Arithmetic, Part 1 · Section 37

Arithmetic · Part 1 · Section 37

p. 137The Rule of Ceres and Virginum.

THis is the moſt uncertaine, and unneceſſary Rule in Arithmetick; being ſeldome uſed except in ſporting queſtions to puzzle young beginners, with eaſie problems; ſuch as follow.

Queſtion 1.

A Caterer bought 8 birds of two ſorts, as Geeſe and Hens for 20 s. the Geeſe coſt 4 s. a piece, the Hens 2 s. a piece; How many did he buy of each ſort?

This may be done by the Rule of Falſe; and alſo thus: multiply the whole number 8, into the leaſt price 2, it produceth 16, which taken from the whole price 20, reſts 4 for a Dividend; which divided by 2, which is the difference of the particular prices, the quotient is 2, for the number of Geeſe; and 6 muſt be for the Hens: the proofe is eaſie.

Queſtion 2.

If 21 Perſons, Men, Women, and Children p. 138ſpend 26 ſhillings; ſo that every man payes 2 s. every woman 1 s. every child 6 d. How many is there of each ſort?

THE RULE.

Multiply the number of Perſons by the leaſt expence, and take the product of it from the whole expence, the reſt ſhall be the Dividend; which divided by the difference betwixt the greateſt and leaſt particular expences: the quotient is a Number which the Number of men (or they which ſpend moſt) comes neer to: but cannot exceed: or of the ſaid Dividend be divided by the ſumme of the greateſt and leaſt expences, the quotient is a Number then which the Number of men or thoſe which ſpend moſt) cannot be much leſſe.

So here 21 multiplyed by 6 d. that is, by ½, the product is 10½, which taken from 26, reſts 15½, for the Dividend: and then taking ½ from 2, reſts 1½ for the Diviſor, and the quotient is 3⅓, which is ſomething more then 10, the Number of men therefore muſt be but 9.

Then turn the Dividend, and the Diviſor both into whole Numbers, by multiplying them by the common Denominator 2, ſo they reduced will be 31 and 3, as before is to be ſeen in the quotient.

Multiply the Diviſor 3, by 9, (which is the Number of Men) the product is 27, which taken from 31, (which is the reduced Dividend) the remaine is 4, for the Number of Women; and the Children muſt be 8.

p. 139Example 1.

But the number of men may be alſo 8, which multiplyed by the reduced Diviſor 3, product is 24, which taken from 31, the remaine is 7 for the women; and then the children muſt be 6.

Example 2.

Or the number of men may be 7, which multiplyed by 3, produceth 21, which taken from 31, remaines 10 for the women; and 4 children.

Example 3.

So here is already ſeen 3 various ſolutions of p. 140this queſtion, which makes this Rule the leſſe to be regarded. But further, the number of men may be 10, and not more, for if you put them 11, that multiplyed by 3, produceth 33, which is greater then 31, from which it ſhould be taken, but I ſay it may be 10, and then there is only one woman, and 5 children: this confirmes the former part of the Rule.

Now for the later part, if the Dividend 31 be divided by the ſumme of the two extream expences (reduced by doubling as the Dividend is) 4, the quotient will be 7¾. And the men may be 7, as hath been ſhewed; but they may be alſo but 6, and fewer they cannot be: as 6 men, 13 women, and 2 children; for if you put them 5, that multiplyed by 3, produceth 15, which taken from 31, there remaines 16 for the women, and ſo there ſhould be no children: which is contrary to the ſuppoſition.

And further, becauſe the quotient was 7¾, the number of men might be ſo, if pure Arithmetical diviſion be onelyonly regarded: and then the women alſo are in number 7¾, and the children 5½, as may eaſily be tryed; I need not exemplifie it.

Queſtion 3.

If there be an Exhibition of 900 l. per annum to 30 perſons: ſome Clerks, ſome Meſſengers, and ſome Door-keepers, at 60 l. each Clerk, 40 l. each meſſenger, and 20 l. each door-keeper, how many muſt there be of each ſort?

Multiply (according to the Rule) 30 by 20, the product is 600, which taken from 900, remains p. 141300 for the Dividend; and 60 want 20, that is 40, for the Diviſor: and the quotient is 7½, and more the Clerks cannot be; Alſo divide by 60 more 20, that is 80, quotient is 3¾, and much fewer the Clerks cannot be.

