Unlock Repeating Decimals: Solve B = 30 * [0.1(6) + 0.(3)]

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Unlock Repeating Decimals: Solve b = 30 * [0.1(6) + 0.(3)]

Hey there, math enthusiasts and curious minds! Ever looked at an equation with repeating decimals and thought, "Whoa, what even is that?" Well, you're not alone, and guess what? It's not nearly as scary as it looks. Today, we're going to dive headfirst into a really cool problem: solving for 'b' in the equation b = 30 * [0.1(6) + 0.(3)]. This isn't just about crunching numbers; it's about understanding the logic, building your problem-solving muscles, and gaining some serious confidence with decimals that seem to go on forever. We'll break it down step-by-step, making sure you grasp every single concept. So, grab your favorite beverage, settle in, and let's turn these tricky repeating decimals into simple, manageable fractions, and ultimately, find our elusive 'b'. This journey will not only help you conquer this specific problem but also equip you with the tools to tackle similar mathematical challenges with a confident smile. Ready to become a repeating decimal pro? Let's get started!

Demystifying Repeating Decimals: Converting 0.1(6) to a Fraction

Alright, guys, let's kick things off with our first repeating decimal: 0.1(6). This notation might look a bit intimidating at first glance, but it simply means that the '6' repeats indefinitely – so it's 0.166666... and so on. Our mission? To transform this never-ending decimal into a neat, tidy, and precise fraction. Why fractions, you ask? Because fractions are often much easier to work with, especially when we're dealing with addition and multiplication, preventing any rounding errors that might crop up if we tried to use truncated decimals. This conversion method is a fundamental skill in algebra and general number theory, and once you get it, you'll feel like a total math wizard. Let's walk through it together, step by meticulous step.

First, we assign our repeating decimal to a variable. Let's call it 'x'.

Step 1: Set up the equation.

  • Let x = 0.1666... (Equation 1)

Now, we need to manipulate this equation so that the repeating part aligns nicely. Since there's one non-repeating digit ('1') before the repeating part, we'll multiply by powers of 10 to shift the decimal point.

Step 2: Shift the decimal to just before the repeating part.

  • Multiply Equation 1 by 10 to move the decimal past the '1'.
  • 10x = 1.6666... (Equation 2)

Great! Now the decimal point is right before the repeating '6's. Next, we want to shift the decimal past one full cycle of the repeating part. Since only the '6' repeats, one cycle is just one digit.

Step 3: Shift the decimal past one full repeating cycle.

  • Multiply Equation 1 by 100 (which is 10 times 10, effectively moving the decimal two places).
  • 100x = 16.6666... (Equation 3)

See what we did there? Both Equation 2 and Equation 3 now have the exact same repeating decimal part after the decimal point (0.6666...). This is the magic trick! Now, we can eliminate those pesky repeating digits by subtracting the two equations.

Step 4: Subtract the equations to eliminate the repeating part.

  • Subtract Equation 2 from Equation 3: 100x - 10x = 16.6666... - 1.6666...

  • This simplifies beautifully to: 90x = 15

We're almost there! All that's left is to solve for 'x' by dividing both sides by 90.

Step 5: Solve for x.

  • x = 15 / 90

And finally, we simplify this fraction to its lowest terms. Both 15 and 90 are divisible by 15.

  • x = 15 ÷ 15 / 90 ÷ 15
  • x = 1/6

Boom! Just like that, 0.1(6) is equivalent to 1/6. Wasn't that satisfying? This method is super reliable for any repeating decimal. It's really important to understand why these steps work. The subtraction step is key because it perfectly cancels out the infinite tails of numbers, leaving us with simple whole numbers to work with. Taking the time to master this conversion will make many future math problems a breeze, guys. It’s about building a strong foundation, and knowing how to handle these numbers gives you a real edge. Plus, getting a precise fraction instead of an approximation is always a win in the world of mathematics, ensuring your calculations are as accurate as possible. So, you've just conquered your first complex repeating decimal conversion!

