Q. How is Conjoined Proportion proved?

A. Make as many single rules of three as the nature of the question requires.

EXAMPLES.

1. If 100 lb. English make 95 lb. Flemish, and 19 lb. Flemish 25 lb. at Bolonia, how many lb. English are equal to 52lb at Bolonia? Answer, 40lb. English.

2. If 25 lb at London be 22lb at Nuremberg; 88 lb at Nuremberg 92 lb at Hamburgh; 46 lb at Hamburgh, 49 lb at Lyons; how many lb at London are equal to 98 lb at Lyons? Answer 100 lb.

3. If 6 braces at Leghorn make 3 ells English, 5 ells English 9 braces at Venice, how many braces at Leghorn will make 45 braces at Venice? Ans. 50 bra. at Leghorn.

4. If 3 ells English make 6 braces at Leghoru, and 150 braces at Leghorn, 135 braces at Venice, how many ells English are equal to 27 braces at Venice P Answer 15 ells English.

CASE 2.

Q. How are questions answered in this case?

A. When it is required to know how many of the last: sort of coin, weight or measure, mentioned in the question, are equal to a given number of the first; then

1. Place the numbers alternately, as in case 1, but let the last number stand on the right hand.

2. Multiply the second rank for a dividend, and the first for a divisor.

EXAMPLES.

1. If 10 lb. at London make 9 lb. at Amsterdam; 90 lb.at Amsterdam 1121b. at Thoulouse? how many lb. at Thoulouse are equal to 50 lb. at London? Ans. 56 lb. at Thoulouse.

2. If 20 braces at Leghorn be equal to 10 vares at Lisbon; 40 vares at Lisbon to 80 braces at Lucca; how many braces at Lucca are equal to 100 braces at Leghorn? Answer, 100 braces at Lucca.

OF ALLIGATION.

Q. How many kinds of Alligation are there?

A. Two alligation medial, and alligation alternate.

OF ALLIGATION MEDIAL.

Q. What is alligation medial?

A. Alligation medial, is when the quantities and prices of several things are given to find the mean price of the mixture compounded of those things.

Q. What is the Rule?

A.

As the whole Composition,

Is to its Total Value:

So is any Part of the Composition
To its mean price.

Q. How is Alligation Medial proved?

A. Find the value of the whole mixture at the mean rate; and if it agrees with the total value of the several quantities at their respective rates, the work is right.

EXAMPLES.

1 A farmer mingled 19 bushels of wheat at 6s. per bushel, and 40 bushels of rye at 4s per bushel, and 12 bushels of barley at 3s. per bushel together; I demand what a bushel of this mixture is worth? Ans. 4s 4d 1qr.41.

2 A farmer mingled 20 bushels of oats at 2s per bushel, and 30 bushels of beans at 2s per bushel, and 20 bushels of peas, at 3s per bushel, together; I demand the worth of a bushel of this mixture? Ans. 2s 3d 1qr.ş.

3 A vintner mingled 5 gallons of Canary, at 8s per gallon, and 6 gallons of Malaga, at 7s per gallon, and 4 gallons of white wine, at 6s per gallon, together; I demand what a gallon of this mixture is worth? Ans. 7s Od 33.

4 A grocer mingled 2 C. of sugar at 56s per C. and 1 C. at 43s per C. and 2 C. at 50s per C. together; 1 demand the price of 3 C. of this mixture? Ans. 71 13s.

5 An alehouse keeper mixed 3 sorts of ale together, viz. 12 gallons at 6d per gallon, 16 gallons at 7d per gallon, and 21 gallons at 9d per gallon: I demand what one gallon of this mixture is worth? Ans. 7d. 2qrs. 23.

22

6 A refiner having 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine, would melt all together: I demand what fineness 1 lb. of this mass shall be? Ans. 6 oz. 13 dwts. 8 grs. fine,

7 A mint-master hath 3 lb. weight of gold, of 22 carrats fine, and 3 lb. of 20 carrats fine: I demand what fineness an oz. of this mixture will bear? Ans. 21 car. fine.

8 An hoster mixing provender for his horses, would put in a quantity of beans at 5s per bushel, with the like quantity of oats, at 3s 6d per bushel: I demand the price of a bush I of this mixture? Ans. 4s 3d.

9 A malster bath several sorts of malt, viz. one sort at

4s 6d. another at 4s. and another at 3s. 6d. per bushel; and he would mix an equal quantity of each together; I demand the price of a bushel of this mixture? Ans. 4s.

