3. If 4 bu. of wheat, worth $11 be mixed with 5 bu. worth $23; what is the mixture worth pr. bu. ?

4. If 6 boxes of raisins, worth $4 pr. box, be mixed with 4 boxes, worth $1 a box; what will the mixture be worth pr. box. ?

Let the following be written.

5. A grocer mixed 81 lbs. of sugar at 19 cts. pr. lb. with 54 lbs. at 14 cts. What pr. lb. was the mixture worth?

81 lbs. at 19 cts., are worth

54 lbs. at 14 cts., are worth

$15.39

7.56

The whole 135 lbs. are worth

2295

$22.95

It needs no reasoning to show that if I divide $22.95, the worth of 135 lbs. by 135, I shall obtain the price of 1 lb. Hence, 17 cts. .Ans. This process is called ALLIGATION MEDIAL.

ALLIGATION is a general name given to the mixing of simple things of different qualities, so as to form a compound of a medium, or mean quality.

It will be observed that ALLIGATION MEDIAL embraces those instances in which the quantities and prices of the simples are given, and the price of the compound required. Hence, its rule is,

DIVIDE THE WHOLE COST OF THE SIMPLES, BY THE WHOLE QUANTITY. 6. If 20 bu. of wheat worth $1.35 pr. bu., be mixed with 10 bu. of wheat worth 90 cts. pr. bu., what will the mixture be worth pr. bu. ?

8

A. $1.20. cts. pr. lb. ; at What was 51 A. $55.965.

7. A grocer mixed equal quantities of sugar at 9 cts. pr. lb; at 10 cts. pr. lb.; and at 12 cts. pr. ib. cwt. worth? 8. If 4 lbs. of gold, 23 carats fine, be mingled with 2 lbs. 17 carats fine, what will be the fineness of the mixture? A. 21 carats.

9. If I mingle 3 kegs of raisins worth 11 cts. pr. lb., and each containing 230 lbs., with 5 kegs worth 13 cts. pr. lb., and each containing 175 lbs., what is the value pr. lb. of the mixture?

10. If 3 barrels of brandy, containing each 29 gals. be mixed, what will be the price pr. gal. of the mixture, supposing the first worth 95 cts. pr. gal., the second $1.10, and the third $1.25 ?

§ XCVI. 1. A merchant has teas at $1 pr. lb., and at $3 pr. lb. He wishes to make such a mixture of them, that the price may be $2 pr. lb. What quantity must he take of each ?

Since $3 exceeds $2 just as much as $1 falls short of $2, the following will be evident.

A pound of the first simple is worth $1, which is a dollar, too small for the man's purpose; and a pound of the second is worth $3, which is a dollar too great for his purpose. But if he puts these two ether, it is plain that the $1 deficiency of the one, will be balanced by the $1 excess of the other, so that the price will be exactly what he wants, that is, $2. The answer, then, is 1 lb. of each, or, generally, any equal quantities of each.

2. A merchant has sugar at 8 cts. and at 11 cts. a pound. He wishes to make a mixture, worth 10 cts. a pound. Required the necessary proportional quantity of each kind.

Here 8 cents falls short of 10 cts. just twice as much as 11 cts. exceeds 10 cts. Hence, it will take two lbs. at 11 cts. to exceed the same number of lbs. at 10 cts. as much as one lb. at 8 cts. falls short of one at 10 cts. Hence, if 2 lbs. at 11 cts. be mixed with 1 lb. at 8 cts. the excess of the one will be exactly balanced by the deficiency of the other, and the price will be 10 cts. as required.

Ans. 2 lb. at 11 cts. and 1 lb. at 8 cts., or, any other quantities having the same ratio.

3. A man has cider at $2 and at $7 a barrel. He mixes in such a manner that the mixture is worth $5 a barrel. What quantities does he take of each ?

