By how many orders of magnitude is 150,000 larger than 50,000?
The actual question is "A large state like Montana might have an area of about 150000 square miles a small state like massachusetts might have an area of about 50000 square miles by how many orders of magnitude is the large state larger than the small state?" Possible answers are 0 through 5. I thought that order of magnitudes were powers of 10 so I keep thinking it must be 0 because one magnitude larger thann 50,000 should be 500,000, right?
The phrase "order of magnitude" is really a way of talking about the relative numbers of digits between two different numbers. In general, the order of magnitude is the number of digits between the leading digit and the decimal place. It can also be thought of as the base 10 exponent of any given number. For example, the number 345 has an order of magnitude of 2 since there are two digits (4 and 5) between the leading digit (3) and the unwritten decimal place at the end. The number 4,586 has an order of magnitude of 3. So there is one order of magnitude difference between 345 and 4,586.
So following this same logic, 150,000 has an order of magnitude of 5. 50,000 has an order of magnitude of 4. So the answer to the question is 1. This makes sense since 150,000 has one extra digit place compared to 50,000 (one hundred thousands versus ten thousands).