Solving For 'a' In 6,(a) = 19/3 A Step-by-Step Guide

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Introduction

In this article, we will delve into the mathematical problem of finding the value of 'a' in the equation 6,(a) = 19/3. This equation involves a recurring decimal, which might seem daunting at first. However, by understanding the underlying principles of recurring decimals and employing a systematic approach, we can easily solve for 'a'. We will break down the problem step by step, providing a clear and concise explanation of each stage. This will not only help you solve this specific equation but also equip you with the knowledge to tackle similar problems involving recurring decimals.

Understanding Recurring Decimals

Before diving into the solution, it's crucial to grasp the concept of recurring decimals. A recurring decimal, also known as a repeating decimal, is a decimal number that has a digit or a block of digits that repeats infinitely. In our equation, 6,(a) represents a recurring decimal where the digit 'a' repeats indefinitely after the decimal point. For instance, if a = 3, then 6,(a) would be 6.3333..., and so on. Understanding this concept is the cornerstone for converting the recurring decimal into a fraction, which is a vital step in solving the equation.

Converting Recurring Decimals to Fractions

The key to solving the equation lies in converting the recurring decimal 6,(a) into a fraction. This is a standard procedure in mathematics, and we'll walk through it meticulously. Let's denote the recurring decimal 6,(a) as 'x'. Therefore, x = 6,(a). The next step involves multiplying 'x' by a power of 10 that shifts the repeating digit to the left of the decimal point. Since only one digit is recurring, we multiply by 10. This gives us 10x = 6(a),(a). Now, we subtract the original equation (x = 6,(a)) from this new equation (10x = 6(a),(a)). This subtraction eliminates the recurring part, leaving us with a simple equation that we can solve for 'a'.

Step-by-Step Solution

Let's translate the above explanation into a step-by-step solution.

  1. Define x: Let x = 6,(a)
  2. Multiply by 10: 10x = 6(a),(a)
  3. Subtract the original equation: 10x - x = 6(a),(a) - 6,(a)
  4. Simplify: This simplifies to 9x = 6(a) - 6, which further simplifies to 9x = 6a. Note that 6(a) represents the number 60 + a.
  5. Substitute the given value: We are given that 6,(a) = 19/3. Since x = 6,(a), we have x = 19/3.
  6. Solve for 'a': Substitute x = 19/3 into the equation 9x = 60 + a - 6. This gives us 9 * (19/3) = 60 + a - 6. Simplifying this equation will lead us to the value of 'a'.

Detailed Calculation

Now, let's perform the detailed calculations to find the value of 'a'. From the previous section, we have the equation 9 * (19/3) = 60 + a - 6. We will simplify this equation step by step.

  1. Simplify the left side: 9 * (19/3) = (9/3) * 19 = 3 * 19 = 57.
  2. Simplify the right side: 60 + a - 6 = 54 + a.
  3. Equate both sides: Now we have the equation 57 = 54 + a.
  4. Isolate 'a': To isolate 'a', we subtract 54 from both sides of the equation: 57 - 54 = a.
  5. Solve for 'a': This gives us a = 3.

Therefore, the value of 'a' in the equation 6,(a) = 19/3 is 3. This means that the recurring decimal is 6.3333..., which is indeed equivalent to 19/3.

Verification

To ensure our solution is correct, let's verify it by substituting a = 3 back into the original equation. We need to check if 6,(3) is equal to 19/3.

We know that 6,(3) represents the recurring decimal 6.3333.... To convert this recurring decimal to a fraction, we can use the same method we discussed earlier:

  1. Let x = 6.3333...
  2. Multiply by 10: 10x = 63.3333...
  3. Subtract the original equation: 10x - x = 63.3333... - 6.3333...
  4. Simplify: 9x = 57
  5. Solve for x: x = 57/9
  6. Further simplify: x = 19/3

This confirms that 6,(3) is indeed equal to 19/3, thus verifying our solution that a = 3.

Alternative Approach

While the method described above is the standard way to solve this type of problem, let's explore an alternative approach to further solidify our understanding. This approach involves directly converting the fraction 19/3 into a mixed number and then into a decimal form.

  1. Convert 19/3 to a mixed number: 19 divided by 3 gives us a quotient of 6 and a remainder of 1. Therefore, 19/3 can be written as the mixed number 6 1/3.
  2. Convert the fraction to a decimal: The fractional part 1/3 is a well-known recurring decimal, which is equal to 0.3333....
  3. Combine the whole number and decimal parts: Combining the whole number 6 with the decimal 0.3333... gives us 6.3333..., which is the recurring decimal 6,(3).

This alternative approach directly shows that when a = 3, the recurring decimal 6,(a) is equal to 19/3, providing another way to verify our solution.

Common Mistakes and How to Avoid Them

When solving problems involving recurring decimals, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

  1. Incorrectly converting the recurring decimal to a fraction: The most common mistake is misapplying the method for converting recurring decimals to fractions. Remember to multiply by the correct power of 10 (depending on the number of recurring digits) and subtract the original equation carefully. Double-check your calculations during this step.
  2. Arithmetic errors: Simple arithmetic errors during the simplification process can lead to incorrect answers. Always double-check your addition, subtraction, multiplication, and division operations.
  3. Misinterpreting the notation: It's crucial to understand the notation of recurring decimals. 6,(a) means 6.aaaa..., not 6 multiplied by 'a'. A clear understanding of the notation is essential for setting up the equations correctly.
  4. Forgetting to verify the solution: Always verify your solution by substituting the value of 'a' back into the original equation. This will help you catch any errors and ensure your answer is correct.

By being mindful of these common mistakes and taking the time to double-check your work, you can improve your accuracy and confidence in solving problems involving recurring decimals.

Practice Problems

To further enhance your understanding and skills in solving problems involving recurring decimals, here are a few practice problems:

  1. Find 'b' from the equation 3,(b) = 10/3.
  2. Solve for 'c' in the equation 1,(c) = 4/3.
  3. Determine the value of 'd' in the equation 9,(d) = 29/3.

Try solving these problems using the methods discussed in this article. Remember to convert the recurring decimals to fractions, set up the equations carefully, and verify your solutions. Practice is key to mastering this concept.

Conclusion

In this comprehensive guide, we have successfully found the value of 'a' in the equation 6,(a) = 19/3. We have explored the concept of recurring decimals, learned how to convert them to fractions, and applied a step-by-step approach to solve the equation. We also verified our solution and discussed an alternative approach. Furthermore, we highlighted common mistakes to avoid and provided practice problems to reinforce your understanding.

By mastering the techniques and concepts presented in this article, you will be well-equipped to tackle a wide range of mathematical problems involving recurring decimals. Remember, practice makes perfect, so keep solving problems and expanding your mathematical knowledge.

This problem exemplifies the importance of understanding recurring decimals and their conversion to fractions in solving equations. The systematic approach and verification steps ensure an accurate solution. Keep practicing, and you'll master these types of problems in no time!