Here $7 exceeds $5 by $2, and $2 falls short of $5 by $3. If he were to take three barrels, then, at the greater price, (viz. $7) he would have 3X$2-$6 too much. But if he were to take two barrels, at the less price, (viz. $2) he would have 2X$3-$6 too little. If, then, he were to put these together, the $6 excess would be exactly balanced by the $6 deficiency, and the medium price would be $5, as required.

Ans. 2 bls. at $2, and 3 bls. at $7, or any other quantities having the same ratio. The correctness of all these conclusions may be verified, or proved, by reversing the question, and proceeding as in the last §. Thus, in the last instance, the inquiry reversed would be, if 2 bls. at $2 be mixed with 3 bls. at $7, what will be the price of the mixture pr. bl. ?

By last $

2X2+3X7
2+3

=25=$5, the given price above.

From the preceding examples, we see, that

THE QUANTITY OF A SIMPLE, TO BE TAKEN, VARIES WITH THE DIFFERENCE BETWEEN ITS PRICE AND THE MIXTURE PRICE; INCREASING AS THIS It is there. DIFFERENCE DIMINISHES, AND DIMINISHING AS IT INCREASES. fore inversely as this difference.

It will be observed of all the questions, in this section, that they require the proportional quantities, which are to be mixed when the prices of the different simples, and the mixture price are given. This is called ALLIGATION ALTERNATE. When but two Simples are given, the process is easy, as shown above. When there are more than two, the case requires further illustration.

4. A merchant mixed grain at the following prices, viz. 70 cts. 95 cts. and $1.00 pr. bu. He made the mixture worth 85 cts. What quantities of each did he take?

First, suppose he had but two kinds to be mixed, at the prices 70 cts. and 95 cts. The differences between these prices and and 85 cts., the mixture price, are 15 and 10. And, as the quantities required are inversely as these differences, we have 10 bu. at 70 cts., and 15 bu. at 95 cts. ; or any other quantities, having the same ratio.

Next, suppose he had only the kinds at 70 cts., and $1. The differences are 15 and 15; and, of course, the quantities required are 15 bu. at 70, and 15 bu. at $1.00, or any other quantities having the same ratio.

In both of the cases above, the compound is worth 85 cts. pr. bu. If then the two compounds be themselves mixed, the price will not be altered, since the quantities mixed, are of the same value per bu.

Hence, he might have mixed 10+15=25 bu. at 70 cts., 15 bu. at 95 cts., and 15 bu. at $1,00, or any other quantities, having the same ratio, and still had his mixture price 85 cts.

The common mode of performing examples of this kind is as follows,
WRITE THE PRICES OF THE SIMPLES,
ONE BELOW ANOTHER, PROCEEDING FROM
THE SMALLEST TO THE GREATEST, AND
WRITING THE MIXTURE PRICE ON ONE SIDE.

70

10+15

85

95

15

Ans.

100

15

THEN CONNECT BY A LINE

EACH PRICE GREATER THAN THE MIXTURE PRICE WITH AT LEAST ONE WHICH IS LESS; AND, ON THE OTHER HAND, EACH ONE THAT IS LESS THAN THE MIXTURE PRICE, WITH ONE, AT LEAST, THAT IS GREATER. TAKE THE DIFFERENCE BETWEEN EACH SIMPLE PRICE AND THE MIXTURE PRICE, AND PLACE IT OPPOSITE EVERY NUMBER, WITH WHICH THAT SIMPLE PRICE IS LINKED. THE NUMBER, OR SUM OF THE NUMBERS STANDING OPPOSITE EACH SIMPLE PRICE, WILL BE THE RELATIVE QUANTITY AT THAT PRICE.

Thus, in the above example, 70 being less than 85; and 95 and 100 being both greater, 70 is linked with both the others. Then 15-85-70 is placed opposite both 95 and 100, and 95-85-10, and 100-85-15 are both placed opposite 70. Then 10+15-25 is the quantity at 70, 15 is the quantity at 75, and 15 the tity at 100.

quan

It will be perceived that we may have, in many cases, a variety of answers all corresponding with the requisition of the question. This results from the various modes in which we may link the simples. Where the simples are numer. ous, and about as many above the mixture price, as below it, the variety is very great. Thus,

5. A grocer wished to mix four qualities of sugar, worth 12 cts., 11 cts., 9 cts. and and 8 cts pr. lb. respectively; so that the mixture should be worth 10 cts. pr. lb. What quantities could be have taken of each ?

