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ALLIGATION. I LXXXII. Alligation is the method of mixing several amples of different qualities, so that the compound, or com position, may be of a mean or middle quality.

When the quantities and prices of the several things or simples are given, to find the mean price or mixture compounded of them, the process is called

ALLIGATION MEDIAL. 1. A farmer mixed together 2 bushels of rye, worth 50 cents a bushel, 4 bushels of corn, worth 60 cents a bushel, and 4 bushels of oats, worth 30 cents a bushel: what is a bushel of this mixture worth?

In this example, it is plain, that, if the cost of the whole be divided by the whole number of bushels, the quotient will be the price of one bushel of tbo mixture.

2 bushels at $,50 cost $1,00
4

$,60 $2,40
4
$,30 . $1,20

$4,60 = 10 = 46 cts., Ans.
10

$4,60 RULE. Divide the whole cost by the whole number of bushels, &c.; the quotient will be the mean price or cost of the mixture.

2. A grocer mixed 10 cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwl., and 8 cwt, at $72 per cwt. : what is 1 cwt. of this mixture worth? and what is 5 cwt worth: A. I cwt, is worth $8, and 5 cwt. is worth $40.

3. A composition was made of 5 lbs. of tea, at $1$ per lb., 9 lbs. at $1,80 por Pb., and 17 lbs. at $14 per lb. : what is a pound of it worth?

A. $1,546% + 4. If 20 bushels of wheat, at $1,35 per bushel, be mixed with 15 bushels of rye, at 85 cents per bushel, what will a bushel of this mixture be worth?

A. $1,135% + 5. If 4 lbs. of gold, of 23 carats fine, be melted with 2 lbs. 17 carats fine, what will be the fineness of this mixture ? A. 21 carats.

ALLIGATION ALTERNATE. { LXXXIII. The process of finding the proportional quantity of each simple, from having the mean price or rate, and the mean prices or rates of the several simples given is called. Alligation Alternate; consequently, it is the reverse of Alligation Medial, and may be proved by it.

1. A farmer has oats, worth 25 cents a bushel, which he wishes to mix with com, worth 50 cents per bushel, so that the mixture may be worth 30 cents per bushel; what proportions or quantities of each must he take?

In this example, it is plain, that, if the price of the corn had boen 35 cents, that is, had it exceeded the price of the mixturc, (30 cents,) just as much as it falls short, he must have taken equal quantities of each sort ; but, since the difference between the price of the corn and the mixture price is 4 times as much as the difference between the price of the oats and the mixtury price, consequently, 4 times as much oats as corn must be taken, that is, 4 to 1, or 4 bushels of oats to 1 of corn. But since we determine this proportion by the differences, hence these differences will represent the same proportion.

These are 20 and 5, that is, 20 bushels of oats to 5 of corn, which are the quantities or proportions required. In determining these differences, it will be found convenient to write them down in the following manner: OPERATION.

It will be recollected, that the difference be

tween 50 and 30 is 20, that is, 20 bushels of oats, Cts. Bushels.

which must, of course, stand at the right of the 30 $,25

Ans.

25, the price of the oats, or, in other words, opposite the price that is connected or linked with

the 50; likewise the difference betwcen 25 and 80 =

- 5, that is, 5 bushels of corn, opposite the 50, (the price of the corn.) The answer, then, is 20 bushcís of pats to 5 bushels of corn, or in that proportion.

By this mode of operation, it will be perceived that there is precisely as much gained by one quantity as there is lost by another, and, therefore, the gain or loss on the whole is equal.

The same will be true of any two ingredients mixed together in the same way. In like manner the proportional quantities of any number of simples may be determined ; for, if a less be linked with a greater than the inean price, there will be an equal balance of loss and guin between every two, consequently as equal balance on the whole.

It is obvious, that this principle of operation will allow a great variety of mswers; for, having found one answer, we may find as many inore as we please, by only inultiplying or dividing each of the quantities found by 2, or 3 or 1, &c.; for, if2 quantities of 2 simples make a balance of loss and gain, as it respects the mean price, so will also the double or treble, the ḥ, or part, of any other ratio of these quantities, and so on to any extent whatever.

