Gym Time Math: Finding Machine Workout Duration

Hey guys! Ever wonder how long you actually spend on each machine at the gym? Today, we're diving into a cool math problem that breaks it down. We've got Leo here, who hit the gym and wants to figure out how much time he dedicated to each of his five weight machines. He also hopped on the treadmill for a bit. If we know his total gym time, we can totally solve for the time spent on each weight machine. This is gonna be a fun one, so let's get to it!

Setting Up the Problem: What Do We Know?

Alright, let's break down what Leo did at the gym. First off, he used five different weight machines. The key here is that he spent the same amount of time on each one. Let's call this unknown amount of time '$x$' hours. So, for each of those five machines, he spent '$x$' hours. That means the total time he spent on all the weight machines combined is $5$ times '$x$', or $5x$ hours. Pretty straightforward, right? Now, after crushing it on the weights, he decided to hit the treadmill. He spent a solid rac{2}{3} of an hour on the treadmill. This is a specific amount of time we know. Finally, we know his total time spent at the gym. He was there for a grand total of 1 rac{3}{4} hours. This total time includes the time on the weight machines and the time on the treadmill. Our mission, should we choose to accept it, is to figure out the value of '$x$', which represents the time spent on each individual weight machine.

To make things even clearer, let's convert those mixed numbers into fractions. 1 rac{3}{4} hours is the same as $(1 imes 4 + 3) / 4$, which equals rac{7}{4} hours. So, Leo spent a total of rac{7}{4} hours at the gym. We've got the time on weights ($5x$), the time on the treadmill (rac{2}{3} hours), and the total gym time (rac{7}{4} hours). Now, how do we put it all together to find '$x$'?

Crafting the Equation: Putting the Pieces Together

So, how do we turn all this info into a mathematical equation? It's like building with LEGOs, guys! We know that the total time spent at the gym is the sum of the time spent on the weight machines and the time spent on the treadmill. We've already figured out that the time spent on the five weight machines is $5x$ hours, and the time on the treadmill is rac{2}{3} hours. The total gym time is rac{7}{4} hours. Therefore, we can write our equation like this: Time on Weight Machines + Time on Treadmill = Total Gym Time. Substituting our values, we get: 5x + rac{2}{3} = rac{7}{4}. This is our equation, the one that holds the secret to finding out how long Leo was on each weight machine. It beautifully represents the situation, combining the unknown variable '$x$' with the known values to equal the total time. This is where the magic happens in algebra, turning a word problem into something we can solve step-by-step.

This equation is super important because it's the foundation for finding our answer. It shows that the combined effort on the weights ($5x$) plus the cardio session (rac{2}{3}) equals the entire duration of his gym visit (rac{7}{4}). Without this correctly formed equation, any attempt to solve for '$x$' would be based on a flawed premise. We've meticulously translated the word problem into mathematical terms, ensuring accuracy. The structure of the equation (5x + rac{2}{3} = rac{7}{4}) is crucial: the term with the variable ($5x$) represents the combined unknown time, the constant term (rac{2}{3}) represents the known treadmill time, and the total (rac{7}{4}) represents the overall duration. It's a perfectly balanced equation, ready for us to manipulate and isolate '$x$'.

Solving for $x$: The Algebraic Adventure

Now for the exciting part – solving for '$x$'! We have our equation: 5x + rac{2}{3} = rac{7}{4}. Our goal is to get '$x$' all by itself on one side of the equation. To do this, we first need to isolate the term with '$x$' ($5x$). We can do this by subtracting rac{2}{3} from both sides of the equation. Remember, whatever you do to one side, you have to do to the other to keep the equation balanced. So, we get: 5x = rac{7}{4} - rac{2}{3}.

Before we can subtract the fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. So, we'll convert both fractions: rac{7}{4} becomes rac{7 imes 3}{4 imes 3} = rac{21}{12}, and rac{2}{3} becomes rac{2 imes 4}{3 imes 4} = rac{8}{12}. Now we can subtract: 5x = rac{21}{12} - rac{8}{12}. This simplifies to 5x = rac{13}{12}.

We're almost there! Now we have $5x$ equaling rac{13}{12}. To find '$x$' alone, we need to divide both sides by 5. Dividing by 5 is the same as multiplying by its reciprocal, which is rac{1}{5}. So, x = rac{13}{12} imes rac{1}{5}. Multiplying the numerators gives us $13 imes 1 = 13$, and multiplying the denominators gives us $12 imes 5 = 60$. Therefore, x = rac{13}{60}.

This value, x = rac{13}{60}, represents the time in hours Leo spent on each of the five weight machines. Isn't that cool? We took a real-world scenario, turned it into an equation, and solved for the unknown. It's a testament to how math helps us understand and quantify our daily activities. The process involved isolating the variable step-by-step, using fraction arithmetic, and finally arriving at a concrete answer. It’s about careful manipulation of numbers and understanding the rules of algebra. We tackled common denominators, subtraction of fractions, and division by a whole number, all leading to the final value of $x$. This solution is precise and directly answers the question posed by Leo's gym session.

The Answer: Leo's Machine Time Revealed!

So, what does this x = rac{13}{60} actually mean in terms of time? Remember, '$x$' is in hours. So, Leo spent rac{13}{60} of an hour on each of the five weight machines. To get a better sense of this, we can convert it into minutes. There are 60 minutes in an hour, so rac{13}{60} of an hour is simply 13 minutes. That's right, guys, Leo spent exactly 13 minutes on each of the five weight machines!

Let's do a quick check to make sure our answer is correct. The total time on the weight machines would be 5 times the time spent on each machine: 5 imes rac{13}{60} hours. This simplifies to rac{65}{60} hours. Now, add the treadmill time: rac{65}{60} + rac{2}{3} hours. To add these, we need a common denominator, which is 60. So, rac{2}{3} becomes rac{2 imes 20}{3 imes 20} = rac{40}{60}. Adding them gives us rac{65}{60} + rac{40}{60} = rac{105}{60} hours.

Now, let's simplify rac{105}{60}. Both numbers are divisible by 15. $105 imes 15 = 7$ and $60 imes 15 = 4$. So, we get rac{7}{4} hours. And hey, remember when we converted Leo's total gym time? It was 1 rac{3}{4} hours, which is indeed rac{7}{4} hours! Our calculation matches the total time given in the problem, so our solution for '$x$' is spot on. This confirmation step is super crucial in math – it tells us we've done our homework right!

Conclusion: Math in the Gym and Beyond

See? Math isn't just for textbooks; it's everywhere, even at the gym! We've successfully figured out that Leo spent 13 minutes on each of the five weight machines, plus his rac{2}{3} hour (or 40 minutes) on the treadmill, totaling 1 rac{3}{4} hours (or 105 minutes). This problem demonstrates how algebra can be used to solve practical, real-world situations. By setting up the correct equation, 5x + rac{2}{3} = rac{7}{4}, and carefully solving for '$x$', we unlocked the answer. It's a great example of how understanding variables, fractions, and algebraic manipulation can help us quantify and understand our daily routines. So next time you're at the gym, think about the math involved – you might be solving a problem without even realizing it! Keep practicing, and you'll become a math whiz in no time. Stay awesome, guys!