1ST. ANS.

2+1=3

Here we have three answers; and as the pair of quantities resulting from each simple linking consists of relative numbers, for which any others bearing the same ratio to each other may be substituted, it follows that the variety of answers is indefinite. Several other answers might be found in this example, directly, by linking.

6. A merchant would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound may be 22 carats fine. What quantity of each must he take? Ans. 2 parts of each of the first three kinds, and 9 of the last.

7. It is required to mix liquors at 75, 88 and 90 cts. pr. qt. with water, so that the mixture shall be worth 60 cts. pr. qt. What quantities must be taken? Ans. 73 qts of water, and 60 of each other liquor.

8. I wish to mix wheat at $2.25 per bu., with some that is $3.00 pr. bu. and some that is $1.75 per bu., so that the compound shall be worth $2.50 per bu. Required the quantities of each.

9. A grocer wishes to mix raisins at 25 cts. pr. lb., with others at 18, 23, 27, 30, and 40 cts. respectively, so that the compound may be worth 29 cts. Required the quantities of each.

We have seen that the quantities obtained by this process are merely relative and that any others might be substituted in their places, which have to each oth er the same ratio. Hence, if it is desired that a particular quantity of one of the ingredients, either greater or less than that found by linking, should be employed, it is only necessary to vary the others, so that they may still bear to it the same ratio. This is done by Simple Proportion. The teacher may also exhibit. the method by analysis.

§ XCVII. 1. I wish to mix 2 gallons of brandy, at $1.50 pr. gal. with rum at 80 cts. pr. gal., so that the mixture may be worth $1.00 pr. gal. What quantity of rum must I take?

100

80.

{1.50

_50

Here I obtain 20 gals. of brandy by linking. This, by the terms of the question, 20 must be diminished down to 2, and there. fore, it is necessary that the 50 gals. of rum should be diminished in the same ratio.

(§ xci.) 20: 2::50:50×2÷20-5 gals. rum Ans.

The proportion then seems to be,

AS THE RELATIVE QUANTITY OF THE LIMITED INGREDIENT, IS TO THE GIVEN QUANTITY; SO IS THE RELATIVE QUANTITY OF EACH OTHER INGREDIENT TO THE ABSOLUTE QUANTITY OF THAT INGREDIENT REQUIRED. And, hence, the rule,

MULTIPLY EACH RELATIVE QUANTITY BY THE GIVEN QUANTITY OF THE LIMITED INGREDIENT; AND DIVIDE THE PRODUCT BY THE RELATIVE QUANTITY OF THE SAME INGREDIENT. THE SEVERAL QUOTIENTS WILL

EXPRESS THE QUANTITIES REQUIred.

2. A merchant has spices at 32 cts. 40 cts. and 64 cts. pr. lb. He wishes to mix 5 lbs. of the first with the others, so that the com. pound may be worth 48 cts. How much of each must he use?

Ans. 5 lb. of the 2d, and 7 lb. 8 oz. of the third.

3. A farmer wishes to mix 16 bu. of rye worth 50 cts. pr. bu. with corn at 40 cts. and oats at 30 cts. pr. bu. so that the mixture may be worth 37 cts. pr. bu. What quantities must he take of each?

When the whole compound is limited; that is, when it is desired that the whole mixture should amount to a certain quantity, the process is somewhat similar. For, since each individual quantity found by linking, is a relative quantity, it is evident, that the sum of the whole can only be a relative sum. Hence, if this sum be increased or diminished, the quantity of each ingredient must be increased or diminished in the same ratio. This is done by Simple Proportion, or by analysis.

