What Time Is It When The Time Passed Is 7/5 Of The Time Remaining In The Day?

Are you ready to dive into a fascinating time-related puzzle, guys? We've got a real brain-teaser on our hands, a classic riddle that blends math and our everyday sense of time. This isn't just about telling time; it's about understanding proportions and how they play out across the 24 hours of a day. So, buckle up, because we're going to unravel this question: "What time is it when the elapsed portion of the day is equal to 7/5 of what remains of the day?"

Understanding the Problem

Before we jump into calculations, let's break down what the problem is really asking. The core concept here is to find the exact moment in a day when the time already gone by is precisely 7/5 of the time that's still left. Think of it like slicing a pie, but instead of equal pieces, one slice is significantly larger than the other – specifically, 7/5 the size. To tackle this, we need to use our understanding of fractions and apply it to the 24-hour cycle of a day. We are essentially dividing the day into two parts, the "elapsed time" and the "remaining time," with a very specific proportional relationship between them.

It's super important to visualize this. Imagine a clock face. As the hours tick by, the elapsed time increases, and the remaining time decreases. Our goal is to pinpoint that precise moment when the ratio of these two segments hits 7/5. This involves a bit of algebraic thinking, but don't worry, we'll walk through it step by step. We need to translate this word problem into a mathematical equation that we can solve. The key is to represent the unknowns with variables and then set up a relationship that reflects the given condition.

Also, remember that we're dealing with a continuous cycle of time. A day starts at midnight and wraps around back to midnight. This cyclical nature is crucial because it means that the "end" of the day is also the "beginning" of the next. So, when we calculate the time, we need to keep in mind that we're working within this 24-hour framework. This kind of problem is not just a mathematical exercise; it's a great way to sharpen your logical thinking and problem-solving skills in general. These are the kinds of challenges that pop up in various fields, from physics to computer science, so getting comfortable with them is a smart move.

Setting Up the Equation

Now, let's get down to the nitty-gritty and translate our time riddle into a workable equation. This is where the magic happens! The first thing we need to do is assign variables. Let's use 'x' to represent the remaining time in the day (in hours). This is the portion of the day we haven't lived through yet. Makes sense, right? Now, according to the problem, the elapsed time, which is the time we have lived through, is 7/5 of the remaining time. So, we can express the elapsed time as (7/5)x.

Think of 'x' as the base unit, and the elapsed time is a fraction of that unit. This allows us to directly relate the two time periods mathematically. Remember, the total time in a day is 24 hours. This is a crucial piece of information because it allows us to create an equation that links the elapsed time and the remaining time. The sum of these two times must equal the total duration of the day. Therefore, we can write the equation as:

(Elapsed Time) + (Remaining Time) = 24 hours

Substituting our expressions, we get:

(7/5)x + x = 24

This is our key equation! It beautifully encapsulates the relationship described in the problem. We have a single equation with one unknown variable ('x'), which means we're well on our way to finding the solution. This step of translating a word problem into a mathematical equation is a fundamental skill in problem-solving, not just in math but in many areas of life. It's about taking a real-world scenario and expressing it in a precise, quantifiable form. Once we have this equation, we can use algebraic techniques to isolate 'x' and determine the remaining time. From there, we can easily calculate the elapsed time and, ultimately, the actual time of day. So, let's keep rolling!

Solving the Equation

Alright, let's roll up our sleeves and solve this equation! We've got (7/5)x + x = 24. Our mission now is to isolate 'x' and find out its value. The first thing we want to do is combine the 'x' terms on the left side of the equation. To do this, we need a common denominator. Think back to your fraction basics! We can rewrite 'x' as (5/5)x. This gives us:

(7/5)x + (5/5)x = 24

Now we can add the fractions because they have the same denominator:

(12/5)x = 24

We're getting closer! Now, to isolate 'x', we need to get rid of that (12/5) coefficient. The way we do that is by multiplying both sides of the equation by the reciprocal of (12/5), which is (5/12). Remember, whatever we do to one side of the equation, we have to do to the other to keep it balanced. So:

(5/12) * (12/5)x = 24 * (5/12)

The (5/12) and (12/5) on the left side cancel each other out, leaving us with just 'x'. On the right side, we have 24 * (5/12). We can simplify this by dividing 24 by 12, which gives us 2. Then, we multiply 2 by 5, which gives us 10. So:

x = 10

Boom! We've found that 'x', the remaining time in the day, is 10 hours. This is a major step. But we're not quite done yet. We need to use this value to figure out the actual time of day. This process highlights the power of algebra in solving real-world problems. By translating the problem into an equation, we were able to systematically work towards the solution. Each step was a logical progression, building upon the previous one. Now that we know the remaining time, we can easily find the elapsed time and answer the original question.

Calculating the Time

Okay, we've cracked the code and figured out that the remaining time in the day is 10 hours. Fantastic work, guys! But what time is it actually? This is the final piece of the puzzle. Remember, 'x' represents the remaining time, and we know that the elapsed time is (7/5)x. Now that we know x = 10, we can calculate the elapsed time:

Elapsed Time = (7/5) * 10

To simplify this, we can divide 10 by 5, which gives us 2. Then, we multiply 7 by 2, which gives us 14. So:

Elapsed Time = 14 hours

So, 14 hours have passed since the beginning of the day. The day starts at midnight (12:00 AM). To find the current time, we simply add 14 hours to midnight. Midnight plus 14 hours is 2:00 PM. This is our answer! The time is 2:00 PM when the elapsed portion of the day is 7/5 of what remains.

This calculation demonstrates how we can use the value of 'x' we found earlier to answer the original question. By understanding the relationship between elapsed time and remaining time, we were able to easily determine the specific time of day. This whole process is a testament to the power of mathematical reasoning. We started with a word problem, translated it into an equation, solved the equation, and then used the solution to answer the original question. It's a beautiful example of how math can help us understand and solve real-world problems. Now, let's recap everything we've done to make sure we've nailed it.

Final Answer and Recap

So, drumroll, please… The final answer is 2:00 PM! That's the magical moment when the elapsed time is exactly 7/5 of the remaining time in the day. We've solved it, guys! Let's take a quick stroll down memory lane and recap how we arrived at this answer. First, we carefully analyzed the problem to understand the relationship between elapsed time and remaining time. We recognized that the elapsed time was a fraction (7/5) of the remaining time.

Then, we translated the word problem into a mathematical equation. We used the variable 'x' to represent the remaining time, expressed the elapsed time as (7/5)x, and set up the equation (7/5)x + x = 24, knowing that the total time in a day is 24 hours. Next, we put on our algebra hats and solved the equation for 'x'. We combined like terms, multiplied by the reciprocal, and found that x = 10 hours. This told us that there were 10 hours remaining in the day. But we weren't done yet! We needed to find the elapsed time, which we calculated as (7/5) * 10 = 14 hours. Finally, we added the elapsed time to midnight (the start of the day) to find the actual time of day, which turned out to be 2:00 PM.

This problem is a classic example of how math can be used to solve everyday puzzles. It required us to combine our understanding of fractions, algebra, and the concept of time. The process we used – translating the problem into an equation, solving the equation, and interpreting the result – is a powerful problem-solving strategy that can be applied to many different situations. So, the next time you encounter a time-related riddle, remember this journey. You've got the skills to crack it! And that, my friends, is something to be proud of.