We have an equal arms two pan balance and need to weigh objects with integral…
2024
We have an equal arms two pan balance and need to weigh objects with integral weights in the range 1 to 40 kilo grams. We have a set of standard weights and can place the weights in any pan. .(i.e) some weights can be in a pan with objects and some weights can be in the other pan. The minimum number of standard weights required is:
- A.
4
- B.
10
- C.
5
- D.
6
Attempted by 219 students.
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Correct answer: A
Answer: 4 (weights 1, 3, 9, 27).
Key idea: Because weights can be placed on either pan, each standard weight can count as −w, 0, or +w. Choosing weights as powers of 3 lets every integer in a range be represented by a combination of these values.
With weights 1, 3 and 9 you can represent any integer from 1 to 13 because 1 + 3 + 9 = 13 (for example, 2 = 3 − 1).
Adding a 27 weight extends the maximum reachable total to 1 + 3 + 9 + 27 = 40, so every integer from 1 to 40 can be formed.
With only three weights the maximum is 13, which is less than 40, so at least four weights are required. Therefore four weights are both necessary and sufficient.
Hence the minimum number of standard weights required is 4, namely 1, 3, 9 and 27.