difference between those prices and the mean rate alternately, that is, placing the difference between the greater price and the mean rate, against the lesser price, and placing the difference between the lesser price and the mean rate, against the greater price, the difference of the prices become mutually changed; and these differences express the relative quantities of each simple, necessary to form the compound, and what is lost on one quantity, is gained on another. Hence the balance of loss and gain between any two, will be equal, consequently the same on the whole.

2. A Merchant has three sorts of tea, viz.: one sort at 5 shillings per lb., another at 7 shillings per lb., and another at 8 shillings per lb., what proportion of each kind must he take to make a mixture worth 6s. per lb. ?

3. How much vinegar at 42cts., 60cts. and 67cts. per gallon, must be mixed together, so that it may be worth 64 cents per gallon?

Ans. 3gals. at 42cts., 3do. at 60c. and 26do. at 67c. 4. How much sugar at 9cts. and 15cts. per lb. must be mixed together, so that the compound may be worth 12 cents per lb.? Ans. An equal quantity of each sort. 5. A Goldsmith mixed together gold of 17, 19, 21, and 24 carats fine, so that the composition was 22 carats fine, what proportion did he take of each ?

Ans. 2 of each of the first 3 sorts, and 9 of the last. 6. A Merchant has spices at 7d. 8d. 10d. and 11d. per lb. which he would mix together so that the whole compo sition may be sold at 9d. per lb., what proportion must he take of each kind?

These four answers arise from the various ways of link、

ing the prices of the ingredients together. Hence we may see that questions in this rule admit of a great variety of answers; for having found one answer, we may take any other numbers which have the same proportion between themselves, as the numbers have which compose the

answer.

CASE II.

When one of the ingredients is limited to a certain quantity, to find what quantity of each of the others must be taken in proportion to the given quantity.

RULE.

1. Link the prices and take their differences as in Case I. which will give the unlimited proportions.

2. Then as the proportion whose quantity is limited, is to its limited quantity, so is each of the other proportions to its required quantity.

EXAMPLES.

1. A man wishes to mix 5 bushels of wheat worth 90 cents per bushel, with rye at 56cts. per bushel, barley at 36cts., and oats at 30cts., so that the composition may be sold for 45cts. per bushel.

15 the proportion whose quantity [is limited.

9

90

cts.

56

30

45

then as 15: 5:: 93 bushels of rye.

15 5:11:3 66

15 5:45: 15 66

barley.

oats.

2. How much water at 0 per gallon, must be mixed with 100 gallons of brandy worth 180cts. per gallon, to reduce it to 150cts. per gallon? Ans. 20gal.

3. How much gold of 15, of 17, and of 22 carats fine, must be mixed with 5oz of 18 carats fine, so that the composition may be 20 carats fine?

Ans. 5oz. of 15, 5oz. of 17, and 25oz. of 22 carats fine. 4. With 95 gallons of wine worth 96cts. per gallon, I mixed wine at 80cts. per gallon, and some water, so that it

stood me in 76cts. per gallon, how much wine and how much water did I take?

Ans. 95gals. wine at 80cts., and 30gals. water

CASE III.

When the whole composition is limited to a given quantity.

RULE.

1. Link the prices and find the proportions as in Case 1. Then, as the sum of the proportion is to the given quantity or whole composition, so is each proportion to its required quantity.

EXAMPLES.

1. A Grocer has sugar at 5cts., 7cts., 10cts. and 13cts. per lb., and would mix them together so as to fill a cask of 200lbs. worth 8cts. per pound.

2. How much water of no value must be mixed with spirits at 90cts. per gallon, so as to fill a cask of 120gals. that may be sold for 60cts. per gallon?

Ans. 40gals. of water, and 80gals. of spirits. 3. A Grocer has teas at 2s., 3s., 5s., and 6s. per lb. and would mix them together so that the composition may be worth 4s. per lb. What quantity must he take of each, to fill a chest that will hold 90lbs. ?

Ans. 30lbs., 15lbs., 15lbs., and 30lbs. 4. How much gold of 15, of 17, of 18, and of 22 carats fine, must be mixed together, to make a composition of 40oz. that will be 20 carats fine?

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

1. What is Alligation?

2. What is Alligation Medial? 3. What is the rule?

Questions.

4. What is Alligation Alternate ? 5. When the mean rate and the rates of the ingredients are given without any limited quantity, how do you find the proportional quantity?

6. When one of the ingredients is

limited to a certain quantity, how do you find what quantity of each of the others must be taken in proportion to the given quantity?

7. When the whole composition is limited to a given quantity, how do you find the proportional quantities of each ingredient?

ARITHMETICAL PROGRESSION.

Any rank, or series of numbers more than two, increasing by a common excess, or decreasing by a common difference, is called an Arithmetical Series or Progression.

The number which is continually added or subtracted, is called the common difference.

When the numbers are formed by a continual addition of the common difference it is called an ascending series; but when they are formed by a continual subtraction of the common difference, they form a descending series.

Thus, {2, 4, 6, 8, 10, is an

ascending series.

10, 8, 6, 4, 2, is a descending series.

The numbers which form the series are called the terms of the series, or progression; the first and last terms of which are called the extremes.

A series in progression includes five parts, viz.:

1. The first term.

2. The last term.

3. The number of terms.
4. The common difference.

5. The sum of all the terms:

by having any three of which given, the other two may be found.

CASE I..

The first term, common difference, and number of terms given, to find the last term.

RULE.

Multiply the number of terms, less 1, by the common difference, and to the product add the first term, and the sum will be the last term.

EXAMPLES.

1. A man bought 17 yards of cloth, giving 5 cents for the first yard, 8 cents for the second, 11 cents for the third, and so on, increasing with a common difference of three cents on each yard; what was the cost of the last yard?

Numb. of terms, less 1,=16

First term,

Last term,.

X3

48 +5

=53

It will be seen that the common difference will always be added one time less than the number of terms. Thus, for the second yard the common difference is added to the first yard one time, for the third yard it is added 2 times; and for the 17th yard, it must be added to the price of the first yard 16 times. 16x3=48, that is the 17th yard will cost 48 cents more than the 1st yard; and 48+5=53, the price of the last.

2. If the first term be 5, and the common difference 3, and the number of terms 100, what is the last term?

Ans. 302. 3. A man, in traveling a certain journey in 11 days, traveled thirteen miles the first day, and increased every day two miles; how many miles did he travel the last day? Ans. 33 miles..

CASE II.

The first term, last term, and number of terms given, to find the common difference.

RULE.

Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common difference.

EXAMPLES.

1. A man bought 17 yards of cloth in arithmetical progression. For the first yard he gave 5 cents, and for the last 53 cents; what was the common difference, or the increase of the price on each succeeding yard?