Prealgebra Question of the Day

1. $250 \times 2^{15}$
2. $250 \times 5^2$
3. $250 \times 2^5$ (correct answer)
4. $250 + 2^5$

Explanation: When you encounter problems about populations that double, triple, or grow by some factor over time, you're working with exponential growth. The key is identifying how many times the growth occurs and what the growth factor is. Here, the bacteria population doubles every 3 hours, starting with 250 bacteria. To find the population after 15 hours, first determine how many doubling periods occur: $15 ÷ 3 = 5$ doubling periods. Since the population doubles each time, you multiply the initial amount by $2^5$ (2 raised to the power of the number of doublings). This gives you $250 × 2^5$, which is answer choice C. Let's see why the other options miss the mark. Choice A uses $250 × 2^{15}$, which incorrectly assumes the population doubles every hour rather than every 3 hours. This would massively overestimate the growth. Choice B gives $250 × 5^2$, which seems to confuse the number of doubling periods (5) with the growth factor (2) – it's using 5 as the base when 2 should be the base. Choice D shows $250 + 2^5$, which adds rather than multiplies, representing linear growth instead of exponential growth. Remember this pattern: for exponential growth problems, always identify the time intervals first, then count how many growth periods fit into your total time. The formula is: initial amount × (growth factor)^(number of periods). Don't let large time values trick you into using them directly as exponents.