Bike Rental Math: Solve Sheenah's P20/hr + P10 Fee Problem

Hey math whizzes and budget-savvy folks! Today, we're diving into a real-world problem that's super common: figuring out costs when you rent something. We've got our friend Sheenah here who decided to rent a bike, and like most rentals, there's a base fee and an hourly charge. It’s a classic scenario, and guys, it’s a fantastic way to practice using inequalities to solve problems. We'll break down exactly how to set up the math and find out just how long Sheenah could have enjoyed her bike ride without breaking the bank. So, buckle up (or, you know, put on your helmet!), because we're about to make math make sense!

Understanding the Bike Rental Scenario

Alright guys, let's get into the nitty-gritty of Sheenah's bike rental adventure. Imagine this: you walk up to a bike rental shop, all excited for a nice ride. The rental place has a deal: you pay a flat fee just to get the bike, and then there's an hourly rate for how long you actually use it. For Sheenah, this meant a P10 fee just to start the rental, plus P20 for every single hour she had the bike. Now, here’s the kicker – she had a budget, and she knew she paid less than P270 in total. This is where the fun begins! We need to figure out what this all means mathematically. That P10 fee is a one-time thing, no matter if she rented for 1 hour or 10 hours. The P20 per hour, though? That’s the variable part, the part that changes based on how long she rides. This is crucial for setting up our inequality. We're not just looking for a single answer; we're looking for a range of possibilities. She paid less than P270, not exactly P270. This little detail is super important in math, as it tells us we're dealing with an inequality, not a simple equation. It means there are many possible numbers of hours she could have rented the bike, as long as the total cost stays under that P270 limit. So, before we even write down a single number, let’s think about what we’re trying to find. We want to know the maximum number of hours Sheenah could have rented the bike. This implies there's a limit, a ceiling on her rental time, dictated by her budget. It's like saying, "How many slices of pizza can I buy if I have $10 and each slice is $2?" You can buy 5, but you can't buy 6 if you only have $10. This bike rental is exactly the same concept, just with a few more numbers and a fixed fee thrown in. We need to translate words into math, which is one of the most powerful skills you can learn. Get ready, because we're about to turn this story into numbers and symbols!

a. Writing the Inequality: Translating Words to Math

Okay guys, let's tackle the first part: writing an inequality that perfectly describes Sheenah's situation. This is where we translate the story into the language of mathematics. Remember, an inequality is like an equation, but instead of saying two things are exactly equal, it says one thing is greater than, less than, greater than or equal to, or less than or equal to another thing. Here, we know Sheenah paid less than P270. This immediately tells us we'll be using a "less than" symbol (<). Now, what makes up the total cost? First, there's that fixed P10 fee. That's a constant, it doesn't change. Then, there's the cost based on time. She pays P20 for each hour. If we let 'h' represent the number of hours Sheenah rented the bike, then the cost from the hourly rate is P20 multiplied by 'h', which we write as 20h. So, the total cost Sheenah paid is the fixed fee plus the hourly cost: P10 + 20h. Since we know this total cost is less than P270, we can put it all together. The inequality that represents this situation is: 10 + 20h < 270. See how that works? We've taken the story – the P10 fee, the P20 per hour, and the total less than P270 – and turned it into a concise mathematical statement. The 'h' is our unknown, the number of hours we want to find. This inequality is the key to unlocking the rest of the problem. It's the mathematical blueprint for Sheenah's rental cost. We're saying that the P10 initial charge, added to P20 for every hour of rental (20h), must always be a value that is strictly smaller than P270. This covers all the conditions mentioned in the problem statement. We’re not saying it’s equal to P270, because the problem explicitly states less than P270. This is a super important distinction in algebra. If it had said "at most P270" or "P270 or less," we would use the "less than or equal to" symbol (≤). But since it's strictly "less than," we use the '<' symbol. This inequality is the perfect representation, and now we're ready to solve it to find out just how many hours Sheenah could have rented that bike!

b. Solving the Inequality: Finding the Maximum Hours

Alright guys, now for the exciting part – solving that inequality and finding out the maximum number of hours Sheenah could have rented the bike! We have our inequality: 10 + 20h < 270. Our goal here is to isolate 'h', the variable representing the number of hours, to see what values of 'h' satisfy this condition. It's very similar to solving a regular equation, we just need to be careful with our steps. First things first, we want to get the term with 'h' (that's 20h) by itself on one side of the inequality. To do that, we need to get rid of that P10 fee. Since it's being added to 20h, we do the opposite: we subtract 10 from both sides of the inequality. Remember, whatever you do to one side, you must do to the other to keep the inequality balanced.

So, we have:

10 + 20h - 10 < 270 - 10

This simplifies to:

20h < 260

Awesome! Now we're closer. We have the cost related to the hours (20h) being less than P260. The next step is to find out what a single hour ('h') costs us in this scenario. Since 'h' is currently being multiplied by 20, we need to do the opposite operation: divide both sides of the inequality by 20.

20h / 20 < 260 / 20

And when we do that division, we get:

h < 13

So, what does h < 13 mean? It means that the number of hours Sheenah rented the bike must be less than 13 hours. Now, the question asks for the maximum number of hours Sheenah could have rented the bike. Since she can't rent for exactly 13 hours (because the cost would then be exactly P270, and she paid less than P270), the maximum whole number of hours she could have rented is the highest integer strictly less than 13. That number is 12 hours. If she rented for 12 hours, the total cost would be P10 (fee) + P20 * 12 (hourly cost) = P10 + P240 = P250. And P250 is indeed less than P270! If she rented for 13 hours, the cost would be P10 + P20 * 13 = P10 + P260 = P270, which is not less than P270. Therefore, the maximum number of full hours Sheenah could have rented the bike is 12. This shows the power of inequalities in defining limits and finding the boundaries of possible solutions in real-life situations. Pretty cool, right guys?

Discussion: Real-World Applications of Inequalities

So, guys, we just solved a pretty neat problem involving Sheenah and her bike rental. We used inequalities to figure out the maximum time she could ride her bike without exceeding her budget. But this isn't just a one-off math exercise; inequalities are everywhere in the real world, and understanding them is super useful. Think about it – whenever there's a limit, a minimum requirement, or a maximum capacity, you're dealing with an inequality. For instance, speed limits on roads are inequalities. If the speed limit is 60 km/h, it means your speed must be less than or equal to 60 km/h (s ≤ 60). You can't go faster! Or consider age restrictions for certain activities. You need to be at least 18 years old to vote, meaning your age must be greater than or equal to 18 (a ≥ 18). You can't vote if you're 17. In budgeting, like Sheenah's case, inequalities are essential. When you plan to spend no more than P500 on groceries, that's an inequality: total cost ≤ 500. It helps you stay within your financial limits. Scientific and engineering fields heavily rely on inequalities too. Designing a bridge requires ensuring that the load it can carry is greater than the maximum expected weight (Load > Expected Weight) to prevent collapse. In computer programming, inequalities are used constantly in decision-making processes (if statements) to control the flow of a program. For example, if (user_input < 0) then display an error message. Even in everyday decisions, like figuring out how many people can fit in your car (total passengers ≤ car capacity), you're implicitly using inequalities. The math we did with Sheenah's bike is a simplified version of these complex real-world applications. It demonstrates how setting up the correct mathematical relationship, whether it's an equation or an inequality, allows us to analyze situations, make informed decisions, and solve problems effectively. So next time you see a sign with a number and a rule, remember you're looking at an inequality in action! Keep practicing, keep questioning, and you'll see math all around you, making sense of the world.