Not to ſtand upon the Fractions (in this caſe of dividing men) the Clerks may be 7, 6, 5, 4, or 3: and the meſſengers 1, 3, 5, 7, or 9, and the Door-keepers 22, 21, 20, 19, or 18, that the Clerks cannot (in whole Numbers) be more then 7, or leſſe then 3, may thus be proved; Firſt, let them be 8, then 8 times 40 is 320, which is more then 300, out of which it ſhould be taken: Secondly, let them be 2, then 2 times 40 is 80, out of 300 remains 220, which divided by 20, gives the quotient 11, for the meſſengers, ſo the Clerks and meſſengers being 13, the remaine thereof to 30, namely, 17, muſt be Door-keepers.

but,

Which is 40 l. too much, therefore the Clerks cannot be two.

Note.

It may be asked, why the remaine 220 ſhould be divided by 20: whereas the like remaine in the former Example, namely 16, was taken (without any diviſion) abſolutely for the number p. 142of Women, or middle number? I anſwer, although the greateſt or firſt Number being found, (as here to be 2) the reſidue of 2 to 30, might be rightly parted into two fit parts in the ſame manner as the firſt queſtion of this Rule was reſolved, or elſe by the Rule of Falſe: yet to give further ſatisfaction, the cauſe of this is, the difference betwixt the two leſſer expences, was there ½, which (before the diviſion) was reduced to 1, which neither multiplies nor divides any Number, but leaves it the ſame: whereas, in this laſt, the middle expence (or exhibition) being 40, and the leaſt 20, the difference of them was 20, by which dividing the Remaine of the laſt ſubtraction: the quotient is ever the Number of the middle perſons. Which may ſerve as an addition to the Rule, where the ſorts of things are but three.

Queſtion 4.

If there be 10 perſons of four ſeveral Countries. Engliſh, French, Dutch, and Spaniſh, to pay a Debt of 1000 l. So that every Engliſh man payes 50 l. every French man 70 l. every Dutch man 130 l. and every Spaniard 150 l. How many is thereof each?

The Dividend, (according to the former Rule) is 500.

Now to make the Diviſor, take his ſumme that payes leaſt (namely 50,) out of each of the other three 150, 130, and 70, and the Remaines will be 100, 80, and 20.

Adde the firſt and laſt for the Diviſor, it is 120.

And the quotient will be 4 2/12, and the Spaniards cannot be more.

p. 143Secondly, adde the firſt and ſecond together for the Diviſor, it is 180, and the quotient is 2 14/18, and the Spaniards cannot be leſſe.

I mean, they cannot be much more than 4, or leſſe then 2: and therefore, ſeeing any one ſolution will ſerve, let them be 3, and by that multiply 100, and take the product out of 500, there remaines 200 for a ſecond Dividend, which divided by (the ſecond remain) 80, the quotient is 2½: therefore the Dutchmen are 2, which multiplyed by 80 make 160, take that out of 200, there remaines 40 for a third Dividend: which divided by (the third remaine) 20, the quotient is 2 for the French-men alſo; and conſequently the Engliſh muſt be 3, becauſe all of them are 10: But the Spaniards may be alſo 4 or 2.

Example.

p. 144The reaſon why the Spaniards and Engliſh, as alſo the Dutch and French are equal in number, is becauſe their payments differ equally from 100, which is the mean ſumme with which 10 men ſhould pay 1000 l. and making it ſo, this queſtion, and many other of this nature may be anſwered by the Rule of Alligation: thus,

And alſo, becauſe the other two differences 30 and 30 are equal: the Number of the French is equal to the Number of the Dutch.

But both thoſe numbers together are 4, becauſe 3 Spaniards, and 3 Engliſh, taken out of 10, the remaine muſt be 4.

Wherefore the Number of the French is 2, and the Dutch alſo are 2.

p. 145Or thus,

If 160 require 10, what 30?

Anſwer is 1 160/160, (that is, in this caſe 2) for the Spaniards; and conſequently 2 Engliſh, and therefore the French and Dutch each 3.

But where any one of the particular ſummes, is equal to the mean ſumme, there this cannot ſo well be done by Alligation.

Example.

If one ſhould buy 12 loaves of bread for 12 pence, ſo that ſome might be two peny; ſome peny, ſome half-peny; and ſome farthing loaves: and it be required to know how many he muſt buy of each; then becauſe of 12 loaves for 12 pence, the mean price is 1, but one of the particulars being alſo 1, there ſhould be no peny loaves, becauſe there is no difference betwixt the mean price, and a peny.

p. 146But it may be found by the Rule of Ceres and Virginum, to be either.

Or elſe,