Ceramic Tile Cost Calculation & Expression Evaluation

Hey guys! Let's break down these math problems together. We've got a ceramic tile cost calculation and an expression evaluation to tackle. Don't worry, we'll go through them step by step so it all makes sense.

B6: Figuring Out the Price of Ceramic Tiles

So, the first problem tells us that the total cost for the tiles and their installation was 524 rubles. The tricky part is that the installation cost was 31% of the tile cost. We need to figure out how much the tiles themselves cost. This is a classic word problem that we can solve using a bit of algebra. Let's dive into how we can crack this one. First things first, we need to define our variables. Let's use 'x' to represent the cost of the ceramic tiles. The installation cost is then 0.31 times the cost of the tiles, which we can write as 0.31x. The total cost, which is the sum of the tile cost and the installation cost, is given as 524 rubles. Therefore, we can set up an equation to represent this situation:

x + 0.31x = 524

Now, let's combine like terms on the left side of the equation:

1. 31x = 524

To solve for x, we need to divide both sides of the equation by 1.31:

x = 524 / 1.31

Performing this division gives us:

x ≈ 400

Therefore, the cost of the ceramic tiles is approximately 400 rubles. To verify our answer, we can calculate the installation cost and add it to the tile cost to see if it matches the total cost given in the problem. The installation cost is 31% of the tile cost, so it would be 0.31 * 400 = 124 rubles. Adding this to the tile cost, we get 400 + 124 = 524 rubles, which matches the total cost given in the problem. This confirms that our calculation is correct. In summary, the key to solving this problem is to set up an equation that represents the relationship between the tile cost, the installation cost, and the total cost. By defining a variable for the tile cost and expressing the installation cost in terms of that variable, we can create an equation that can be easily solved using basic algebraic principles. Once we have solved for the tile cost, we can verify our answer by calculating the installation cost and checking if the sum of the two costs matches the given total cost. This step-by-step approach not only helps us find the correct answer but also ensures that we understand the logic and reasoning behind the solution.

B7: Evaluating the Expression (a² - a) / (a² - a²)

Okay, let's move on to the second problem. This one asks us to find the value of the expression (a² - a) / (a² - a²) when a = something (the value of 'a' seems to be missing in the original prompt, but we'll address that!). The first thing we should always do when we see an expression like this is to try and simplify it. Simplifying beforehand makes the calculation much easier and reduces the risk of making mistakes. So, let's see what we can do with this fraction. Before we can evaluate the expression, we should simplify it as much as possible. The given expression is (a² - a) / (a² - a²). Notice that the denominator has a term a² - a², which simplifies to 0. This is a critical observation because division by zero is undefined in mathematics. Therefore, regardless of the value of 'a', the expression (a² - a) / (a² - a²) is undefined. If we were to proceed without noticing this, we might attempt to factor the numerator and denominator and then cancel out common terms. However, since the denominator is zero, any such simplification would be misleading. The expression (a² - a) can be factored as a(a - 1). If the denominator were something other than zero, we could explore whether there are any common factors between the numerator and the denominator that could be canceled out. However, in this case, the denominator being zero makes the entire expression undefined. This highlights an important principle in mathematics: always check for potential divisions by zero before proceeding with any calculations or simplifications. Division by zero leads to undefined results and can invalidate any subsequent steps in the solution. In summary, the key to this problem is recognizing that the denominator simplifies to zero, making the expression undefined. This understanding requires a careful examination of the given expression and an awareness of the fundamental rules of arithmetic. No further calculations or substitutions are necessary once we identify the division by zero. This type of problem often appears in mathematical assessments to test students' understanding of basic algebraic principles and their ability to identify potential pitfalls in calculations. Always remember to simplify and check for undefined conditions before proceeding with any evaluations.

Dealing with Division by Zero

It looks like there's a sneaky issue here: a² - a² is always zero! This means we're trying to divide by zero, which is a big no-no in math. Division by zero is undefined. So, no matter what value 'a' is, the answer will always be undefined. This is a crucial thing to watch out for in math problems – always check for potential division by zero! In mathematics, division by zero is a major issue because it violates the fundamental rules of arithmetic. The concept of division is based on the idea of splitting a quantity into equal parts. When we divide a number by zero, we are essentially asking how many times zero can fit into that number, which doesn't make sense. For example, if we try to divide 10 by 0, we are asking how many zeros we need to add together to get 10. There is no such number, because no matter how many zeros we add, we will never reach 10. This is why division by zero is undefined. In the context of the expression (a² - a) / (a² - a²), the denominator a² - a² simplifies to zero regardless of the value of a. This means that we are attempting to divide the numerator (a² - a) by zero, which is undefined. Therefore, the entire expression is undefined for all values of a. It is important to recognize this division by zero before attempting any further calculations or simplifications. Trying to manipulate the expression without addressing the division by zero can lead to incorrect results and a misunderstanding of the mathematical principles involved. Division by zero is not just a theoretical issue; it can also cause problems in real-world applications of mathematics. For example, in computer programming, attempting to divide by zero will typically result in an error message or a program crash. This is because computers are designed to follow the rules of mathematics, and division by zero is not a valid operation. Therefore, programmers must be careful to avoid division by zero in their code.

Key Takeaways

• Careful Reading: Always read the problem carefully and make sure you understand what it's asking.
• Define Variables: Use variables to represent unknowns. It makes setting up equations much easier.
• Simplify First: Always try to simplify expressions before plugging in numbers.
• Watch for Division by Zero: This is a huge one! Division by zero makes the whole thing undefined.

So, there you have it! We've tackled a ceramic tile cost problem and a tricky expression evaluation. Remember, math is all about breaking things down into smaller steps and thinking logically. Keep practicing, and you'll get the hang of it! Remember that practice makes perfect, and the more you work through problems like these, the more confident you'll become in your math skills. Whether it's figuring out the cost of tiles or evaluating algebraic expressions, the key is to approach each problem with a clear strategy and a willingness to break it down into manageable steps. And don't forget, it's okay to make mistakes – they're just opportunities to learn and grow!