I'^EB. 9, 1883.] 



• KNOWLEDGE • 



93 





<^r inatt)ematttal Column. 



DOUBLE RULE OF TUREE, OR COMPOUND PROPOUTION. 

 By Richard A. Pboctbb. 



T)l E rules commonly piven iii our books of arithmetic for Double 

 Rule of Throe are not bo clear that the student applies them 

 readily and understandingly. I have fonnd in practice- as, for 

 instance, in explaining Coraponnd Proportion to my children — the 

 following reasoning effective : — 



In every case of simple proportion, \vc have to determine what a 

 certain quantity becomes when increased or diminished in a certain 

 degree or ratio represented by two other numbers. Even when 

 composite concrete quantities are given, wc can always represent 

 the ratio of these quantities by numbers, when we have reduced 

 them to the same denomination. (In the very nature of the case 

 they are of the same kind.) 



For instance, suppose we have such a question as this, — 



If 7 net. 2qrs. 3 !b. of somethiny or other cost £2. 10s., )ww iniich 

 will 5 cut. 1 qr. 5 lb. east ? 



We see at once that the dimitmtion of the quantity will diminish 

 the cost in the same ratio or degree ; so that our answer is 

 f 10s X "*' ""<''.^' ll>' IS there are in 5 ewt. 1 qr. 5 lb. 

 as many lb. as there are in 7 cwt. 2qrs. 31b. 



Again, suppose wc have ench a question as this, — 



If a company of men uorkimj 6 h. 15 m. a day do a certain piece of 

 work in 3J days, in what time will the same company, working 

 7 h. 17 in. a day, do the job ? 



Here we see, as readily, that the increase in the time of working 

 per day will diminish the duration of the job in the same ratio or 

 degree ; so that our answer is 



OS J as manv min. as there are in Gh. 15 m. 



ojtdaysx : 



as many min. as there arc in 7 h. 17 m. 



In one case decrea.se brings proportionate decrease, and increase 

 brings proportionate increase ; in the other, increase brings i)r()- 

 portionate decrease, and decrease brings proportionate increase. 

 One is a case of direct proportion, the other a case of inverse pro- 

 portion. But the use of these terms does not make the reasoning 

 simpler or more obvious ; while applying technically-expressed rules 

 is likely to make the student go wrong where, if he were left to his 

 own reasoning, he hardly cnuld do so. 



The case is similar with Double Rule of Three, and with Com- 

 pound Proportion generally : only we get more than one multiplier 

 like the multiplier in each of the above examples. 



Thus, suppose the question asked, — 



If 10 men wfjrkiiig (Uirf. build 17 ft. of a wall, how much of the 

 same kiml of wall will 17 men build, working 5 hrs ^ 



Here it is obvious that the increase in the number of men will 

 tend to increase the length of wall in the same degree ; while the 

 diminution in the hours of working will tend to dimini.s/i the length 

 of wall in the same degree ; so that our answer is 



,,.. 17 5 



1 1 leet X — X - : 

 10 6 

 all we have to do is to see that these multipliers are increasing 

 ones or diminishing ones, according to the obvioua effect of the 

 change of wlu'cli the question tells us. 

 Suppose, instead, the question, — 



If 10 nien working 6hrs. a day build a certain icall in 17 days, in 

 what time will 17 men workiiig 5 hrs. a day build the wall f 



Here it is obvious that the increase in the number of men tends 

 to decrease the time in the same degree, and that the decrease in 

 the hoars of working tends to increase the time in the same degree, 



SO that our answer is 17 days x — • x -. 

 ■^ 17 & 



Again, take this question, — 



If a wall 17 ft. long is built by 10 men in Hhrs., in what lime will 

 a wall 23 ft. long be built by 13 men ? 



Here it is obvious that the increase in the length of the wall 

 mast tend to increase the time in the same ratio, while the ivcrea.-^e 

 in the number of men mnst tend to decrease the time in the 

 same ratio. Thus the answer is 



6hrsx?5xl^ 



17 13 



In the first of these three cases both ratios are direct; in the 



second both are inverse ; in the third one is direct and the other 



inverse. But in none of the three can any doubt or difficulty arise. 



(To be continued.) 



it^m W!Al)iit Column* 



By " FiVK OF Olubs." 



To see the absurdities of the prejudices against it, we need only 

 remark that science is simply a higher d*!Vclopment of common 

 knowledge ; and that if science is repudiated, all knowledge must 

 be repudiated along with it. — Hebbert Spencer. 



