Nap Time At The Retirement Home: Math Problem!

Hey guys! Let's dive into a fun little math problem about nap times in a retirement home. This one involves fractions, so get ready to dust off those fraction skills! We're figuring out how many of the residents didn't take a nap, based on some info about the napping habits of the others. So, let's break it down and solve this together!

Understanding the Nap Situation

So, here's the scoop: In this cozy retirement home, we've got some interesting napping patterns among the residents. Specifically, half of the residents (both men and women) enjoyed a nice, long nap of more than one hour. Sounds relaxing, right? Then, we have another group: three out of every eight residents preferred a shorter snooze, clocking in at less than an hour. Finally, we know that there are 12 residents who decided to skip nap time altogether. Our mission, should we choose to accept it, is to figure out what fraction of the total residents these non-nappers represent.

Now, before we start crunching numbers, let's think about what this tells us. We have three groups of residents: long nappers, short nappers, and non-nappers. We know the fraction of long nappers (1/2) and short nappers (3/8), and we know the exact number of non-nappers (12). To find the fraction of non-nappers, we'll first need to determine the total number of residents. This will involve a bit of fraction manipulation and some good old-fashioned problem-solving. So, grab your thinking caps, and let's get started!

Setting Up the Math

Okay, let's get our math hats on and start setting up the problem. We know that half of the residents took a long nap, which we can represent as 1/2. We also know that three out of eight residents took a shorter nap, represented as 3/8. And, of course, we know that 12 residents skipped their naps altogether. The key here is to figure out what fraction of the total residents these 12 non-nappers represent. To do that, we first need to find the fraction of residents who did nap. We can do this by adding the fractions of the long nappers and the short nappers.

So, we add 1/2 + 3/8. To add these fractions, we need a common denominator. The least common multiple of 2 and 8 is 8, so we'll convert 1/2 to have a denominator of 8. That means 1/2 becomes 4/8. Now we can add: 4/8 + 3/8 = 7/8. So, 7/8 of the residents took a nap (either a long one or a short one). This means that the remaining fraction of residents did not nap. Since the whole group of residents is represented by the fraction 1 (or 8/8), we can subtract the fraction of nappers from 1 to find the fraction of non-nappers. That's 1 - 7/8 = 1/8. Aha! So, 1/8 of the residents did not take a nap. And we know that this 1/8 represents 12 people. Now we're getting somewhere!

Cracking the Code: Finding the Total

Alright, let's keep this train rolling! We've figured out that 1/8 of the total residents didn't nap, and we know that this fraction corresponds to 12 residents. So, the equation to find the total number of residents looks like this: (1/8) * (Total Residents) = 12. To find the total number of residents, we need to isolate "Total Residents" on one side of the equation. We can do this by multiplying both sides of the equation by 8. Why 8? Because multiplying (1/8) by 8 gives us 1, which leaves us with just "Total Residents" on the left side of the equation. So, here's the math: 8 * (1/8) * (Total Residents) = 8 * 12. This simplifies to Total Residents = 96. Ta-da! We now know that there are a total of 96 residents in the retirement home.

Now that we know the total number of residents, we can double-check our work and make sure everything adds up correctly. We know that 1/2 of the residents took a long nap. Half of 96 is 48, so 48 residents took a long nap. We also know that 3/8 of the residents took a short nap. To find 3/8 of 96, we can multiply (3/8) * 96. This equals 36. So, 36 residents took a short nap. And we know that 12 residents didn't nap at all. Let's add those numbers up: 48 (long naps) + 36 (short naps) + 12 (no naps) = 96. Bingo! It all adds up. This confirms that our calculations are correct and we have accurately determined the total number of residents in the retirement home.

The Final Answer

Okay, guys, let's bring it all home and nail this question! The original question was: What fraction of the residents did not take a nap? We've already done the heavy lifting, so this is the easy part. We know that 1/8 of the residents did not nap, and we also know that this represents 12 residents out of a total of 96. So, the fraction of residents who did not take a nap is 1/8. Woohoo! We solved it! This problem walked us through fractions, figuring out totals, and using those totals to confirm our calculations. Math can be sneaky, but we tackled it head-on and came out victorious. Great job, everyone!

So, to recap: In the retirement home, half the residents took a long nap, three-eighths took a short nap, and 12 skipped nap time. After working through the math, we found that these 12 residents represent 1/8 of the total residents. Thus, the answer to the question, "What fraction of the residents did not take a nap?" is 1/8. Go us! Now, who's up for a nap?