Proor. We will now ascertain the correctness of the foregoing operation by the last rule, thus :

20 bushels of oats, at 25 cents per bushel, = $5,00
5...

- $2,50
25

25 ) 7,50 (30

75 Ans. 30 cents, the price of the mixture.

corn, at 50

Hence we derive the following

RULE.
Reduce the several prices to the same denomination.

Connect, by a line, each price that is less than the mean rate, with one or mcre that is greater, and each price greater than the mean rate with one or more that is less.

Place the difference between the mean rate and that of each of the simples opposite the price with which they are connected.

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Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be seva eral, their sum will express the quantity.

2. A merchant has several sorts of tea, somo at 10 s, some at 11 s., some at 18 s., and some at 24 8. per lb.; what proportions of sach must be taken to naké a composition worth 12 s. per lb. ?

OPERATIONA
Ibs.

Ibs.
110

-2+1=3 11 12

=1 Ans.

Ans. 13

13

-1+2=3 14

14

-2 =2) 3. How much wine, at 5 g. per gallon, and 3 8. per gallon, must be mixed together, that the compound may be worth 4 s. per gallon?

A. An equal quantity of each sort. 4. How much corn, at 42 cents, 60 cents, 67 centid, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushel? A. 14 bushels at 42 cents, 3 bushels at 60 cents, 4 busheis at 07 cents, and 2 bushels at 78 cents.

5. A grocer would mix different quantities of sugar; viz. one at 20, one at 23, and one at 26 cents per lb. ; what quantity o' each sort must be taken to make a mixture worth D cents per lb.?

A. 5 at 20 cents,&:23 cents, and 2 at 20 cents. 6. A jeweller wishes to procure gold of 20 carats fine, from gold of 16, 19, 21, and 24 carats fino; what quantity of cach must be take?

A. 4 at 14, 1 at 19, 1 at 21, and 4 at 24. We have seen that we can take 3 times, 4 times, 1; 4, or any proportion of each quantity, 19 form a mixture. llence, when the quantity of one simple is given, to find the proportional quantities of any compound whatever, after having found the proportional quantities by the last rule, we have the following

RULL. As the PROPORTIONAL QUA.TITY of that price whose quantity is given : is to EACH PROPORTIONAL QUANTITI :: so is the GIVEN QUANTITY : to the QUANTITIES or PRO PORTIONS of the compound required.

7. A grocer wishes to mix 1 gallow of brandy, worth 15 s. per gallon, with rum worth 8 s., so that the mixture may be worth 10 s. per gallon; how mucha rum must be taken?

By the last rule, the differences are 5 to 2; that is, the proportions are 2 of brandy to 5 of rum; hence he mus take 24 gallons of rum for every gallon of brandy. A. 24 gallons.

8. A person wishes to mix 10 bushels of wheat, at 70 cents per bushel, with rye at 48 cents, corn at 36 centa, and barley at 30 conts per bushel, so that a bushel of this mixture may be worth 38 cents ; what quantity of each must be taken? We find by the last rule, thai tho proport ons are 8, 2, 10, and 32. Then, as 8 : 6 :: 10 : 24 bishels of rye.

8: 10 :: 10 : 121 b. shels of corn. Ans.

8: 32 :: 10 : 40 bushels of barley. 9. How much water must be mixed with 100 gallons of rum, worth 90 conta per gallon, to reduce it to 75 cents per gallon? A. 20 gallons.

10. A grocor mixes tcas at $1,20, $i, and 60 cents, with 20 lbs. at 40 cente per lb. ; how much of each sort must he take to make the composition worth 20 conts per lb.? A. 20 at $1,20, 10 at $1, and 10 at 60 cents.

11. A grocer has currants at 4 cents, 6 cents, 9 cents, and 11 cents per Ib.; and he wishes to make a mixture of 240 lbs., worth 8 cents per lh. ; how many currants of each kinil must he take ?-In this example, we can find the proportional quantities by linking, as before; then it is plain that their sum will be in the same proportion to any part of their sum, as the whole compound is to any part of the compound, which exactly accords with the principle of Fellowship.

Hence we have the following

RULE. As the sum of the PROPORTIONAL QUANTITIES found by linking, as before : is to EACH PROPORTIONAL QUANTITY :: so is the WHOLE QUANTITY or compound required : to the REQUIRED QUANTITY of each.

