A man selling fruit has only three weights. But with them he can weigh any whole number of kilograms (from 1 kg to 13 kg) inclusive on his balance.
What weights does he have?
Responses
math - Kalyan, Friday, September 5, 2008 at 4:38am
The man had 1kg, 3kg and 9 kg weights.
How do you set it up to get this answer? Any ideas?
You can prove to yourself that 1,3 and 9 work. With them, you can form counterbalances of
1, 2 (3-1), 3, 4 (3+1), 5 (9-1-3),6 (9-3), 7 (9+1-3), 8 (9-1), 10 (9+1), 11 (9+3-1), 12 (9+3) and 13 (9+3+1)
A minus sign means putting the weight on the same side of the balance as the weighed object.
Proving that other combinations will not work is harder, but is not necessary.
To determine the three weights that the man has, we can approach it by trial and error.
1. Start with the smallest possible weight, which is 1kg. Place it on one side of the balance.
- If the other side of the balance is empty, it means that the weight on the left side is 1kg.
- If the other side of the balance is not empty, it means that the weight on the left side is either 2kg or 3kg.
2. If the weight on the left side is 2kg, this means that the weight on the right side can only be either 1kg or 3kg.
- If the right side has a weight of 1kg, it means the weight on the left side is 2kg.
- If the right side has a weight of 3kg, it means the weight on the left side is 1kg.
3. If the weight on the left side is 3kg, this means that the weight on the right side can only be either 1kg or 2kg.
- If the right side has a weight of 1kg, it means the weight on the left side is 3kg.
- If the right side has a weight of 2kg, it means the weight on the left side is 1kg.
4. If the weight on the left side is neither 2kg nor 3kg, it means that the weight on the left side is 9kg.
By going through this trial and error process, we can see that the man has weights of 1kg, 3kg, and 9kg.
To find the weights the man has, we can use a method called the "balance scale method." We want to find three weights that can be combined to weigh any whole number of kilograms from 1 kg to 13 kg inclusive.
1. Start by placing the 1 kg weight on one side of the balance scale.
2. On the other side, add the 3 kg weight. This balances the weights at a total of 4 kg.
3. Now, remove the 1 kg weight and add the 9 kg weight on the same side as the 3 kg weight.
4. The total weight on the side with the 9 kg weight now becomes 12 kg.
5. Finally, remove the 3 kg weight and place the 1 kg weight back on the same side as the 9 kg weight.
6. Now, the total weight on this side becomes 10 kg.
By following these steps, we can see that by using the 1 kg, 3 kg, and 9 kg weights, the man can measure any whole number of kilograms from 1 kg to 13 kg inclusive.
So, the weights the man has are 1 kg, 3 kg, and 9 kg.