Fkb. 16, 1883.] 



♦ KNOWLEDGE ♦ 



107 



w 





^ir iflatt)tmatical Column. 



COMPOLXD PROPORTIOX. 

 (Concluded from p. 93.) 

 E mav, however, change the way of arranging otr numbers 

 for multiplying and dividing — though I do not think we im- 

 prove them — so as to be more in accordance M'ith what is taught 

 in the books and expected (I suppose) in examinations. The nume- 

 rators in our fractions are of course what we multiply by, in effect, 

 and the denominators are what we divide by. Thus, instead of 

 pntting down fractions as we did, with numerator at the top and 

 denominator at the bottom, we may set all our multipliers, or what 

 wonld be the numerators in the other arrangeni*'nt, in one column, 

 and all our divisors in another ; while we may set to the right of both 

 columns the ([uantity which has thus to be multiplied and divided. 

 Thus our tirat sum in compound proportion might be arranged 



17 ft. 



6 hrs. 



Tliis is the usual way of arranging the figures ; and though I 

 prefer the other, as more in accordance with common sense and 

 ■with practice in other cases, it is easy to apply common sense to 

 this arrangement too. Thus, we first write Mult, over the multi- 

 plying nniubei-s, and Div. over the dividing numbers, — say in first 

 question. Then we reason thus : — changing 10 men to 17 will 

 increase the work dorie in tliix deijiee, so that we must inulliphi by 

 the larijer and divide by the i^maller number ; so we set 17 under 

 Mnlt. and 10 under Div. Again, changing G hrs. to 5 hrs. of working 

 will tend to diminish the work done in the same degree, therefore 

 we must multiply by the sviaUer and divide by the larger number ; 

 so wc put 5 under. Mult, and G under Div. Smce, however, our 

 answer is thus seen to be 



, 17x5 

 10x6 



it seems better to write it down so at once, pntting 17 and 5 above 

 the line, 10 and 6 below, instead of under tbe respective headings 

 of the other arrangement. 



The same method applies to every case of double, triple, or 

 multiple rule of three, or compound proportion. Xo technically- 

 worded rule whatever is wanted, but the simple consideration of 

 the effect which the various changes indicated in the question must 

 produce. 



I take a somewhat less simple problem, to show the working of 

 this common-sense process of reasoning. I do not use columns, 

 though those who are going in for examination may use that plan 

 as more commonly known to examiners. I simply set all the 

 multipliers ahnve a line, and all the dividers vnderthc line as in the 

 tisnal way of treating the multiplication of fractions or ratios. 



Suppose the following problem, — 



Problem. If 7 men working 5 hrg. 15 m. a day for 16 days dig 13i 

 acres, in what time will 5 «ien, working 6hrs. 18m. o day, dig 12^ 

 Mres. 



Noting that 5h. 15m. = 315m., 6h. 18m. = 378m. 

 13i = 13t% = '^ ; and 12^ = 12^^ = V';^ 

 our answer is 



,„, 7x315x148 



16 days x ' 



' 5 X 378 159 

 We write down first the 16 days, becanse it is about days that our 

 problem deals ; we see that reducing the number of men from 7 to 5 

 will increase the time in the same degree, so we put 7 adore the line 

 and 5 below; we sec that increasing the number of minutes from 

 315 to 378 will diminish the time in the same degree, so we put 315 

 <j6ofe the line and 378 below ; and lastly, we see that diminishing 

 the nnmbcr of 12ths of an acre from 159 to H8 will in the same 

 degree diminish the time, so we put 148a6oie the line and 169 belnv. 



Reducing, we find our answer to be 



16x63x148 8x7x118 ,-,-„,,.„ 



= — l/iiSdav'S. 



54x159 3x159 ■•■'■' 



And with the same ease and certainty of going right can all 

 problems in compound proportion be solved, if common sense be 

 applied to determine whether each several change lends to increase 

 or to dtn.iHi,-/! the result sought, the largerot the two numbers being 

 put above the line in the case of increase and below the line in the 

 case of decrease. 



: 17 ft. 



&UV ©Kbist Column. 



By " Five of Clubs." 



B. 



Hca rts — Xone. 

 Clubs— A, 10, 9, 7, 4, 3, 2 

 Diamonds — Q,Kn, 9, 7. _. 

 Spades— A, Q. • Y 



A. 



Hearts— A. 



Clubs— 8. 



Diamonds — A, 10, 8, G, 3. 



Spades— K, 10, 9, 6, 3, 2. 



<? 9 



Hearls—Kn. 9,8,6,3,2. 

 Clubs— Q, Kn, G, 5. 

 Diamonds — 1, 3. 

 B\. Spades — 5. 



Z. 



//,«i(s—K,y, 10, 7,5,4. 

 Clubs— K. 

 Diamonds— K, 2. 

 Spades— Kn, 8, 7, 4. 



NOTES ON THE PLAY. 



1. .4 leads the anti-penultimate. 

 Z begins a signal. 



2. r properly leads the Ace of 

 his long suit. S begins a signal. 



3. 1' changes his suit. To force 

 his partner by continuing Clubs 

 would be contrary to an important 

 Whist principle, and manifestly 

 dangerous. B bfgins to signal in 

 Diamonds. 



4. Z completes his signal, but 



5. B properly leads trumps, de- 

 spite the signal, having held six 

 originally- The chances are against 

 Z holding more than five, and B is 

 leading through his strength. Z 

 signals in trumps after opponent's 

 lead, and plays the ante-penulti- 

 mate. He, as it were, shouts his 

 strength in trumps. 



G. .4's not returning trumps may 

 not, after Z's play at trick 5, mean 

 that he has none left. But to all 

 the others it should be clear that 

 the chances are he has none. Y 

 rightly discards a Club, as B holds 

 both the best. Z knows I's suit. 



7. If .4 leads a spade here, forc- 

 ing his partner, and B then un- 

 wisely resumes the trump lead, 



Y Z will make two by tricks (for 

 Z can safely finesse trump 10). 

 ]3ut if B, being e.o forced, force 

 in return with Club Queen, Y Z 

 will only make the odd trick. A 

 rightly leads a Diamond, not only 

 to avoid (if possible) forcing his 

 partner, but because he leads 

 through Y'b strength. 



8. B properly refrains from lead- 

 ing trumps again. 



The rest of the game plays itself; 



Y Z cannot prevent A B from 

 making one more trick. 



• The game described in part by 

 " Q. T. ^'.," at p. 93, suggests an 

 interesting and instructive illustra- 

 tive game, showing how Y Z, who 

 lost two by tricks when 1' forced 

 his partner at trick 3, would have 

 won the odd trick had 1' then led 

 a Diamond : and two by tricks un- 

 less A B played correctly. We 

 have fairly distributed the other 

 cards, giving to B six trumps as 

 the best way of balancing Z's 

 strength in trumps, and giving A 

 the Ace of Diamonds, Z the King. 

 It is noteworthy how much signal- 

 ling there is of all kinds. 



