the third Term is, in respect to the firft, the leffer will the fourth Term be, in refpect to the fecond. Example 2. If 8 Men can do a Piece of Work in 12 Days, How many Days will 16 Men require to do the fame Work? Here it is plain the fourth Term must be less than the fecond, because 16 Men undoubtedly can do the fame Work in lefs Time than 8 Men can. From thefe Confiderations, compared with thofe in page 85. it will be eafy to perceive, whether the Terms of any propofed Question are in Direct or Reciprocal Proportion. For when, according to the true Meaning and Defign of any Question in Proportion, More requires More, or Lefs requires Lefs, the Terms are in Direct Proportion; as in this laft Section. But if More require Lefs, or Less require More (as above) then the Terms will be in Reciprocal Proportion. The Manner of placing down the propofed Terms is the fame in both Rules, viz. The firft Term in the Suppofition must be of the fame Kind and Denomination with the third Term which moves the Queftion; and the Term fought must be of the fame Kind and Denomination with the second Term in the Suppofition. As in the two laft Examples. Men Days Men Days Thus, in { Example 1. 16 : 6 :: 8 Example 2. : The Queftion being truly ftated, obferve this Theorem. Multiply the first and fecond Terms together, and Theorer. divide the Product by the third Term, the Quotient will be the Answer required. Thus in the fecond Example 12 x 896. Then 16) 96 (= 6 Days, the Answer required. That is, 16 Men may do the fame Work in 6 Days as 8 Men can do in 12 Days. Now the Reafon of this Operation (and confequently of the Theorem) is grounded upon this Confideration; viz. If 8 Men require 12 Days to do the Work, it is plain that one Man would require 8 Times 12 Days 96 Days to do the fame Work; but if one Man can do it in 96 Days, moft certain 16 Men can do it in one 16th Part of that Time. Therefore 96 divided by 16 will give the Anfwer required, viz. 16) 96 (6 as before, &c. = Quest. 3. Suppofe 800 Soldiers were befieged in a Town, and their Victuals were computed to ferve them two Months (or 56 Days) How many of thofe Soldiers must depart the Garrison, that the fame Victuals may ferve the remaining Soldiers 5 Months. The The Queftion truly stated will stand Thus, 2 : 800 :: 5 : 2 5) 1600 (320 : So many Soldiers may ftay in the Garrison. Confequently, 800-320-480 Soldiers that muft go out of the Garrifon, which is the Answer required. Question 4. A borrowed of his Friend B 2501. for fix Months, promifing to do him the like Kindness upon Demand: Some Time after B defires A to lend him 4007. the Question is, how long B must keep the 400l. to be fully fatisfied for his former Kindness to A. 4) 84 (21 Days. Anfw. 3 Months, 21 Days. Question 5. If a Penny White Loaf ought to weigh eight Ounces Troy Weight, when Wheat is fold for fix Shillings fix Pence the Bufhel, what muft it weigh when Wheat is fold for four Shillings the Bushel? Thus 6 s. 6 d. 78 d. 8 oz. 4 s. 48 d. to the Anfwer. 8 48) 624 (13 oz. the Answer required. 48 144 144 (0) The Proof of this Inverfe Rule is eafily deduced from it's Operations; viz. The Product of the firft and fecond Terms, muft be equal to the Product of the third and fourth Terms. Note, Any Queftion that falls under this Inverfe Rule or Reciprocal Proportion, may be fo ftated as to have it's Terms in Direct Proportion; by only changing the Places of the first and third Terms in the Queftion. Thus, Question Question 6. If a Field will feed eighteen Horfes for seven Weeks: how long will it feed forty-two Horfes at the fame Rate of feeding. Firft, 18 Horfes: 7 Weeks :: 42 Horfes: 3 Weeks. Here the Terms are stated inversely, as before. Otherwife thus, 42 Horfes: 7 Weeks :: 18 Horfes : 3 Weeks. Then 18 x 7126. And 126423 Weeks. The Anfwer required. Sect. 3. Of Compound Proportion; commonly called The Double Rule of Three. Compound Proportion (as it is here meant) is, when there are five Numbers given to find out a fixth Proportional; and this is generally performed by a Double Pofition; that is, by stating and working the Question at two Operations, either in Direct or Reciprocal Proportion, according as the Queftion requires. And therefore it is called, The Double Golden Rule, or Double Rule of Three. The Double Rule Direct is, when the fixth Term or Number fought, is found by two Operations, both of them in Direct Proportion. Example 1. If a hundred Pounds gain fix Pounds Intereft in twelve Months; how much will three hundred Pounds gain in nine Months, at the fame Rate? 12) 162 (137. 