Taming the Simpler Repeater: Converting 0.(3) to a Fraction

Okay, team, with 0.1(6) successfully converted, let's move on to its slightly simpler cousin: 0.(3). This one is a classic, and you might even have a hunch what its fractional form is already! But even for the seemingly obvious ones, it’s crucial to understand the systematic approach we used before. Why? Because the method is universal, and once you nail it, you can apply it to any repeating decimal, no matter how complex it appears. The '0.(3)' simply means that the '3' repeats forever: 0.333333... This is a prime example of a pure repeating decimal, where the repeating part starts immediately after the decimal point, making its conversion often a little quicker than the mixed repeating decimal we just handled. Let's dive in and confirm its fractional identity, reinforcing the powerful technique we just learned.

Just like last time, we'll start by assigning our repeating decimal to a variable. Let's use 'y' this time.

Step 1: Set up the equation.

  • Let y = 0.3333... (Equation 1)

Now, because the repeating part starts immediately after the decimal point, we don't need that first step of shifting the decimal past any non-repeating digits. We can jump straight to shifting the decimal past one full cycle of the repeating part. Since only the '3' repeats, one cycle is just one digit.

Step 2: Shift the decimal past one full repeating cycle.

  • Multiply Equation 1 by 10 to move the decimal past the first '3'.
  • 10y = 3.3333... (Equation 2)

Notice how both Equation 1 and Equation 2 now have the exact same repeating decimal part (0.3333...) after the decimal point. This is the perfect setup for subtraction! By subtracting the original equation from this new one, we cleanly eliminate the infinite string of threes.

Step 3: Subtract the equations to eliminate the repeating part.

  • Subtract Equation 1 from Equation 2: 10y - y = 3.3333... - 0.3333...

  • This simplifies very nicely to: 9y = 3

Almost there, folks! The next move is to isolate 'y' by dividing both sides by 9.

Step 4: Solve for y.

  • y = 3 / 9

And, as always, we simplify the fraction to its lowest terms. Both 3 and 9 are divisible by 3.

  • y = 3 ÷ 3 / 9 ÷ 3
  • y = 1/3

There it is! As expected, 0.(3) is indeed equivalent to 1/3. Wasn't that a breeze? This conversion is a classic for a reason – it perfectly illustrates the elegance of this algebraic method. Mastering this simple case solidifies your understanding, making more complex conversions less daunting. Understanding repeating decimals and their fractional equivalents is a cornerstone of number sense, and it helps you appreciate the relationship between different forms of numbers. It’s also super useful for quickly estimating or performing mental math. For instance, knowing 0.(3) is 1/3 helps you immediately recognize that 0.(6) would be 2/3, and so on. This foundational knowledge really builds your mathematical intuition, empowering you to tackle problems with confidence and precision. Keep up the great work, guys – you're doing awesome!

The Power of Addition: Combining Our Fractions

Alright, fantastic job on converting those repeating decimals into their fractional forms! Now that we've transformed 0.1(6) into 1/6 and 0.(3) into 1/3, the next logical step in solving our equation, b = 30 * [0.1(6) + 0.(3)], is to tackle the expression inside the brackets: [1/6 + 1/3]. This part is all about adding fractions, a skill that’s absolutely fundamental in mathematics. It might seem basic to some, but often, the simplest steps are where crucial mistakes can sneak in. So, we'll go through this carefully, ensuring we nail the concept of finding a common denominator, which is the secret sauce to successfully adding or subtracting fractions. Without a common denominator, our fractions are like apples and oranges – we can't directly combine them. Let's make sure our "apples" and "oranges" are compatible!

Our task is to add 1/6 and 1/3.

Step 1: Identify the denominators.

  • Our denominators are 6 and 3.

To add these fractions, we need to find a common denominator. The best common denominator to use is the Least Common Multiple (LCM) of the two denominators. This keeps our numbers as small as possible and makes calculations easier.

Step 2: Find the Least Common Multiple (LCM) of the denominators.

  • Multiples of 6: 6, 12, 18...
  • Multiples of 3: 3, 6, 9, 12...

Looking at these lists, the smallest number that appears in both is 6. So, our LCM, and thus our common denominator, is 6. This is perfect because 1/6 already has this denominator!

Step 3: Convert the fractions to equivalent fractions with the common denominator.

  • The first fraction, 1/6, already has the common denominator. So, it stays as 1/6.

  • For the second fraction, 1/3, we need to change its denominator to 6. To do this, we ask ourselves: "What do I multiply 3 by to get 6?" The answer is 2. Whatever we do to the denominator, we must also do to the numerator to keep the fraction equivalent. So, we multiply both the numerator and denominator of 1/3 by 2:

    (1 * 2) / (3 * 2) = 2/6

Awesome! Now both our fractions have the same denominator. We have 1/6 and 2/6.

