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What is the sum of the infinite decreasing series of which
2. 2 is the first term and 1 is the common ratio ?
(a.) It is sometimes desirable to mix articles of the same kind but of different values together, so as to obtain a compound differing in value from any of its ingredients. The problems connected with this subject are called problems in ALLIGATION.
(b.) They are usually considered as coming under one of two classes, viz.
(c.) ALLIGATION MEDIAL, in which the quantities of the several ingredients and their prices are given, and we are required to find the price of the mixture per pound, per gallon, or per bushel.
(d.) ALLIGATION ALTERNATE, in which the prices of the various ingredients are given, and we are required to find what quantities of each must be taken to make a mixture having a given value per pound, per bushel, or per gallon.
1. A trader mixed together 12 lb. of sugar worth 8 cents per lb., 10 lb. worth 9 cents per lb., and 17 lb. worth 12 cents per lb. What was the mixture worth per lb.?
SOLUTION.- 12 lb. at 8 cents are worth 96 cents. 10 lb. at 9
90 17 lb. at 12
Hence, the 39 lb. are worth
$3.90, and 1 lb. is worth 10 cents.
2. A liquor dealer mixed together 100 gallons of liquor worth 35 cents per gallon, 100 gallons worth 42 cents per gallon, 100 gallons worth 40 cents per gallon, and 100 gallons of water. What was the mixture worth per gallon ?
3. A man bought 500 gallons of vinegar at 12 cents per gallon, and 400 gallons at 124 cents per gallon, and, after mixing the whole with 100 gallons of water, sold it at 13 cents per gallon, What did he gain per gallon on the mixture, and how much did he gain in all ?
Note. — The above questions are in ALLIGATION MEDIAL. Those which follow are in ALLIGATION ALTERNATE.
(e.) Problems in Alligation Alternate are solved by determining what proportion of the various kinds must be taken in order to make the gains on those kinds which are worth less than the mean price of the mixture, equal the losses on those which are worth
The things to be done, then, are 1st. To determine the gain or loss on a unit of each kind.
2d. To select such number of units as may be convenient of kinds which cost either less or more than the mean price, and determine the gain or loss on them.
3d. To determine what quantities of the remaining kinds shall be taken to counterbalance this gain or loss?
4. A merchant has sugars at 6, 8, 11, and 13 cents per pound, and he wishes to make a mixture which shall be worth just 10 cents per pound. How many pounds of each kind may he take ?
SOLUTION.-By selling the mixture at 10 cents per pound, he gains 4 cents per pound on the first kind and 2 cents per pound on the second; while he loses 1 cent per pound on the third kind and 3 cents per pound on the fourth. Suppose that he takes 12 lb. of the first and 9 lb. of the second; then he will gain 12 times 4 conts, or 48 cents, on the first, and 9 times 2 cents, or 18 cents, on the second, and his total gains will be 48 + 18, or 66 cents. Ho must now take enough of the others to lose 66 cents. Suppose that be takes 15 lb. of that at 11 cents; he will lose 15 cents on it, and he must take enough of that at 13 cents to lose 66 — 15, or 51 cents.
But as be loses 3 cents on one pound, he must take as many pounds as there are times 3 in 51, which are 17 times. Hence, he must take 17 lb. of the last kind.
The following is a convenient form of writing the work -
Sum of Gains
66 cts. gain.
66 cts. loss.
3........... 17 lb. lose 51 cts.
PROOF. - See if 12 lb. at 6 cents per lb., 9 lb. at 8 cents per lb., 15 lb. at 11 cents per lb., and 17 lb. at 13 cents per lb., will make a mixture worth 10 cents per pound.
Note. - There is no limit to the number of answers which may be obtained to such problems as the above, for however many or few pounds of one kind we may take, we can take enough of other kinds to counterbalance the gain or loss.
