Solving The Equation: 20/(3m) = 2 - 4/(3m)

Hey guys! Today, we're diving into a fun math problem: solving the equation $\frac{20}{3m} = 2 - \frac{4}{3m}$. Don't worry, it might look a little intimidating at first, but we'll break it down step by step so it's super easy to understand. We'll cover everything from the initial setup to the final solution, making sure you grasp every concept along the way. So, grab your calculators (or your brainpower!) and let's get started!

Understanding the Equation

Okay, so let's first take a good look at our equation: $\frac{20}{3m} = 2 - \frac{4}{3m}$. The key here is to recognize that we're dealing with fractions and a variable, m, in the denominator. This means we need to be a little careful about how we manipulate the equation. Our goal is to isolate m on one side of the equation, but we can't do that directly with m in the denominator. So, the first order of business is to get rid of those fractions! Eliminating fractions often simplifies the equation and makes it much easier to work with. Think of it like decluttering your workspace before starting a big project—you want everything neat and tidy before you dive in. In our case, tidying up means getting rid of the denominators.

To do this, we need to find the least common denominator (LCD) of all the fractions in the equation. In this case, we have two fractions: $\frac{20}{3m}$ and $\frac{4}{3m}$. Both of these have the same denominator, which is 3m. That makes our job a bit easier! If they were different, we'd need to find the smallest expression that both denominators could divide into evenly. But since they're the same, our LCD is simply 3m. Now, why is finding the LCD so crucial? Well, it's because we're going to multiply both sides of the equation by this LCD. This clever trick will cancel out the denominators in our fractions, leaving us with a much simpler equation to solve. It’s like having a magic wand that makes fractions disappear! This step is fundamental in solving equations with fractions, and it's a technique you'll use again and again in algebra. So, make sure you understand this part perfectly before moving on. Trust me, mastering this technique will save you a lot of headaches down the road.

Step-by-Step Solution

Alright, let's get into the nitty-gritty and solve this equation step-by-step. This is where the rubber meets the road, guys! Remember our equation? It's $\frac{20}{3m} = 2 - \frac{4}{3m}$. We've already identified the LCD as 3m, so let's put that knowledge to work.

Step 1: Multiply both sides of the equation by the LCD. This is the crucial step where we eliminate the fractions. We're going to multiply both the left side and the right side of the equation by 3m. This ensures that we maintain the balance of the equation – what we do to one side, we must do to the other! So, we have:

3m * $\frac{20}{3m}$ = 3m * (2 - $\frac{4}{3m}$)

Now, let's break this down. On the left side, the 3m in the numerator and the 3m in the denominator cancel each other out. This is exactly what we wanted! We're left with just 20 on the left side. Nice and clean!

On the right side, we need to distribute the 3m to both terms inside the parentheses. Remember the distributive property? It's super important here. So, we have:

3m * 2 - 3m * $\frac{4}{3m}$

This simplifies to:

6m - 4

Why -4? Because in the second term, the 3m in the numerator and the 3m in the denominator cancel out, leaving us with just 4, and we still have the minus sign in front. So, our equation now looks like this:

20 = 6m - 4

See how much simpler it is already? We've gotten rid of those pesky fractions and now we have a straightforward linear equation. This is a huge win! The next steps involve isolating m, and we're well on our way.

Step 2: Isolate the term with m. Our goal now is to get the term with m (which is 6m) by itself on one side of the equation. To do this, we need to get rid of the -4 on the right side. How do we do that? We add 4 to both sides of the equation! This is another key principle of equation solving: we can add (or subtract) the same value from both sides without changing the solution.

So, we add 4 to both sides:

20 + 4 = 6m - 4 + 4

This simplifies to:

24 = 6m

Fantastic! We've successfully isolated the term with m. We're almost there, guys. Just one more step to go!

Step 3: Solve for m. Now we have the equation 24 = 6m. To find the value of m, we need to get m all by itself. It's currently being multiplied by 6, so what's the opposite of multiplication? Division! We're going to divide both sides of the equation by 6.

