Question 29. Several Merchants enter into Partnerfhip, every one put into the Stock 65 times as many Pounds as there were Partners; with that Stock they traded and gained as many Pounds per 100l. as there were Partners. Now if 107. ros. be added to, and fubftracted from, their Gain, the Product of that Sum and Difference will be 64911. 65. 3 d. Quare, How many Merchants there were, &c. a= the Number of Merchants. Let 11 1 x 65 265 a 2 xa 365 aa And 4 100: a:: 65 a à: by the Queftion. 100 8 × toooo 94225 a6 – iso2500=64913125 9+10 4225 a666015625 104225 11 |a6 = II 6 f2a = 66015625 4225 15625 156255 the Number of Merchants. 12 x 651365a= 325 the Number of Pounds each put in. Question 30. Three Merchants join Stocks together; the firft Man's Stock was less than the fecond Man's by 131. the fecond and third Man's Stock was 1751. in trading they gain 481. more than their whole Stock was; the firft Man's proportional Part of the Gain was 78. What was each Man's Stock and Paft of the Gain? Let a,,y reprefent each Man's Stock. Then{ And and $1 aeys the whole Stock. 3a+13=} by the Question. 30+2+)=175+0 6 and 2 6 and 27s+48=223+a,.. But 8 175+a: 223 +a::a: 78 per Question. 9-78 a 10 aa + 145 a 13650 10 Ċ 112a + 145 a +5256,25 = 18906,25 II w2 12a +72,5✔ 18906,25 = 137,5 12-72,5 13 a 137,572,5 = 65 3, 14 e=a+13=78 4-14 15 y=97 Then 16 65: 78 :: 78: 934 12 5. ➡e's Gain. Proof{ 18 116.8s. +934. 12s. +787. =2881. the Gain. 19 65+78 +97 240. the whole Stock. 18-19/201288-240-48 the Gain more than the Stock. Question 31. A Father at his Death left his three Sons his Money in this manner; to the eldest he gave half of it, wanting 44 Pounds; to the second he gave one third of it, and 14 Pounds more; to the youngest he gave the Remainder, which was less than the Share of the fecond Son, by 82 Pounds: What was each Son's Share? Let a, e, y be the three Shares, and z the whole Sum. 7 x 2 86x=42+32-588 89%=588, the whole Sum that was left. 2 and 9 10 a 14844250, the eldeft Son's Share. 3 and 9 IT 24+14=210, the fecond Son's Share. 4 and 9 12 =242 +14-82 128, the youngest, &c. Question 32. A Man playing at Hazard or Dice, won the firft Throw juft fo much Money as he had in his Pocket, the fecond fecond Throw he won the Square Root of what he then had, and five Shillings more; the third Throw he won the Square of all he then had; after which his whole Sum was 1127. 16s. What Money had he when he began to play? Suppofe | a= his firft Sum. Then X 2 22 a his Sum after the first Throw. 4 @2 And 35+ 2 a the Winnings at the 2d Throw. 42a+5+√2a the Sum after the 2d Throw. 54aa+22a+25 +49√ 20: +102a= the Winnings at the 3d Throw: and therefore 4+5 64aa +24a+30+48√2a+11√2a=2256 Shil. But to avoid thefe Surd Quantities, let us, inftead of fuppofing a the first Sum, make a second Trial, viz. Let 1 x 2 Then 2+3 24 aa the Sum after the first Throw. 32a+5= the Sum won at the zd Throw. 44aa+za+5= his Sum after the 2d Throw. 516a+ +16a3+44aa+20a +25= the Winnings at the 3d Throw; and therefore 4+5 6·16a+ +16a3 +48aa +22a +30=2256 Shil. Yet again, to avoid these high Equations, let us make a third Suppofition; thus, Let I x 2 Then 2+3 Subfti. the first Sum, 2 2a a the Sum after the firft Throw. 3a+5= the Winnings at the 2d Throw. 52 6ee the Winnings at the 3d Throw. Then u 8 9+9,5 =√2256,2547,5 9-0,5 10 e 47 5 and 1011aa+a+5= 47 Note, In refolving of the last Queftion, I have made three different Suppofitions for the Thing fought, purely as an Inftance, to fhew the young Learner how well he ought to confider the Nature of the Queftion, when he firft ftates it, and make Choice of representing the Things fought, fo as to avoid running it into Surds, if poffible, viz. as in the first Suppofition of a the first Sum, &. Not but that fuch Equations may be folved, as fhall be fhew'd in the next Chapter. However, it is moft like an Artift to perform Things of this Nature the nearest and easieft Way they can be done. Question 33. Suppofe there were two equal Circles, whofe Peripheries (viz. Circumferences) are divided into 44310 equal Parts; and that thofe Circles were fo placed upon one Axis, as to move the contrary Way to each other; and suppose one of them to move but one of thefe equal Parts the first Day, two Parts the fecond Day, three Parts the third Day, and fo on in Arithmetical Progreffion, viz. 1, 2, 3, 4, 5, &c. and the other to move every Day the Cube of thofe Parts, 1, 8, 27, 64, 125, &c. of the fame Parts; How many Parts, and how many Days must each Circle move, before the fame two Points meet that were together when they began to move? In order to give a ready Solution to this Question (or any other in this Kind) it will be convenient to premife this Lemma. LEMM A. The Sum of any Series of Cubes whofe Roots are in Arithmetick Progreffion (the firft Term, and common Difference being Unity or 1) is equal to the Square of the Sum of all thofe Roots. As in these Terms in Arith. &c. their Cubes. 2a a the Sum of all the Parts the 1 Circle moves. the Sum of all the Parts the 2d moves Confequen. 3 aa+a=44310 by the Quest. (per Lem. Let a Then 4 ա 23 5-0,5 51+0,5=√ 44310,25 210,5 6a2 210 {the Number of Parts the first Circle muft move. 6@2 7aa=44100 Circle moves. Next to find the Number of Days they moved; there is given the firft Term =1, the common Difference = 1, and the Sum of all the Terms 210, thence to find the laft Term, which in this Cafe is the fame with the Number of all the Terms. Let a the first Term, e1 the common Difference, and S= 210 the Sum of all the Terms, to find y = the laft Term; as per Sect. 1. Chap. 6. Then yy+ey=25+aa-ae by the 16 Step, Page 186. that is, yy+y=210×2=420 &c. Hence y=20 the Number of Days required. I fhall now proceed to give an Example or two of the Method ufed in arguing about unlimited Questions; viz. fuch Questions which admit of various Anfwers, fuch as thofe in Alligation Alternate promised in Page 117. In order to fhorten that Work, it will be convenient for the Learner to know the two Signs of Comparifon, and. The Sign is of eater than; as ba fignifies that b is greater than a. The Sign is of Leffer than; as b、d fignifies that b is leffer than d, &c. EXAMPLE 1. Question 34. A Tobacconist hath three Sorts of Tobacco, viz. one of 2 s. 8 d. the Pound, another of 20 d. the Pound, and a third Sort of 16 d. the Pound; of thefe he would make a Mixture to contain 56 Pound, that may be fold for 22 d. the Pound: How much of each Sort may he take? Let a the Quantity of that worth 32 Pence the Pound, e= that of 20 Pence the Pound, and y = that of 16 Pence the Pound; |