 1.00 90 25 90cts., and 48 at $1.25, would give us 1.25 38+10 a mixture of the desired average value. But as we have 30 bushels at 62cts., we must takes or of these proportionate quantities, and we have 30 bushels at 90cts. and 573bu. at $1.25, for the Answer. 8. How much water must be mixed with 40 gallons of syrup, at 50cts. a gallon, to make the whole worth 37įcts. a gallon ? 9. A grocer has 10 gallons of wine at 75cts., 12gal. at 90cts., and @gal. at $1.00, with which he would ix brandy at $1.25, and water, so as to make a mixture worth 95cts. a gallon. How much of each must he take? 10. If a cubic inch of gold weighs 11.11oz. and a cubic inch of silver 6.040%., what quantity is contained in a mass of 630%., of which 1 cubic inch weighs 8.35oz. ? 11. How much molasses, at 50 cents, and water, must be mixed with 15 gallons at 371 cents, 28 gallons at 25 cents, and 19 gallons at 33 cents, to make a mixture worth 31 cents a gallon ? 12. A grocer has an order for 1501b, of tea, at 90 cents per lb., but having none at that price, he would mix some at 75 cents, some at 87cents, and some at $1.00 per pound. How much of each sort must he take ? |