3. A wine merchant mixes 12 gallons of wine, at 45. rod. per gallon, with 24 gallons, at 5s. 6d. and 16 gallons, at 6s. 34d.-What is a gallon of this compofition worth? Anf. 58. 7d.

4. A goldsmith melted together 8oz. of gold of 22 carats fine, 1lb. 8oz. of 21 carats fine, and ooz. of 18 carats fine Pray what is the quality, or fineness of the compofition?

8×22+20X21+10X18

8+20+10

10 carats fine, Anf

5. A refiner melts 5 lb of gold, of 20 carats fine, with 81b. of 18 carats fine: How much alloy muft he put to it, to make it 22 carats fine?

22 ·5 × 20+8 X 18÷5+8=373;

3

T3

Anf. It is not fine enough, by 3 carats fo that no alloy must be added, but more gold.

ALLIGATION ALT: N TE*

Is the method of finding what quantity of each of the ingredients, whofe rates are given, will compofe a mixture of a given rate; fo that is the reverfe of Alligation Medial and may be proved by it.

CASE

* By connecting the lefs rate with the greater, and placing the diff rence between them and the mean rate alternately, or one after the other in turn, the quantities refuting are fuch that there is precifels as much gained by one quantity as is loft by the other; and therefore the gain and os upon the whole are equal and exactly the proposed rate

In ke manner let the number of fimpies be what it may, and with how many foever each one is linked, fince it is always a lefs with a greater than the mean price, there will be an equal balance of lofs and gain between every two, and confequently an equal balance on the whole

It is obvious from the rule, that questions of his fort admit of a great variety of anfvers; for having found one answer, we may fild a many more as we pleafe, by only multiplying or dividing each of the quantities found, by 2, 3, 1, &c

Ifay one of the fimples be of little or no value with refpect to the rest, rate is fuppofed to be nothing, as water mixed with wine, and alloy with gold and filver.

E E

CASE I.

The whole work of this cafe confifts in linking the er tremes truly together, and taking the differences between then and he mean price, which differences are the quantities fought.

RULE 1. Place the feveral prices of the fimples, be ing reduced to one denomination, in a column under each other, the leaft uppermoft, and fo gradually downwards they increase; with a line of connection at the left hand, and the mean price at the left hand of all.

2. Connect with a continued line the price of each fimple, or ingredient, which is lefs than that of the compound, with one or any number of those, which are greater than the compound, and each greater rate or price with one, or any number, of the lefs.

3 Place the difference between the mean price (or mixture rate) and that of each of the fimples oppofite to the rates with which they are connected.

4 Then, if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be feveral, their fum will be the quantity.

EXAMPLES.

1. A merchant has spices, fome at Is. 6d. per lb. some at 2s. fome at 4s. and fome at 5s. per lb-How much of each fort muft he mix, that he may fell the mixture at 3s. 4d. per lb. ?

d.

Ib s. d.

d

ს.

s. d.

Note. Thefe five answers arise from as many various ways of linking the rates of the ingredients together..

2. *A merchant has Canary wine, at 3s, per gallon; Sherry at 2s. 1d. and Claret at s. 5d. per gallon: How much of each fort muft he take to fell it at 2s 4d. per gallon?

36

3+11

s. d. 14 at 30

per gall.

3. A goldfmith would mix gold of 19 carats fine, with fome of 16, 18, 23 and 24 carats fine, fo that the compound may be 21 carats fine; What quantity of each must he take?

[16.
18-

21 19
23

24

4. It.

Note, The fecond queftion admits but one way of linking, and fo but of one anfwer; yet all numbers in the fame pro portion between themselves, as the numbers which compose the anfwer, will like wife fatisfy the condition of the question

4. It is required to mix feveral forts of wine at 35.55. and 78. gallon, with water, that the mixture may be worth 4s. per gallon: How much of each fort must t mixture confift of?

Anf. 3 gal. water, 1 gal. wine, at 3s. I ditto at 5s. and 4 ditto at 7 s.

CASE II.

When the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture, are given to find the feveral quantities of the rest in proportin to the quantity given.

RULE. Take the differences between each price and the mean rate and place them alternately, as in Cafe 1. Then, as the difference standing against that fimpie, whofe quantity is given, is to that quantity; fo is each of the other differences, feverally, to the feveral quantities required.

EXAMPLES.

1. A merchant has 40lb. of tea, at 6s. per lb. which he will mix with fome at 5s. 8d. fome at 5s. 2d. and fome at 4s. 6d. How much of each fort must he take, to mix with the 40lb. that he may fell the mixture at 5s. 5d. per lb.

2. How much gold of 16, 20 and 24 carats fine, and how much alloy, muft be mixed with ooz. of 18 carats fine, that the compofition may be 22 carats fine?

Anf. 100z. of 16 carats fine, 10 of 20, 170 of 24, and io of alloy.

ALTERNATION TOTAL.

CASE III,

When the rates of the feveral ingredients, the quantity to be compounded, and the mean rate of the whole mixture are given to find how much of each fort will make up the quantity.

RULE. Place the differences between the mean rate, and the feveral prices alternately, as in Cafe ift. Then, As the fum of the quantities, or differences thus determined, is to the given quantity or whole compofition; fo is the difference of each rate, to the required quan tity of each rate.

EXAMPLES.

1. Suppofe I have 4 forts of currants, at 8d. 12d. 18d. and 22d. per lb.-the worst would not fell and the beft were too dear; I therefore concluded to mix 120lb. and fo much of each fort, as to fell them at 16d. per lb. -How much of each fort must I take?

120

pr. lb.

2. A goldfmith has feveral forts of gold, viz. of 19, 17, 20 and 22 carats fine, and would melt together, of all these forts, fo much as may make a mass of 400Z.

E E 2