This is a prep question for the National Science Olympiad (sample paper). Four bags were to be weighed but the scale could weigh only weights in excess of 100 kg. If the bags were weighed in pairs and the weights were found to be 103, 105, 106, 107 and 109, then the weight of the lightest bag is  options are (A) 50 kg  (B) 51 kg  (C)  49 kg  (D) 52 kg

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Let a,b,c,d be the weights of the bags in order from least to greatest: A+B = 103 and C+D = 109 and B + D = 107 so C= B+2 So a+c = 105 so a+d must be 106 so d=c+1 So 2c=108 c=54 therefore a = 51 And since a is the lightest, a is 51 The only pair we did not expressly state is b+c. B must be 52 since a+b is 103. So b+c is also 106, so all our totals check out! Recap: a 51 b 52 c 54 d 55
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