4. A flour merchant, having flour at $4, $6, and $8, pr. bar. sold 120 bar. at the average price of $6.50 pr. bar. How many barrels of each kind must he have sold, in order not to have gained nor lost?

NOTE. This case is evidently the same as if the flour had been mixed, as far as calculation is concerned.

Here, by linking, I obtain 150 & 150 & 300, the several quantities of the Simples. The sum is 600. Now, by the terms of the question, this must be diminished down to 120, and of course, the quantity of each Simple, must be diminished in the same ratio.

( xXCI.) 600 120: 150: 120X150÷600-30 at $4, and also at $6, since its relative quantity is likewise 150.

Also

600 120 300: 120X300-600=60 at $8.

The proportion, then, is,

AS THE SUM OF THE RELATIVE QUANTITIES, IS TO THE GIVEN QUANTITY; SO IS EACH RELATIVE QUANTITY, TO THE ABSOLUTE QUANTITY OF THAT INGREDIENT REQUIRED. And the rule,

MULTIPLY EACH RELATIVE QUANTITY BY THE GIVEN QUANTITY, AND DIVIDE THE PRODUCT BY THE SUM OF THE RELATIVE QUANTITIES. THE SEVERAL QUOTIENTS WILL EXPRESS THE QUANTITIES REQUIRED.

It may be mentioned, though perhaps the pupil has always made the discovery himself, that this process is precisely the same with that employed in Simple Fellowship.

5. How much gold, that is 15, 17, 18, and 22 carats fine, must be mixed together to form a compound of 40 lb. 20 carats fine? Ans. 5 lb. of 15, 17, and 18, and 25 lb. of 22.

6. A man would mix 100 lbs. of sugar, at 8 cts. 10 cts. and at 14 cts. pr. lb. so that the compound may be worth 12 cts. pr. lb. What quantity of each must he use ?

Ans. 20 lb. at 8, and at 10 cts., and 60 lb. at 14 cts.

7. A grocer has currants at 5 cts. 7 cts. 10 cts. and 12 cts., and he wishes to mix them so as to sell them at 8 cts. pr. lb. How many lb. of each sort must he take?

This and some of the following admit of a great variety of answers.

8. A farmer mixed meal of the value of 25 cts. pr. bu. with other kinds of the values of 30 cts., 35 cts. and 18 cts. The com. pound contained 4 bu. and was worth 28 cts. pr. bu. What quantity did he take of each?

9. If a grocer make a mixture of teas of the following prices, pr. lb. viz. $1.25, $1.40, $1.63, and $1.75, so that the mixture may be sold at $1.50; how much does he take of each kind, supposing the whole mixture to contain 120 lb. ?

10. If a grocer fill 3 wine hogsheads with water and liquors at $1.20, $1.30, and $1.40 a gallon; how much does he take of each, supposing the compound worth $1.35 pr. gal. ?

11. In a brewery there are in an upper room 3 vats, each capable of containing 120 gals., and in a lower room 1 vat capable of hold. ing 192 gals. The lower vat is empty and the three upper ones are filled with beer, worth respectively 12 cts. 20 cts. and 30 cts. pr. gal. Pipes are set running at the same moment, from each upper vat to the lower, and in 1 hour, exactly, it is filled. The owner then finds, on mixing what remains in the upper vats, that the compound is worth 25 cts pr. gallon. What is the value of the mixture in the lower vat, and in what time would each pipe separately have filled it ?

A. Price, $0.1681.-Times, for first two, 1 h. 8 m.-for last, 16 h. 12. Suppose there are vats situated as the above, the upper ones being of the same dimensions, and containing beer of the values 14 cts. 18 cts. and 32 cts. pr. gal. Suppose the lower one filled as before, in an hour, and that there remains of the first kind of beer, 40 gals., the price of the mixture in the room above, being 25 cts., as before. Required the dimensions of the lower vat, the value pr. gal. of the mixture contained in it, and the time in which each pipe will fill it.