We will now apply this rule in performing the last question. 4

-3

10:3:: 240 : 72 lbs., at 4 cts. 6

10:1:: 240 : 24 lbs., at 6 cts.

Then,
9
-2

10 : 2 :: 240 : 48 lbs., at 9 cts.
10:4 :: 240 : 96 lbs., at 11 cts.

Ans.

10 12. A grocer, having sugars at 8 cents, 12 cents, and 16 cents per pound, wishes to make a composition of 120 lbs., worth 13 cents per pound, without gain or loss; what quantity of each must be taken?

A. 30 lbs. at 8, 30 lbs. at 12, and 60 lbs. at 16. 13. How much water, at per gallon, must be mixed with wine, at 80 oents per gallon, so as to fill a vessel of 90 gallons, which may be offered at 50 cents per gallon? A. 565 gallons of wino, and 33 gallons of water.

14. How much gold, of 15, 17, 18, and 22 carats fine, must be mixed together, to form a composition of 40 ounces of 20 carats fine ?

A. 5 oz. of 15, of 17, of 18, and 25 oz. of 22.

INVOLUTION.

I LXXXIV. Q. How much does 2, multiplied into itself, or by 2, make ?

Q. How much does 2, multiplied into itself, or by 2, and that product by % make ?

Q. When a number is multiplied into itself once or more, in this masnor, what is the process called?

A. Involution, or the Raising of Powers.
Q. What is the number, before it is multiplied into itself, called?
A. The first power, or root.
Q. What are the several products called?
A. Powers.

Q. In multiplying 1 by 6, that is, 6 into itself, making 36, we use 6 twice ,
what, then, is 35 called ?
A. The second power, or square of 6.
Q. What is the second power, or square of 8? 10 ? 12?
A. 64. 100. 144.

Q. In multiplying 3 by 3, making 9, and the 9 also by 3, making 27, we use the 3 three times; what, then, is the 21 called?

A. The third power, or cube of 3.
Q. What is the thiru power of 2? 3? 4?
A. 8. 27. 64.

Q. What is the figure, or number, called, which denotes the power, aus X power, 2d power, &c.?

A. The index, or exponent.

Q. When it is required, for instance, to find the third power of 3, what is the index, and what is the power ?

A. 3'is the index, 27 the power.

Q. This index is sometimes written over the number to be multiplied, thu g? ; what, then, is the power denoted by 24 ?

A. 2 X 2 X 2 X 2=16.

Q. When a figure has a small one at the right of it, thus, 6o, what does it moan?

A. The 5th power of 6, or that 6 must be raised to the

5th power.

1. How much is 122, or the square of 12? A. 144.
2. How much is 4?, or tho square of 4? A. 16.
3. How much is 10%, or the square of 10 ? A. 100.
4. How much is 43, or tho cube of 4? A. 64.
5. How much is 14, or the 4th power of 1? A. l.
6. What is the biquadrate, or 4th power, of 3? A. 81.
7. What is the square of 1? ? ? A. 1. .
8. What is the cube of 4 ? ? ? 4. · ਗੁੰਬ: 6
9. What is the square of ,5? 1,2?. A. 25. 1,44.
10. Involve 2 to the ad power; 2 to the 3d power. A. 4, 8.
11. Involve to the 2d power; fo to the 2d power. A. 6. Ito
12. Involve to the 20 power. A. = 1.
13. Involve to the 20 power. A. g.
14. What is 3“, or the square of 1? AT:
15. What is the value of 2 ? A. 25
16. What is the value of f*? A. BT:

Exercises for the Slate.
1. What is the square of 900 ? A. 810000.
2. What is the cube of 211? A. 9393931.
3. What is the biquadrate, or 4th power, of 80 ? A. 40960000.
4. What is the sursolid, or the 5th power, of 7? A. 16807.

5. Involve 13, , g, each to the 3d power. A. 1733, 43. 593 T9 6. What is the square of 51 ? A. 301. 7. What is the square of 163 ? A. 2724. 8. What is the value of 85 ? A. 32768. 9. What is the value of 104? A. 10000. 10. What is the value of 6? A. 1296. u. What is the cube of 25 ? A. 15625

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