Arithmetic, Part 1 · Section 35

Arithmetic · Part 1 · Section 35

p. 124The Rule of Alligation.

THis hath its name from binding, tying, or uniting many particulars in one Maſſe or Summe, the nature of it will be underſtood in working ſome Queſtions or Examples.

Queſtion 1.

A Corn Maſter would mix 4 ſorts of graine together, viz. wheat at 4 s. the buſhel, wheat at 4 s. 0 d. the buſhel; Rie, at 3 s. the buſhel, and Barly at 2 s. 8 d. the buſhel; ſo as to make 15 quarters in all, to be ſold at 3 s. 6 d. the buſhel; How much muſt he take of each?

THE RULE.

Multiply the whole Maſſe to be made, by any particular difference; and divide the product by the ſumme of all the differences, the quotient ſhall be the juſt quantity of that particular kind, whoſe price ſtandeth againſt the difference you worked with.

Example.

Firſt turn the quarters into buſhels, by ſaying, 8 times 15 is 120, then for the quantity of the firſt ſort at 4½: multiply 120 by ⅚, the product is 100, which divided by 2⅚, the quotient is 35 5/7 buſhels of that ſort at 4 s. 6 d. and working ſo for every of the other; they will be found to be thus:

Now to prove this right; firſt multiply the whole Maſſe 120 buſhels, by the deſired price 3½ s. omitting the denominations, the ſumme is 420 s.

But ſo much all ſhould be worth in ſhillings. And therefore the queſtion in rightly ſolved.

Queſt. 2.

One hath 6 ſorts of Fruits at ſeveral prices, Dates at 2 s. Almonds at 1 s. 4 d. Currance at 10 d. Raiſons at 5 d. Prunes at 4 d. and Figs at 3 d. the pound; and would take of every ſort ſome to make a mixed quantity of 30 l. weight to ſell one with another for 9 d. the pound, how much muſt he take of each?

So 30, multiplyed by 6, produceth 180, which Divided by 38, the quotient is 4 22/38 of a pound weight: and ſo much muſt be taken of Dates, at 24 d.

Secondly, 5 times 30 is 150, which divided by 38, the quotient is 3 36/38 for Almonds. And working after the ſame manner with 4, 1, 7, 15, their reſpective quantities will be found to be theſe;

That is 26 252/32, and the reduction of the fraction will make it 30, as it ought to be, and by comparing the prices of theſe particulars added, with the price of 30 li. weight, at 9 d. per l. weight, which makes 470 d. this may be proved like the former.

But that the Reader may be perfect in it, I will doe it here alſo, as followeth.

Say firſt, 24 times 4 is 96, and 24 times 28 p. 128is 672; for the firſt, ſet them thus: 96, 672.

Now this 1634 being the ſumme of the Numerators of Fractions, whoſe common Denominator is 38, muſt be divided by 38, and the quotient will be 43, which added to the whole number 227: the ſumme is 270. And ſo much is 30, multiplyed by 9, which ſhewes the work to be right.

The Combination, or linking of Numbers may be varied at pleaſure, as whereas above I linked 24, and 3, alſo 16 with 4, and 10 with 5, it might have been 24 with 5, and 16 with 4, and 10 with 3. Or 14 with 4, and 16 with 3, and 10 with 4, of which diverſity of linking would follow diverſity of ſolutions, but all true, as the Reader may eaſily prove by himſelf.

Likewiſe, if the Numbers to be linked were 3, 5, 7, or any odde Number, one of them may be linked to two ſeverally, to make the worke even.

p. 129Example 3.

If the Numbers were 12, 10, 8, 6, and 4, and the mean or common price required were 9, you might firſt linke them as you ſee here, takeing 12 twice, or elſe you might take any other twice as you ſhall like; and ſo the work will be every way right, though not the ſame; if the differences be rightly ſet off and orderly uſed, as is taught before in the firſt queſtion.

Queſtion 3.

A Goldſmith would mix 3 ſorts of ſilver, A. B. C. A is 10 d. weight better; B 7 d. weight better; and C, 4 d. weight better, to make an Ingot of 50 li. weight, which ſhould be in fineſſe 8 d. weight better: How much muſt be taken of each?

Which is equal to 50, the quantity required,

That is, 1600/9, which is equal to 400, and if the whole Ingot 50, be multiplyed by the betterneſſe required, namely, by 8, they ſhall produce 400 alſo: So this is proved.

In every Alligation, or linking of two numbers, this is evident, that if the ſumme of the numbers linked be greater then the mean number required taken ſo many times as there are numbers to be linked, the queſtion would be abſurd; and the reſolution-thereof impoſſible. And this ſhall ſerve for the rule of Alligation.