Proving A Math Equation: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a cool problem: showing that (2-1/2)(3-1/3)(4-1/4) equals 3/5. Sounds fun, right? Don't worry, it's not as scary as it looks. We'll break it down step-by-step, making sure everyone can follow along. This is a great exercise to boost your understanding of fractions and basic arithmetic. Let's get started, shall we? This problem is perfect for practicing your fraction skills. We'll be doing some calculations, so grab your pencils and let's roll!

To begin, we'll start by making sure we understand what the question is asking. We need to demonstrate that when we multiply the three terms given together, the final result will be equivalent to the fraction 3/5. The problem is a classic example of simplifying expressions and dealing with fractions, a fundamental skill in mathematics. The beauty of this problem lies in its simplicity. It requires no complex formulas or advanced concepts. Just pure, unadulterated arithmetic. We'll address each part of the expression individually, simplifying it before combining them. This structured approach makes it easier to keep track of our progress and reduces the chances of making mistakes. It also reinforces the idea of following a logical order of operations, which is crucial in mathematics. We're going to break down each part of the expression, making it a piece of cake. Ready to transform some fractions?

First, we need to handle those mixed numbers within the parentheses. This means converting each mixed number (a whole number and a fraction) into a single, improper fraction. Remember, an improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This step is essential because it allows us to simplify the expression by multiplying fractions. We can't directly multiply a mixed number. We must first convert it into a simple fraction, thus making it easier for us to calculate. Now, this is where the magic happens! To transform a mixed number into an improper fraction, we multiply the whole number by the denominator and then add the numerator. The denominator remains the same. Easy peasy, right?

Let's go through it. For the first term, (2-1/2), we multiply 2 by 2 (the denominator), which equals 4. Then we subtract 1 from 4 that equals 3, and the denominator remains 2, giving us 3/2. Similarly, for the second term, (3-1/3), we multiply 3 by 3, which equals 9, then subtract 1 from 9. Hence, it will result in 8/3. Moving to the third term, (4-1/4), we multiply 4 by 4, which equals 16, then subtract 1, and the denominator remains the same, which will be 15/4. With each step, the problem becomes easier and easier. See, it's not so difficult, is it? We are making good progress. We're not just solving a math problem; we are building our confidence in our math skills. We will keep practicing until it becomes second nature to us. Remember, practice makes perfect! So, let's keep the momentum going and get this done!

Step-by-Step Simplification

Alright, let's break down each part of the equation step by step. We'll focus on converting those mixed numbers and simplifying them into fractions. Don't worry, I'll walk you through it, so you don't miss a beat. By the end, we'll have a much easier equation to handle. It is all about precision and accuracy. We want to be sure that our results are valid.

Converting Mixed Numbers to Fractions

Let's start with (2 - 1/2). As we discussed, this is a mixed number. We can convert it into an improper fraction. To do that, we multiply the whole number (2) by the denominator (2), which gives us 4. Then, we add the numerator (1). This provides us with 3/2. Now we have a simplified fraction to work with. Following this same approach, we will work with other fractions. Easy, right?

Next, let’s simplify (3 - 1/3). We multiply the whole number (3) by the denominator (3), which gives us 9. Now, subtract 1 to get 8/3. This is our second fraction, and we are ready to combine it with the first one. Now, let’s move on to the third set. We're on a roll. It's time to take on (4 - 1/4). Following the same steps, we multiply 4 by 4, which equals 16, and subtract 1. Thus we get 15/4. There you go! Now we have converted all of the mixed numbers into improper fractions. We're making great progress, guys! Let's get to the fun part now!

Multiplying the Fractions

Now comes the fun part: multiplying the fractions! We now have (3/2) * (8/3) * (15/4). To multiply fractions, we simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Let's start with the numerators: 3 * 8 * 15 = 360. Next, we multiply the denominators: 2 * 3 * 4 = 24. This gives us 360/24. But wait, we are not done yet. We have one more step to finish the work. The beauty of this is that it allows us to simplify the expressions by canceling out the common factors, which will make the numbers easier to work with. Does it feel like we are almost there? Yes! Let us go on!

Simplifying the Resulting Fraction

We have the fraction 360/24. Now, we need to simplify this fraction. Simplification means reducing the fraction to its lowest terms. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 360 and 24 is 24. So, we divide both the numerator and the denominator by 24. Therefore, 360 / 24 = 15, and 24 / 24 = 1. This gives us 15/1, which simplifies to 15. Wait a minute! Something is wrong? Let's check our steps again. Where did we go wrong?

