Probability: Two-Digit Number In [√85, √850] Range
Hey guys! Let's dive into a fun probability problem. We're going to figure out the chances of picking a two-digit number that falls within a specific range defined by square roots. Sounds a bit tricky, but we'll break it down step by step. So, grab your thinking caps, and let's get started!
Understanding the Problem
Okay, so the main question we need to answer is: What is the probability that a randomly selected two-digit natural number falls within the range of [√85, √850]? To solve this, we first need to understand what the question is asking. Essentially, we're dealing with probability, which is the measure of how likely an event is to occur. In this case, the event is selecting a two-digit number within a certain range. A natural number is a positive whole number (1, 2, 3, ...), and a two-digit number is any number from 10 to 99. The range [√85, √850] involves square roots, which might seem intimidating, but don't worry, we'll simplify them. The square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3 because 3 * 3 = 9.
To calculate probability, we need two things: the number of favorable outcomes (the numbers that fit our criteria) and the total number of possible outcomes (all two-digit numbers). Think of it like this: if you have a bag of marbles, some red and some blue, the probability of picking a red marble is the number of red marbles divided by the total number of marbles. We'll apply the same logic here. First, we'll figure out how many two-digit numbers there are in total. Then, we'll figure out how many of those numbers fall within the range [√85, √850]. Finally, we'll divide the favorable outcomes by the total outcomes to get the probability. So, let's get started with the first step: figuring out the range of numbers we're dealing with and simplifying those square roots. This will give us a clearer picture of what numbers actually fit the criteria.
Determining the Range
Now, let's figure out the range [√85, √850]. This means we need to find the approximate values of these square roots. Grab your calculators, guys, or use your estimation skills! The square root of 85 (√85) is between 9 and 10 because 9 squared (9²) is 81, and 10 squared (10²) is 100. Since 85 is closer to 81, √85 will be closer to 9. A calculator gives us approximately 9.22. Similarly, the square root of 850 (√850) is between 29 and 30 because 29 squared (29²) is 841, and 30 squared (30²) is 900. Since 850 is very close to 841, √850 will be slightly more than 29. A calculator gives us approximately 29.15.
So, the range [√85, √850] is approximately [9.22, 29.15]. Now, here's the crucial part: we're looking for natural numbers within this range. Natural numbers are whole numbers (1, 2, 3, and so on). This means we need to consider the whole numbers that fall between 9.22 and 29.15. The smallest whole number greater than 9.22 is 10, and the largest whole number less than 29.15 is 29. Therefore, the natural numbers in the range [√85, √850] are the whole numbers from 10 to 29, inclusive. This is a significant step because it transforms the problem from dealing with square roots to dealing with a simple range of whole numbers. We've effectively translated the original mathematical expression into something much easier to work with. Now, we need to figure out how many numbers are in this range. This will tell us the number of favorable outcomes for our probability calculation. We're getting closer to the solution!
Counting Favorable Outcomes
Alright, guys, let's count how many numbers are within the range we've established: the natural numbers from 10 to 29, inclusive. This is like counting the numbers on a specific section of a number line. You might be tempted to simply subtract 10 from 29 (29 - 10 = 19), but remember, we need to include both 10 and 29 in our count. Think of it this way: if you were counting the numbers from 1 to 3 (1, 2, 3), there are three numbers, not two (3 - 1 = 2). To get the correct count, we need to add 1 to the difference. So, the number of natural numbers from 10 to 29 is (29 - 10) + 1 = 19 + 1 = 20.
This means there are 20 favorable outcomes – 20 numbers that meet our criteria of being between √85 and √850. These numbers are 10, 11, 12, ..., 28, 29. To double-check, you could even list them out and count them, but the formula (largest number - smallest number) + 1 is a much quicker and reliable method, especially when dealing with larger ranges. Now that we know the number of favorable outcomes, we need to determine the total number of possible outcomes. Remember, we're selecting a two-digit natural number. This means we need to figure out how many two-digit numbers there are in total. This is the next piece of the puzzle, and once we have it, we'll be able to calculate the probability.
Determining Total Possible Outcomes
Now, let's figure out the total number of possible outcomes. We're selecting a two-digit natural number, so we need to determine how many two-digit numbers exist. The smallest two-digit number is 10, and the largest is 99. We can use the same logic we used earlier to count the numbers in the range from 10 to 29. The total number of two-digit numbers is (largest number - smallest number) + 1. So, in this case, it's (99 - 10) + 1 = 89 + 1 = 90. Therefore, there are 90 two-digit natural numbers in total. These numbers are 10, 11, 12, all the way up to 99.
Think of it this way: there are 100 numbers from 0 to 99. We exclude the single-digit numbers (0 to 9), which are 10 numbers in total. So, 100 - 10 = 90 two-digit numbers. Now we have all the pieces we need to calculate the probability. We know the number of favorable outcomes (20 numbers between √85 and √850) and the total number of possible outcomes (90 two-digit numbers). The final step is to divide the favorable outcomes by the total possible outcomes. Get ready, guys, we're about to solve the problem!
Calculating the Probability
Okay, the moment we've been waiting for! Let's calculate the probability. Remember, probability is the number of favorable outcomes divided by the total number of possible outcomes. We've already figured out both of these: We have 20 favorable outcomes (the two-digit numbers between √85 and √850) and 90 total possible outcomes (all two-digit numbers). So, the probability is 20 / 90. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10. 20 divided by 10 is 2, and 90 divided by 10 is 9. Therefore, the simplified probability is 2/9.
This means that there is a 2 out of 9 chance, or approximately a 22.22% chance, that a randomly selected two-digit number will fall within the range [√85, √850]. So, there you have it! We've successfully calculated the probability. To recap, we first determined the range of numbers by simplifying the square roots. Then, we counted the number of favorable outcomes within that range and the total number of possible outcomes. Finally, we divided the favorable outcomes by the total possible outcomes to get the probability. Guys, that was a fun problem! We tackled square roots, ranges, and probability – you've leveled up your math skills today. Remember, probability problems often involve breaking down the question into smaller, manageable steps. Understanding the terminology (like natural numbers and square roots) is crucial, and taking the time to carefully count favorable and total outcomes will lead you to the correct answer.
Conclusion
In conclusion, the probability that a randomly selected two-digit natural number falls within the range of [√85, √850] is 2/9. We arrived at this answer by first simplifying the range using square root approximations, then identifying the natural numbers within that range. We counted 20 such numbers. Next, we determined that there are 90 total two-digit natural numbers. Finally, we calculated the probability by dividing the favorable outcomes (20) by the total possible outcomes (90), simplifying the fraction to 2/9. This problem demonstrates how probability calculations often involve a combination of different mathematical concepts, such as number theory (natural numbers) and algebra (square roots). By carefully breaking down the problem into smaller steps and using appropriate formulas and techniques, we can arrive at a clear and accurate solution. Remember to always double-check your work and ensure that your answer makes logical sense within the context of the problem. Great job, guys! You conquered this probability challenge!