Step 4: Add the fractions.

  • With common denominators, adding fractions is super straightforward: you just add the numerators and keep the denominator the same.

  • 1/6 + 2/6 = (1 + 2) / 6

  • = 3/6

We're not quite done yet! Always remember to simplify your fractions to their lowest terms if possible. Both 3 and 6 are divisible by 3.

Step 5: Simplify the resulting fraction.

  • 3 ÷ 3 / 6 ÷ 3 = 1/2

So, the sum of 0.1(6) and 0.(3), or rather 1/6 + 1/3, is equal to 1/2. How cool is that? This step might seem elementary, but it's a critical juncture in our problem. Getting this right sets us up perfectly for the final calculation. Learning to add fractions efficiently and accurately is a skill that will serve you well in countless mathematical contexts, from geometry to advanced calculus. It reinforces your number sense and your ability to manipulate expressions, ensuring you build a robust foundation. Remember, meticulousness in these steps pays off big time in the final answer!

The Grand Calculation: Multiplying by 30 to Find 'b'

We're in the home stretch, guys! We've successfully converted our repeating decimals into precise fractions (0.1(6) became 1/6, and 0.(3) became 1/3), and we've masterfully added those fractions to get a simplified result of 1/2. Now, it's time for the grand finale: taking that sum and multiplying it by 30 to finally solve for 'b' in our original equation, b = 30 * [0.1(6) + 0.(3)]. This step involves straightforward multiplication, but it’s also a great opportunity to highlight the power of simplification and smart arithmetic. Many times, you can make calculations much easier by doing a little bit of cancellation before you multiply, especially when dealing with fractions and whole numbers. Let's make sure we do this efficiently and accurately to clinch our answer for 'b'.

Our equation now looks like this:

b = 30 * (1/2)

When you multiply a whole number by a fraction, you can think of the whole number as having a denominator of 1. So, 30 can be written as 30/1. This often makes the multiplication process clearer, especially when we're looking for opportunities to simplify.

Step 1: Rewrite the whole number as a fraction.

  • b = (30/1) * (1/2)

Now, for multiplying fractions, the rule is simple: multiply the numerators together and multiply the denominators together. However, before we jump into that, let's look for any chances to cross-cancel. This is a brilliant trick that simplifies your numbers before you multiply, making the final multiplication much easier and reducing the need for simplifying a large fraction at the end. In our case, 30 in the numerator and 2 in the denominator share a common factor, which is 2.

Step 2: Look for opportunities to cross-cancel.

  • Divide 30 by 2, which gives you 15.
  • Divide 2 by 2, which gives you 1.

So, our expression now transforms into:

  • b = (15/1) * (1/1)

See how much cleaner that looks? This simplification step is often overlooked but can save you a ton of work and prevent errors with larger numbers. It’s all about working smarter, not harder!

Step 3: Perform the multiplication.

  • Now, multiply the new numerators: 15 * 1 = 15.

  • And multiply the new denominators: 1 * 1 = 1.

  • b = 15/1

Step 4: State the final value of 'b'.

  • b = 15

There you have it! Through a series of logical, well-executed steps, we've successfully found that b equals 15. This final multiplication step ties everything together. It shows how converting repeating decimals to fractions, finding common denominators, and adding fractions are all essential components of solving a seemingly complex problem. Understanding these operations and knowing when to simplify (like with cross-cancellation) are invaluable skills in mathematics. It's not just about getting the right answer, but also about appreciating the elegance and efficiency of the process. Every time you correctly apply these principles, you're building a stronger, more resilient mathematical mind. Great work, everyone – you've mastered a truly insightful problem!

Beyond the Numbers: Why Understanding This Equation Matters

Alright, my friends, we've done it! We've tackled the equation b = 30 * [0.1(6) + 0.(3)] from start to finish, converting those tricky repeating decimals into fractions, adding them up, and finally arriving at b = 15. But here's the kicker: this whole exercise wasn't just about finding that one answer. Nope, it was about so much more! It was about sharpening your problem-solving skills, boosting your mathematical understanding, and honestly, just proving to yourself that you can take on seemingly complex challenges and break them down into manageable pieces. This isn't just