Pbopek Leao. — Having no trumps, should the rule [of leaiiing- 

 Ace hold in a very long suit ? 



Hand. A homo game, played for love. My partner dealt. 

 Trump U. 5. Score.— Three all. 



My hand was: — Hearts (tramps) — none; two Spades — Ace, 

 Queen; seven Clubs — Ace, Nine, and small ones ; four Diamonds — 

 Queen, Kuave, Kiue, Seven. 



Trick 1. — Spades led; I took with Qooen. 



Trick 2. — I led Ace of Clubs ; partner played King ; adversaries, 

 small cards. 



Trick 3 — I led a small Club ; partner trumped, and was ovei- 

 trunipod. 



Trick 4. — Spades led again ; my Ace was tramped, and we lost 

 by two tricks. 



We then re-played the hand carefully, leading a small Club at 

 Trick 2, and won by two tricks. My partner held sis tramps, 

 headed by King and Queen. Ace was with leader. 



Should I, having no trumps, have led the Club Ace, or given 

 partner the chance of taking trick and then leading trumps if hc 

 could ? 



P.S. — Not playing by rule had just before led to disaster, thus : — 

 We were (my son and myself) a treble and four on against a single. 

 My partner led a small Diamond. I had Ace, King, and Tliree. 

 I took with the King, and rashly returned him Ace before leading 

 from six trumps. My Ace was trumped, and we lost the odd trick, 

 and next hand lost the game and rubber. Q. T. V. 



[I can see no reason for departing from rule when you led first. 

 There were two chances to one against your partner holding the 

 King. It was unfortunate, of course, that he held the King un- 

 guarded, but that is all that can be said. You could not know that 

 a small Club would turn out well ; it might have turned out very 

 badly. Why, however, did you go on with Clubs Y You were liot 

 only forcing your partner, who alone could have any trump 

 strength to save your game, but forcing him under unfavourable 

 conditions. You should have led a small Diamond. Your game 

 was lost unless your partner had great strength in trumps and 

 good strength in Diamonds. Your right-hand opponent had 

 indicated length in Spades — ])robably great length, the cards being 

 unevenly divided. The wiiming Clubs were with the enemy. 

 Your best chance, after the second I rick, was to lead a Diamond. — 

 Five of Clubs.] 



Learning WniST. — Is it, in your opinion, possible for four unfor- 

 tunate beings, whose whist education has been neglected, by playing 

 together, to learn the scientific game, referring occasionally to 

 Cavendish as to the play F The process of learning by playing 

 with experienced " whistists" is rather too painful to all concerned 

 to recommend itself, save as a last resoiu'ce. Please give us the 

 benefit of yoiu- advice in this matter. Four ok Hearts. 



[Nothing easier. Let the learners agree to study the simple 

 rules of play, and call each other's attention to mistakes as noticed. 

 The art of conducting a game well comes later. Cavendish, how- 

 ever, is rather too technical for beginners. — Five of Clubs.] 



Exposed Card. — A player having an exposed card, not being able 

 to follow suit, contends ho has a right to play any card which may 

 best suit his purpose in lien of the exposed one, nnless the exposed 

 card is within reasonable time called by his ailversaries; is this so or 

 not y If so, it then follows the offending player may take advan- 

 tage, and can dispute what is " reasonable time." Queki.st. 



[In pra<tiee no ditliculty need occur. If the exposed card is not 

 at once called when it is the owner's turn to play, he should wait to 

 ascertain whether the opponents wish to call it. But he has no 

 more right to hnrry their decision than a player has to hurry any 

 one in playing to a trick. There is no law forbidding anyone tu 

 wait an unreasonable time before playing. Neither is there any 

 law forbidding players to wait an unreasonable time before deciding 

 whether to call or not. In practice, no such law is necessary. But 

 the player who, having an exposed card, plays without waiting 

 to know whether it is called, is no more "in his right" than one 

 who, getting impatient, plays out of turn. — Five of Clubs.] 



M. G. B. — Your opponents, who, when the score was two all and 

 yoji made two by tricks and two by honours, denied you the game 

 on the plea that tricks count before honours, so that yuur score was 

 made four by your two tricks, and you could not count honoiu's, 

 know — let us hope — very little of the game. If they do know the 

 game, and were not joking, my advice to you is not to play with 

 them any more. — Five of Clubs. 