Below is a part of two other solutions to the last problem. Let the pupil supply the missing number, and prove the correctness of his result. Quantities
$2.30 gain. $.24
21 lb. gain $ .42 17 lb. lose $.17
93 lb. lose $ .93
$ loss. Ib. lose $
Sum of Gains
Sum of Gains and Losses.
11 lb. gain 8:44} $.68 gain.
17 lb. lose 8.51} $.68 loss.
5. I have grains worth respectively 42, 46, 52, and 60 cents per bushel. How many bushels of each kind may I take to make a mixture worth 50 cents per bushel ?
6. I have coffees worth respectively 10, 13, and 16 cents per pound. How many pounds of each kind may be taken to make a mixture worth 12 cents per pound?
7. A trader wishes to mix 100 gallons of molasses worth 35 cents per gallon, with enough water to make the mixture worth 30 cents per gallon? How many gallons of water must be take ?
8. How many gallons of oil worth 55 cents per quart must be mixed with 50 gallons worth 46 cents per quart, to make a mixture worth 50 cents per quart?
9. flow many pounds of tea worth 40 cents per pound must be mixed with 12 lb. worth 30 cents per lb., 10 lb. worth 48 cents per Ib., and 20 lb. worth 55 cents per lb., to make a mixture worth 45 cents per lb.?
10. I have raisins at 8, 9, 12, and 13 cents per lb. How many pounds of each kind must I take to make a mixture of 100 pounds worth 10 cents per pound?
Suggestion.—First get a set of answers, as though the number of pounds in the mixture was not specified. Then find how many times as much must be taken to give the required quantity.
11. I have spices at 20, 25, 33, 35, and 37 cents per lb. How many pounds of each may be taken to produce a mixture of 250 Ib. worth 30 cents per lb. ?
12. I have oats worth 40 cents per bushel, corn worth 60 cents per bushel, and barley worth 90 cents per bushel, how many bushels of each may I take to make a mixture of 100 bushels worth 65 cents per bushel ?
150. MISCELLANEOUS PROBLEMS.
1. A sum of money was divided between A and B, in such a way that A had $846, which was $39 more than of what B had. How many dollars had B?
2. How many feet of boards 1 inch thick will it take to make a box, the outside dimensions of which are 8 ft. 7' long, 6 ft. 5 wide, and 2 ft. high ?
3. Multiply .125 by 22.5, and extract the square root of the result.
4. If the minuend is 4788, and the subtrahend is twice the remainder, what is the remainder ?
5. If the minuend is 963, and the remainder is 127 more than the subtrahend, what is the remainder ?
6. The sum of the minuend and the subtrahend is 1785, which is 849 more than the remainder. What is the subtrahend and what the remainder ?
7. If the subtrahend is 100 more than twice the remainder, and their sum is 400, what is the subtrahend and what the remainder?
8. If the excess of the subtrahend over the remainder is of the subtrahend, and the remainder is 96, what is the subtrahend and what the minuend ?
9. I bought apples at the rate of 3 for 2 cents, and sold them at the rate of 2 for 3 cents. What per cent did I gain ?
10. What is the value of 1 + +++1++ + 1 + 18?
11. If my house-lot were 15 feet longer than it now is, it would contain 1200 more square feet than now; while, if it were 15 feet wider, it would contain 1275 more square feet than now. How many square feet does it contain ?
12. I bought goods to the amount of $1000, giving in payment my note at 6 mouths. When 2 months had passed, I sold the goods for $1000 cash, and “shaved” the note I had given for them at 2 per cent per month discount. How many dollars did I gain by the transaction ?
13. If the least common multiple of two numbers is their product, what is their greatest common divisor ?
14. A farmer bought 6ş yards of cloth, giving in part payment 123 dozen of eggs at 16 cents per dozen. Now, allowing that the eggs were sufficient to pay for 1} yards of cloth, what was the value of the cloth which he bought?
15. I bought 2 house-lots, paying for the second as much as for the first. I paid $248 more for the first than for the second. What was the cost of each ?
16. What number is that which, increased by } of itself, gives for a result 150 less than 3 times itself ?