:

10. A brewer had several sorts of ale, viz. one sort at 20s per barrel, another at 25s. a third at 30s, and a fourth at 35s per barrel and he would mix an equal quantity of each together: I demand the price of a barrel, and also of a gallon of this mixture? Ans. 27s 6d per barrel, and 10d Igr. per gallon.

OF ALLIGATION ALTERNATE.

Q. What is Alligation Alternate ?

A. Alligation Alternate is, when the rate of several things are given to find such quantities of them as are necessary to make a mixture, which may bear a certain rate propounded.

Q. How are the rates of prices of the given things to be ordered?

A. They must be placed one over

mean

the other, and the propounded price of rate 7 the composition against them: thus,

4 Prices

5 of the

6 Sim

8 ples. 2 Link the several rates together, in such sort, that one greater than the mean rate may be coupled to another which is less.

3. Take the differences between the mean rate, and the several prices, and place them each against his yoke-fellow and for the rest, observe the following cases.

CASE 1.

Q. What do you observe in this first case?

A. When the prices of the several things, together with. the mean rate of the mixture are given, without any quantity to find how much of each ingredient is required to compose the mixture: take the difference between each price, and the mean rate, and set them alternately, and they will be the quantities required.

Q. How are the operations in this and the following cases proved?

A. They are all proved by Alligation Medial.

EXAMPLES.

1. How much rye at 4s per bushel, barley at 3s per bushel. and oats at 2s per bushel, will make a mixture worth 28 6d per bushel? Ans. 6 bushels of rye, 6 bushels of barley, and 24 bushels of oats.

2. How many raisins of the sun at 7d per lb. and Malaga

raisins at 4d per lb. may be mixed together for 6d per lb? Ans. 2lb. of raisins of the Sun, and 1lb. Malaga raisins. Note. Questions in this rule do frequently admit of an infinite .variety of answers, and all in whole numbers as in this last example, where 2 and 1 do answer the question, yet any other 2 numbers will as truly do the like, that are in the same proportion.

3. A grocer would mix three sorts of sugar together, viz. one sort of 10d. per lb. another at 7d. and another at 6d. how much of each sort must he take, that the whole mixture may be sold for 8d per lb.

lb. d. lb. d lb. d.

Ans. 3 at 10: 2 at 7 and 2 at 6 per

lb.

4. A malster hath several sorts of malt, viz. one sort at 4s per bushel, another at 3s. 6d. a third at 3s and a fourth at 2s per bushel, and he is desirous to mix so much of each sort together, that the whole may be sold at 2s 6d per bushel; I demand how much he must take of each sort ? Bush. s. B. s. d. B. s, B. S.

Ans. 6 at 4; 6 at 3 6; 6 at 3 and 36 at 2 per bushel. 5. A druggist hath several sorts of tea, viz. one sort at 12s per Ib. another at 11s. a third at 9s and a fourth at 8s per lb. I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10s

per lb?

Answ.

2 at 11

2 at 11

2 An.

3 An.

1 at 9

2 at 9

2 at 9

Note. These seven answers arise from as many different ways of linking the rates of the simples together.

6. How much alloy must I mix with bullion of 10 oz. fine to abase the same to 8 oz. fine? Ans. to every 8 oz.

of bullion of 10 oz. fine, put 2 oz. of alloy, and that will abase it to 8 oz. fine.

CASE 2.

OF ALTERNATION PARTIAL.

Q. What do you observe in this second case?

A. When the rates of all the things, the quantity of but one of them, and the mean rate of the whole mixture are given to find the several quantities of the rest in proportion to the quantity given; take the difference between each price, and the mean rate, and place them alternately as in case 1. Then say,

As the difference of the same name with the quantity given,

Is to the rest of the differences severally;

So is the quantity given,

To the several quantities required.

EXAMPLES.

1. A man being determined to mix 10 bushels of wheat, at 4s. per bushel, with rye at 3s. with barley at 2s. and with oats at Is. per bushel; I demand how much rye, barley and oats, must be mixed with the 10 bushels of wheat, that the whole may be sold at 28d per bushel?

2. A man being determined to mix 12 bushels of oats at 18d per bushel with barley at 2s. 6d. with rye at 3s. and with wheat at 4s per bushel-I demand how much barley, rye and wheat must be mixed with the 12 bushels of oats, that it may bear the price of 22d. per bushel? Answ. 1 bushel of each sort.