Pan Balance With Weights – 5
A pan balance tells you when its two sides are carrying the same amount of weight or whether one side is heavier than the other.
THE CHALLENGE
You would like to be able to weigh anything that is a whole number of ounces from 1 ounce up to 15 ounces. If you can only put the weights on one side of the pan balance, what is the fewest number of weights that will make this work and what will the weights be?
EXPLORATION
How does your answer change if you want to be able to weigh anything between 1 ounce and 25 ounces? How about other ranges? Are there some ranges where there is only one best answer, and other ranges where there are many?
Notes
THE CHALLENGE
The simplest answer is to always use powers of 2. This uses the ideas behind the base 2 number system, though you do not need to tell your students about that. To be able to weigh all amounts from 1 to 15, use the four weights 1, 2, 4, and 8.
EXPLORATION
For the numbers that are a power of 2 minus 1, there is only one answer – the powers of 2. For other numbers, there is a choice. For example, if you want to cover the numbers from 1 to 14, you can use 1, 2, 4, and 8, but you can also use 1, 2, 4, and 7.
A nice thing about using powers of 2 is that there is always exactly one way to weigh each amount. If you use 1, 2, 4, and 7, then a weight of 7 can be weighed using one weight of 7 or using the three weights 1, 2, and 4