Q 2. A grocer mixed the following teas together, viz. 15lbs. at 8s. per lb. 20lbs at 75. 4d. per 1b, 10lbs. at 6s. 8d. per lb. and 24lbs. at 4s. per lb. what is one pound of this mixture worth?-Anfwer 6s. 24d. 18.

Qu. 3. A vintner mixes-5 gallons of wine at 7s. per gallon, with 9 gallons at Ss. 6d. per gallon, and 14 gallons at 5s. 10d. per gallon; what is one gallon of this mixture worth-Anfwer 6s. 10 d. 14.

*

Qu. 4. A goldfmith melts 101 ounces of gold bullion, of 14 carats fine, with 152 ounces of 18 carats fine, how many carats fine is this mixture?-Anfwer 1638 carats fine.

Alligation alternate is the method of finding what quantity of fimples, whofe rates or prices are given, will form a mixture of a certain given rate or price.

Rate 1. Write the rates or prices of the feveral fimples under each other. 2. Connect with a curve line the rate or price of each fimple that is lets than the rate or price of the mixture, with one or more of these rates or prices that are greater than that of the mixture; and each greater rate with one or more that are lefs: and place the rate or price of the mixture on the left hand of the rates or prices. 3. Write the difference between the rate of the mixtures and the rate of each fimple oppofite the rate with which fuch fimple is connected or linked..

Then these differences which stand opposite any rate is the quantity which that rate requires to form a mixture of the given rate; but if there is more than one difference opposite any fimple, their fum is the true difference.

*Gold is generally mixed with copper or fome other base metal, which is called the allay; and the gold is faid to be fo many carats fine as it contains pure gold: thus, if an article weighs 24 carats, and contains 22 carats of gold, and 2 of allay, it is faid to be 22 carats fine. What a carat is may be seen, page 135.

Example

Example: 1. A grocer would mix teas at 45. per lb. 75. per lb. 9s. per lb. and 10. per lb. in fuch proportion that the mixture may be worth 6s. per lb.; what quantity of each

In this example, I first state the work as before directed, placing the prices of the teas in a column over each other, with 6, the given price of the mixture, on the left hand.

Secondly, I connect the prices with each other by curve lines; 4 the top figure, being lefs than the rate of the mixture, I connect with 7, 9, and 10, becaufe they are all greater than 6, the rate of the mixture.

Thirdly, I find the difference between 6, the price of the mixture, and 4, that of the first fimple, which is 2; I therefore place 2 oppofite the 7, 9, and 10, as the 4 is linked to all of them. Then I find the difference between the 6 and the next figure 7, which is 1, I therefore place 1 oppofite the 4, being the figure to which the 7 is linked. Then the difference between the 6 and 9 is 3, which I place alfo oppofite the 4 (the 9 being linked thereto), and the difference between 6 and 10, which is 4, I also place oppofite the 4 (as the zo is alfo linked to it). Thefe differences, fo placed, contain the true proportion of each fort of tea at the price oppofite to each, that fhould be taken to form a mixture at the defired rate.

But oppofite the 4 there are three differences, viz. 1, 3, and 4, which are to be added together, as feen in the last column. Thus, there must be 8lb. of tea at 45% per lb. 2lb. at 75. per lb. 2lb. at 9s. per ib. and 2lb. at 10s, per lb.; and the whole quantity of the mixture is 14lb. at 6s. per lb.

Thefe

These questions are proved in the fame manner as thofe in alligation medial, viz. by finding the total value of all the fimples in their feparate ftate, and the total value of the mixture; and if these two values be equal, the work is right.

I

Total 4

Qu. 2. A farmer mixed wheat at 4s. the bufhel, with rye at 35. the bushel, and barley at 18d. the bufhel, how much must he mix of each to fell the whole mixture at 22d. the bufhel? Anfwer, 40 bushels at 18. per bufhel, 4 bushels ́at 35. per bushel, and 4 bushels at 41. per bufhel.

Qu. 3. A goldfmith has gold to melt of 24 carats fine, 21 carats fine, 19 carats fine, and 16 carats fine; how much of each muft he take to form an article of gold that shall make 17 carats fine?-Anfwer, i of 24, 1 of 21, 1 of 19, -and-13 of 16 carats fine.

When the whole mixture is limited to a certain quantity, after finding the quantity of each of the fimples as before, fay (by the rule of three), as the fun of the quantities is to the given quantity, fo is the quantity of each fimple to the required quantity of each.

Example

Example 4. A vintner is defirous to mix 5 forts of wine together: viz. at 115. per gallon, 10s. per gallon, 9s. per gallon, 75. per gallon, and 6s. per gallon, in such proportion as to make 40 gallons of wine, worth 8. per gallon: how much of each fort must he take?

In this example, after linking the prices together, as before directed, I have the quantity of each wine to form a mixture of 8s. per gallon; but the whole quantity of the mixture thus found is only 11 gallons, whereas it should be 40 gallons, therefore I say,

Gall.

Gall. Gall.

Gall. Pints.

As 11 to 2 fo is 40 the quantity required to 7 27

Question 5. A grocer has fugar at 10d. 8d. 6d. and 4d. per lb. of which he would make a mixture to consist of 60lb. and worth 5d. per lb.; how much of each fort must he take? Kinfruer, 5lb. at iod. gib. at 8d. 5lb. at 6d. and 45lb. at 4d. per lb.

Sometimes it is required to take a certain quantity of any one fimple to mix with the others, and which generally alters the quantities of the other fimples. To find what proportion of the others is requifite, I fay (by the rule of VOL. I. Cc three),

three), as the quantity of that fimple whofe particular quantity is given is to the given quantity, fo is the found quantity of any other simple to the quantity required.

Example 6. A grocer would mix raifins at 11d. per lb 10d. per lb. 9d. per lb. and 6d. per lb. with 120lb. at 7d. per lb.; how much of each fort must he take, that the whole may be worth Sd. per lb.?

Hence the feveral quantities requifite are as placed in the example; but to find the quantities which should be taken of each fort, I fay (by the rule of three), as 5lb. (the quantity there found) is to 120lb. (the quantity required to be taken), so is 3lb. (the quantity at tid. per lb.) to 72lb. (the quantity that should be taken).

As 5 is to 120 fo is 3 to the quantity required

3

72

• Question 7, A vintner mixes wine at 125. 10s. and 6. per gallon, with 20 gallons at 45 per gallon; how much of each fort must he take to make the mixture worth 8s. per gallon?—Answer, 20 gallons at 125, 10 gallons at 105, and 10 gallons at 65,

From the foregoing examples, it is evident that there are Several ways of working this rule, according to the method.

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