Math Problems Solved: Equations, Functions, Probability

Hey guys! Let's dive into some cool math problems today. We'll break down equations, functions, and even a bit of probability. Math can seem intimidating, but we'll tackle it together step by step. So grab your thinking caps, and let's get started!

1. Proving an Arithmetic Equation

Let's kick things off by proving that $(0.3 + 0.4) \times 10 + 2 \times 0.5 = 8$. This might look like a jumble of numbers and operations, but it's all about following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

First, we need to deal with what's inside the parentheses. We have $0.3 + 0.4$, which is a straightforward addition. Adding these decimal numbers, we get $0.7$. So, our equation now looks like this: $(0.7) \times 10 + 2 \times 0.5 = 8$.

Next up are the multiplication operations. We have two of them: $0.7 \times 10$ and $2 \times 0.5$. Let's tackle them one at a time. Multiplying $0.7$ by $10$ is the same as shifting the decimal point one place to the right, which gives us $7$. Now, let's multiply $2$ by $0.5$. Remember that $0.5$ is the same as one-half, so $2 \times 0.5$ is the same as half of $2$, which is $1$. Great! Our equation has simplified further to $7 + 1 = 8$.

Finally, we have a simple addition left. Adding $7$ and $1$ gives us $8$. So, the left side of the equation equals $8$, which is exactly what we wanted to show! Therefore, $(0.3 + 0.4) \times 10 + 2 \times 0.5$ indeed equals $8$. We've successfully proven the equation, and it wasn't so scary after all, right? The key is to break it down into smaller, manageable steps and follow the order of operations. By doing this, even complex-looking problems become much easier to solve.

2. Evaluating a Function

Now, let's dive into functions. We're given the function $f: R \rightarrow R$, which simply means that the function $f$ takes real numbers as input and produces real numbers as output. The function itself is defined as $f(x) = 2x - 1$. Our task is to show that $f(1) + f(2) = 4$.

To do this, we need to figure out what $f(1)$ and $f(2)$ are. This involves substituting the values $1$ and $2$ for $x$ in the function's definition. Let's start with $f(1)$. Substituting $x = 1$ into the function, we get $f(1) = 2(1) - 1$. Simplifying this, we have $f(1) = 2 - 1$, which equals $1$. So, we've found that $f(1) = 1$.

Next, let's find $f(2)$. This time, we substitute $x = 2$ into the function, giving us $f(2) = 2(2) - 1$. Simplifying, we get $f(2) = 4 - 1$, which equals $3$. So, we know that $f(2) = 3$.

Now that we have both $f(1)$ and $f(2)$, we can add them together. We have $f(1) + f(2) = 1 + 3$. Adding $1$ and $3$ gives us $4$. Therefore, $f(1) + f(2)$ indeed equals $4$, just as we needed to show. Evaluating functions is all about substituting the given input values into the function's definition and simplifying. With a bit of practice, it becomes second nature.

3. Solving a Logarithmic Equation

Let's move on to solving the logarithmic equation $\log_5(2x + 1) = \log_5 5$. Logarithmic equations might seem a bit intimidating, but they're actually quite manageable with the right approach. The key here is to understand the properties of logarithms.

Notice that we have logarithms with the same base (base 5) on both sides of the equation. This is great news because it means we can use a fundamental property of logarithms: if $\log_b a = \log_b c$, then $a = c$. In other words, if the logarithms of two expressions are equal and they have the same base, then the expressions themselves must be equal.

Applying this property to our equation, $\log_5(2x + 1) = \log_5 5$, we can drop the logarithms and simply equate the arguments (the expressions inside the logarithms). This gives us the equation $2x + 1 = 5$. See how much simpler that is? The logarithmic equation has transformed into a linear equation that we can easily solve.

Now, let's isolate $x$. To do this, we first subtract $1$ from both sides of the equation: $2x + 1 - 1 = 5 - 1$, which simplifies to $2x = 4$. Next, we divide both sides by $2$ to get $x$ by itself: $\frac{2x}{2} = \frac{4}{2}$. This simplifies to $x = 2$. So, we've found a potential solution: $x = 2$.

However, with logarithmic equations, it's crucial to check our solution. We need to make sure that plugging $x = 2$ back into the original equation doesn't result in taking the logarithm of a non-positive number (since logarithms are only defined for positive arguments). Let's check it out: $\log_5(2(2) + 1) = \log_5(4 + 1) = \log_5 5$. This is perfectly valid! So, our solution $x = 2$ is indeed correct. The solution to the equation $\log_5(2x + 1) = \log_5 5$ is $x = 2$. Remember to always check your solutions in logarithmic equations to avoid potential errors.

4. Calculating Probability

Let's wrap things up with a probability problem. While the original prompt was incomplete, let's consider a classic probability scenario to illustrate the concept. Suppose we have a bag containing 5 red balls and 3 blue balls. We want to calculate the probability of randomly selecting a red ball from the bag.

Probability, at its core, is about quantifying the likelihood of an event occurring. It's defined as the ratio of favorable outcomes to the total possible outcomes. In our case, the event we're interested in is selecting a red ball. So, we need to determine the number of favorable outcomes (number of red balls) and the total number of possible outcomes (total number of balls).

We have 5 red balls, which represent our favorable outcomes. The total number of balls in the bag is $5 \text{ (red)} + 3 \text{ (blue)} = 8$. This represents the total possible outcomes. Now we can calculate the probability.

The probability of selecting a red ball is the number of red balls divided by the total number of balls: $\frac{5}{8}$. This fraction represents the probability, and it tells us that there's a 5 out of 8 chance of picking a red ball. We can also express this probability as a decimal or a percentage. Dividing $5$ by $8$, we get $0.625$. Multiplying this by $100$, we get $62.5\%$. So, there's a $62.5\%$ probability of selecting a red ball.

In probability problems, always identify the favorable outcomes and the total possible outcomes. Once you have these, calculating the probability is a straightforward division. Remember, probability is a powerful tool for understanding the likelihood of events in various situations, from simple games to complex scientific experiments.

Conclusion

So, there you have it! We've tackled a range of math problems today, from proving equations to solving logarithmic equations and calculating probabilities. The key takeaway is that even complex problems can be broken down into smaller, more manageable steps. With a little practice and the right approach, you can conquer any math challenge that comes your way. Keep practicing, keep exploring, and most importantly, have fun with math! You guys got this!