$\frac{24}{6}$ = $\frac{6m}{6}$

This simplifies to:

4 = m

And there you have it! We've solved the equation. m = 4. Hooray!

Checking the Solution

Okay, we've found our solution, m = 4. But how do we know if it's actually correct? This is where checking our solution comes in. It's like proofreading your work before submitting it – you want to make sure you haven't made any mistakes. Checking our solution is a crucial step in problem-solving, and it can save you from a lot of grief down the road. It’s a fantastic habit to get into, and it gives you confidence that your answer is right.

So, to check our solution, we're going to plug m = 4 back into the original equation: $\frac{20}{3m} = 2 - \frac{4}{3m}$. We'll substitute 4 for m wherever it appears in the equation.

This gives us:

$\frac{20}{3 * 4}$ = 2 - $\frac{4}{3 * 4}$

Let's simplify both sides separately. On the left side, we have:

$\frac{20}{12}$

This can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4. So, we get:

$\frac{5}{3}$

Now, let's look at the right side. We have:

2 - $\frac{4}{12}$

First, we can simplify the fraction $\frac{4}{12}$ by dividing both the numerator and denominator by 4, giving us:

$\frac{1}{3}$

So, the right side becomes:

2 - $\frac{1}{3}$

To subtract these, we need a common denominator. We can rewrite 2 as $\frac{6}{3}$, so we have:

$\frac{6}{3}$ - $\frac{1}{3}$

This simplifies to:

$\frac{5}{3}$

Now, let's compare both sides of the equation. We have $\frac{5}{3}$ on the left side and $\frac{5}{3}$ on the right side. They are equal! This means our solution, m = 4, is correct. We've aced it!

Common Mistakes to Avoid

Alright, let's talk about some common pitfalls that students often encounter when solving equations like this. Knowing these mistakes ahead of time can help you avoid them and ensure you get the correct answer. It’s like having a map that shows you where the potholes are on the road – you can steer clear and have a smoother journey!

1. Forgetting to Distribute: This is a classic mistake! When you multiply one side of the equation by the LCD, remember to distribute it to every term on that side. In our equation, we had to multiply 3m by both 2 and -$\frac{4}{3m}$. If you forget to distribute, you'll end up with an incorrect equation and an incorrect solution. So, always double-check that you've distributed correctly.

2. Incorrectly Simplifying Fractions: Fractions can be tricky, and it's easy to make a mistake when simplifying them. Make sure you're dividing both the numerator and the denominator by the same number. Also, remember to look for the greatest common divisor to simplify the fraction completely. If you simplify incorrectly, it can throw off your entire solution.

3. Arithmetic Errors: Simple arithmetic errors can happen to anyone, especially under pressure. A wrong addition, subtraction, multiplication, or division can lead to the wrong answer. This is why it's so important to show your work and double-check each step. It's also a good idea to use a calculator for more complex calculations to minimize the risk of errors.

4. Not Checking the Solution: We've already emphasized the importance of checking your solution, but it's worth mentioning again. Skipping this step is a big mistake! Plugging your solution back into the original equation is the best way to catch any errors you might have made along the way. If the equation doesn't balance, you know you need to go back and review your work.

5. Dividing by Zero: This is a major no-no in mathematics! You can never divide by zero. When solving equations, be careful that your solution doesn't make the denominator of any fraction equal to zero. In our equation, we had 3m in the denominator. If we had found a solution that made 3m equal to zero (like m = 0), that solution would be invalid. Always be mindful of this rule.

Conclusion

So, there you have it, guys! We've successfully solved the equation $\frac{20}{3m} = 2 - \frac{4}{3m}$ step by step. We started by understanding the equation, then we eliminated the fractions by multiplying by the LCD, isolated the term with m, and finally solved for m. We even checked our solution to make sure it was correct. We also discussed some common mistakes to avoid, which should help you tackle similar problems with confidence. Remember, practice makes perfect! The more you solve these types of equations, the easier they will become. Keep up the great work, and you'll be a math whiz in no time!