10s. The Anfwer required. I fuppofe the Learner will eafily conceive the Reason of thefe two Operations. For, firft it is plain by Direct Proportion, that if 1001. gain 67. in twelve Months, 300l. will gain 187, in the fame Time, and at the fame Rate, And And by the fame Rule it is plain, that if 12 Months will produce or give 187. Intereft for 300l. then 9 Months must needs give 13 for the fame Sum, viz. 300 1. The Double Rule of Three Inverfe is, when the fixth Term or Number fought is found at two Operations (as before). But one of them requires an Answer in Reciprocal Proportion. Question 2. If 6 Bufhels of Oats will ferve 4 Horfes 8 Days, How many Days will 21 Bufhels ferve 16 Horfes, at the fame Rate of feeding?. This Question being parted into two Pofitions, the firft will be thus: If 6 Bushels of Oats will ferve 4 Horfes 8 Days, How many Days will 21 Bufhels ferve them? Here it is plain, that 21 Bufhels will ferve them longer than 6 Bufhels; therefore the firft Pofition falls in Direct Proportion. 6) 168 (28 Days That is, if 6 Bushels will ferve 4 Horfes 8 Days, 21 Bushels will ferve them 28 Days. The next Pofition must be to find how long the said 21 Bushels will ferve 16 Horfes at the fame Rate of feeding: it is plain, that 21 Bufhels cannot ferve 16 Horfes fo many Days as they will ferve 4 Horses; therefore this fecond Pofition falls in Reciprocal Proportion. Horfes. Days. Horfes. Days. Thus, 4: 28: 16 7 the Answer required. After the like Manner any Question in the Double Rule of Three may be answered by two fingle Pofitions, if Care be taken in ftating them right, viz. Whether their Operation must be performed by the fingle Rule Direct, or Inverse. But all Queftions in this Double Rule, where five Numbers are proposed to find a fixth, may more eafily and readily be answered by one general Theorem; which comprifeth both the Direct and Inverse Rules; without confidering either of them being deduced from the fingle Operations before-going. But first you muft carefully note, that in all Queftions of this Nature, three of the five propofed Terms are always conditional and and fuppofed; and that the other two move the Queftion. As for Inftance in Example 1. Viz. If 100l. will gain 67. in 12 Months: these three Terms are only fuppofed or conditional. Then comes the Question; What will 300. gain in 9 Months? Now, in Order to raise the general Theorem, let us fuppofe, inftead of Numbers, these Letters. P100. The Principal. Viz. Let T= 12. The Time. }{ In the Suppofition of any proposed Question. And, t= 9. The Time. }{ The Principal. The three Terms wherein the Que ftion lies. G = Then P:G::: by the first Extream. That is, 100: 6 :: 300: { The Product of the two Means divided Got That is, the Product of the Extreams is equal to that of the Means. Confequently, Tg P=Gpt is the Theorem. This Theorem affords two Rules, by which all Questions in this Double Rule of Three, or rather of five Numbers, may be refolved; due Regard being had to the true placing down of the proposed Terms, which must be thus: Always place the three conditional Terms in this Order; let that Number which is the principal Caufe of Gain, Lofs, or Action, &c. (viz. P.) be put in the firft Place; that Number which denotes the Space of Time, or Diftance of Place, &c. (viz. T.) be put in the fecond Place. And that Number which is the Gain, Lofs, or Action, &c. (viz. G.) be put in the third Place. Now according to these Directions, the conditional Terms of the laft Queftion will ftand thus; P. T. G. That done, place the other two Terms which move the Queftion, underneath thofe of the fame Name, Thus, P. T. G. 2p. t. Then |