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Ten-balls-were-manufactured-nine-of-them-have-the-same-mass-while-just-one-of-them-has-a-slightly-higher-or-slightly-lower-mass-Given-is-just-a-beam-balance-and-no-weights-comparing-the-masses-of




Question Number 16450 by ajfour last updated on 22/Jun/17
Ten balls were manufactured,   nine of them have the same  mass, while just one of them  has a slightly higher or slightly  lower mass. Given is just a beam  balance and no weights. comparing  the masses of balls only, with the  help of the balance , and in just  3 weighings explain how to judge  which is the defective one and  whether it is heavier or lighter  than the rest (as the case may be).
$${Ten}\:{balls}\:{were}\:{manufactured}, \\ $$$$\:{nine}\:{of}\:{them}\:{have}\:{the}\:{same} \\ $$$${mass},\:{while}\:{just}\:{one}\:{of}\:{them} \\ $$$${has}\:{a}\:{slightly}\:{higher}\:{or}\:{slightly} \\ $$$${lower}\:{mass}.\:{Given}\:{is}\:{just}\:{a}\:{beam} \\ $$$${balance}\:{and}\:{no}\:{weights}.\:{comparing} \\ $$$${the}\:{masses}\:{of}\:{balls}\:{only},\:{with}\:{the} \\ $$$${help}\:{of}\:{the}\:{balance}\:,\:{and}\:{in}\:{just} \\ $$$$\mathrm{3}\:{weighings}\:{explain}\:{how}\:{to}\:{judge} \\ $$$${which}\:{is}\:{the}\:{defective}\:{one}\:{and} \\ $$$${whether}\:{it}\:{is}\:{heavier}\:{or}\:{lighter} \\ $$$${than}\:{the}\:{rest}\:\left({as}\:{the}\:{case}\:{may}\:{be}\right). \\ $$
Commented by prakash jain last updated on 22/Jun/17
10 balls=b_1 ,...,b_(10)   set A=b_1 ,b_2 ,b_3   set B=b_4 ,b_5 ,b_6   set C=b_7 ,b_8 ,b_9   2 weighs compare A&B and A&C   case 1: A=B=C          b_(10)  is fault.          measure b_(10)  with b_1  to find out          if b_(10)  is heavier to lighter.  case 2: one of group is unequal          at we know the defective set          and whether it is heavier to lighter          than the other. Let us say set is A.          compare b_1  and b_2  if equal b_3  is defectivd            if equal b_3  is defectivd            if not equal the heavier or lighter            ball is defective. (we already               identified the nature of defecr)
$$\mathrm{10}\:\mathrm{balls}={b}_{\mathrm{1}} ,…,{b}_{\mathrm{10}} \\ $$$${set}\:\mathrm{A}={b}_{\mathrm{1}} ,{b}_{\mathrm{2}} ,{b}_{\mathrm{3}} \\ $$$${set}\:{B}={b}_{\mathrm{4}} ,{b}_{\mathrm{5}} ,{b}_{\mathrm{6}} \\ $$$${set}\:{C}={b}_{\mathrm{7}} ,{b}_{\mathrm{8}} ,{b}_{\mathrm{9}} \\ $$$$\mathrm{2}\:\mathrm{weighs}\:\mathrm{compare}\:\mathrm{A\&B}\:\mathrm{and}\:\mathrm{A\&C}\: \\ $$$$\mathrm{case}\:\mathrm{1}:\:\mathrm{A}=\mathrm{B}=\mathrm{C} \\ $$$$\:\:\:\:\:\:\:\:{b}_{\mathrm{10}} \:\mathrm{is}\:\mathrm{fault}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{measure}\:{b}_{\mathrm{10}} \:\mathrm{with}\:{b}_{\mathrm{1}} \:\mathrm{to}\:\mathrm{find}\:\mathrm{out} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{if}\:{b}_{\mathrm{10}} \:\mathrm{is}\:\mathrm{heavier}\:\mathrm{to}\:\mathrm{lighter}. \\ $$$$\mathrm{case}\:\mathrm{2}:\:\mathrm{one}\:\mathrm{of}\:\mathrm{group}\:\mathrm{is}\:\mathrm{unequal} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{at}\:\mathrm{we}\:\mathrm{know}\:\mathrm{the}\:\mathrm{defective}\:\mathrm{set} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{and}\:\mathrm{whether}\:\mathrm{it}\:\mathrm{is}\:\mathrm{heavier}\:\mathrm{to}\:\mathrm{lighter} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{than}\:\mathrm{the}\:\mathrm{other}.\:\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{set}\:\mathrm{is}\:\mathrm{A}. \\ $$$$\:\:\:\:\:\:\:\:\mathrm{compare}\:{b}_{\mathrm{1}} \:\mathrm{and}\:{b}_{\mathrm{2}} \:\mathrm{if}\:\mathrm{equal}\:{b}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{defectivd} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\mathrm{equal}\:{b}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{defectivd} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\mathrm{not}\:\mathrm{equal}\:\mathrm{the}\:\mathrm{heavier}\:\mathrm{or}\:\mathrm{lighter} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{ball}\:\mathrm{is}\:\mathrm{defective}.\:\left(\mathrm{we}\:\mathrm{already}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{identified}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{defecr}\right) \\ $$
Commented by prakash jain last updated on 22/Jun/17
Extension: How many measurement are required  if there are 3n+1 balls and  1 is defective.
$$\mathrm{Extension}:\:\mathrm{How}\:\mathrm{many}\:\mathrm{measurement}\:\mathrm{are}\:\mathrm{required} \\ $$$$\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:\mathrm{3}{n}+\mathrm{1}\:\mathrm{balls}\:\mathrm{and} \\ $$$$\mathrm{1}\:\mathrm{is}\:\mathrm{defective}. \\ $$
Commented by ajfour last updated on 23/Jun/17
 Great solving sir, good noticing   ability. Thanks.
$$\:{Great}\:{solving}\:{sir},\:{good}\:{noticing} \\ $$$$\:{ability}.\:{Thanks}. \\ $$

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