Let's review our previous steps. We multiplied 2 by 2 and subtracted 1 to get 3/2. We then multiplied 3 by 3 and subtracted 1 to get 8/3. Finally, we multiplied 4 by 4 and subtracted 1 to get 15/4. The issue arises when we multiply those improper fractions. Let's redo those steps.

(3/2) * (8/3) * (15/4). Let's start by multiplying 3/2 by 8/3. We will then get (3 * 8)/(2 * 3) = 24/6. Now we can divide 24 by 6 to get 4. So now we can multiply 4 by 15/4. Now we get (4 * 15)/4 = 60/4. If we divide 60 by 4, we get 15. So we were right. Something is not right. We need to go back and check our problem once again.

(2 - 1/2) = 3/2 (3 - 1/3) = 8/3 (4 - 1/4) = 15/4 (3/2) * (8/3) * (15/4) = 3/5

It looks like there is a problem with the original question. If the original question is (2-1/2)(3-1/3)(4-1/4), the result will not be 3/5. It is our result, and the result is 15. The question seems to have a typo, and it is impossible to solve it with the information that we have. We will solve it in the next chapter. Let's go!

Let's Correct The Problem

Okay, guys, it looks like there might be a typo in the original question. The correct version should be a bit different, to get the final answer. We'll walk through how to adapt the problem slightly to achieve the desired result of 3/5. This will not only clarify any confusion but also provide us with an excellent opportunity to reinforce our mathematical skills. This adjustment allows us to solve the problem and also emphasizes the importance of precision when working with mathematical expressions. Ready to dive back in?

Rewriting the Equation

The original equation seemed to be (2-1/2) * (3-1/3) * (4-1/4) = 3/5. As we discovered, this does not yield 3/5. We will now rewrite the equation so that the answer equals 3/5. Let's change the original question. To achieve the 3/5 answer, we will work with slightly different numbers. This helps demonstrate how simple adjustments to numbers can dramatically change the outcome. So, let’s rewrite the equation as follows. We are going to rewrite the problem to prove that (1/2) * (3/3) * (2/5) = 3/5. Ready to find the solution?

Step-by-Step Solution

Now, let's step by step look at the solution. First, we will solve (1/2) * (3/3) * (2/5). Remember, to multiply fractions, you simply multiply the numerators together and the denominators together. Let's start this by multiplying 1/2 by 3/3. When we do that, we get (1 * 3) / (2 * 3), which gives us 3/6. Then, we can multiply this fraction by 2/5, to get the final result. In this stage, we have the expression of 3/6 * 2/5. We will multiply this, and we will get (3 * 2)/(6 * 5) = 6/30. We can simplify this fraction. What is the GCD of 6 and 30? The greatest common divisor is 6. So let's divide both numbers by 6. Thus, 6/6 = 1 and 30/6 = 5. Now we have 1/5. This is not our result, so we will need to change the numbers so that it becomes 3/5.

(2 - 1/2) = 3/2 (3 - 1/3) = 8/3 (4 - 1/4) = 15/4

Let's check the result again. When we solve (3/2) * (8/3) * (15/4), we get 30. Something is still wrong. The problem has to be rewritten. The rewritten equation has to be: (1/2) * (3/3) * (2/5). Then the result will be 1/5. Let us change the numbers once more. The real problem is (3/5). What do we need to do? Let's try (3/5). Therefore, the question should be (3/2) * (1/3) * (2/5). Let's go!

So let's see. We will multiply 3/2 by 1/3 and then by 2/5. Let's see the result. Now we can multiply the fractions. (3 * 1 * 2)/(2 * 3 * 5) = 6/30. We now can simplify the fraction. The greatest common divisor of the numbers is 6. Therefore 6/6=1, 30/6=5. The result is 1/5. This is not the answer too. We can then assume that the answer to this problem is not 3/5. There is no solution to the question. Sorry guys!

Conclusion

So, guys, what did we learn today? We tried to solve the equation (2-1/2)(3-1/3)(4-1/4) = 3/5. We learned about converting mixed numbers into improper fractions. We practiced multiplying fractions, and we tried to find out the solution. Also, we found out that the question has some mistakes. The correct result is not 3/5. We have to correct the problem to get the result. Remember, math is all about practice and patience. Keep practicing, and you will become math experts. Do not give up!

Hopefully, this step-by-step guide helped you better understand how to solve equations. Keep practicing, and you will become a math master! See you next time!"