Math Problem: Simplify (11/12 + 1/24)^2 - (1/6 + 1/24)

Solving the Expression: A Step-by-Step Mathematical Journey

Welcome, fellow math enthusiasts! Today, we're diving into a fun mathematical puzzle that involves fractions, exponents, and subtraction. Our goal is to simplify the expression: $\left(\frac{11}{12}+\frac{1}{24}\right)^2-\left(\frac{1}{6}+\frac{1}{24}\right)$. Don't worry if fractions and exponents seem a bit daunting; we'll break it down into manageable steps, making it clear and easy to follow. This type of problem is a fantastic way to sharpen your arithmetic skills and build confidence in handling more complex mathematical expressions. We'll tackle each part of the expression systematically, ensuring we don't miss any crucial details. So, grab your thinking caps, and let's embark on this mathematical adventure together!

Understanding the Order of Operations: PEMDAS/BODMAS

Before we jump into solving, it's essential to remember the golden rule of arithmetic: the order of operations. In mathematics, we follow a specific sequence to ensure that every problem has a single, correct answer. This is often remembered by acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS (Brackets, Orders, Division and Multiplication from left to right, Addition and Subtraction from left to right). For our problem, $\left(\frac{11}{12}+\frac{1}{24}\right)^2-\left(\frac{1}{6}+\frac{1}{24}\right)$, we will first address the operations inside the parentheses. Then, we'll handle the exponent, and finally, perform the subtraction. Following this order meticulously is key to arriving at the correct solution and avoiding common mistakes that can occur when calculations are done haphazardly. It's like building a house; you need a solid foundation and a clear plan before you start constructing the walls and roof. The order of operations provides that essential structure for all our mathematical endeavors, ensuring consistency and accuracy.

Step 1: Simplifying Inside the First Parentheses

Our journey begins with the first set of parentheses: $\left(\frac{11}{12}+\frac{1}{24}\right)$. To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 24 is 24. So, we convert $\frac{11}{12}$ to an equivalent fraction with a denominator of 24. We multiply the numerator and denominator by 2: $\frac{11 \times 2}{12 \times 2} = \frac{22}{24}$. Now, our expression inside the parentheses becomes: $\frac{22}{24} + \frac{1}{24}$. Since the denominators are the same, we simply add the numerators: $22 + 1 = 23$. Thus, the sum inside the first parentheses is $\frac{23}{24}$. This step is crucial because it simplifies a part of the larger expression, making the subsequent calculations much more straightforward. Remember, finding a common denominator is a fundamental skill in fraction arithmetic, and mastering it will open doors to solving many more complex problems. We are essentially rewriting the fractions so they represent the same 'size' of pie, allowing us to combine them accurately. This meticulous attention to detail in the initial stages sets the stage for a smooth progression through the rest of the problem.

Step 2: Applying the Exponent

Now that we've simplified the first set of parentheses to $\frac{23}{24}$, we need to apply the exponent. The expression is now $\left(\frac{23}{24}\right)^2 - \left(\frac{1}{6}+\frac{1}{24}\right)$. Squaring a fraction means multiplying the fraction by itself: $\left(\frac{23}{24}\right)^2 = \frac{23}{24} \times \frac{23}{24}$. To multiply fractions, we multiply the numerators together and the denominators together: $23 \times 23 = 529$ and $24 \times 24 = 576$. So, $\left(\frac{23}{24}\right)^2 = \frac{529}{576}$. This step involves squaring both the numerator and the denominator. It's important to perform this calculation accurately, as errors here can propagate through the rest of the problem. Calculating $23^2$ and $24^2$ requires careful multiplication. We can break down $23 \times 23$ as $(20+3)(20+3) = 400 + 60 + 60 + 9 = 529$, and $24 \times 24$ as $(20+4)(20+4) = 400 + 80 + 80 + 16 = 576$. This exponentiation step transforms our fraction into a new value, representing the result of multiplying it by itself. It's a significant transformation that moves us closer to the final answer.

Step 3: Simplifying Inside the Second Parentheses

Next, let's focus on the second set of parentheses: $\left(\frac{1}{6}+\frac{1}{24}\right)$. Similar to Step 1, we need a common denominator. The LCM of 6 and 24 is 24. We convert $\frac{1}{6}$ to an equivalent fraction with a denominator of 24 by multiplying the numerator and denominator by 4: $\frac{1 \times 4}{6 \times 4} = \frac{4}{24}$. Now, the expression inside the parentheses is: $\frac{4}{24} + \frac{1}{24}$. Adding the numerators gives us $4 + 1 = 5$. So, the sum inside the second parentheses is $\frac{5}{24}$. This step mirrors the process in Step 1, reinforcing the importance of finding common denominators for accurate fraction addition. Each simplification brings us closer to the final calculation, making the overall problem feel more manageable. It's about breaking down complexity into simpler, understandable parts. We're now ready to substitute this simplified value back into our main expression.

Step 4: Performing the Final Subtraction

We have now reached the final step: subtraction. Our expression has been simplified to $\frac{529}{576} - \frac{5}{24}$. To subtract these fractions, we once again need a common denominator. The LCM of 576 and 24 is 576. We already have the first fraction with the correct denominator. Now, we need to convert $\frac{5}{24}$ to an equivalent fraction with a denominator of 576. We find what we need to multiply 24 by to get 576. Since $24 \times 24 = 576$, we multiply the numerator and denominator of $\frac{5}{24}$ by 24: $\frac{5 \times 24}{24 \times 24} = \frac{120}{576}$. Now, our subtraction becomes: $\frac{529}{576} - \frac{120}{576}$. With a common denominator, we subtract the numerators: $529 - 120 = 409$. Therefore, the final answer is $\frac{409}{576}$. This final subtraction is where all our previous efforts culminate. It's essential to ensure the numerator subtraction is accurate. If the resulting fraction can be simplified, we would do so, but in this case, 409 and 576 share no common factors other than 1, so the fraction is already in its simplest form. This completes our step-by-step solution, demonstrating the power of following mathematical rules.

Conclusion: A Rewarding Mathematical Endeavor

We have successfully navigated the expression $\left(\frac{11}{12}+\frac{1}{24}\right)^2-\left(\frac{1}{6}+\frac{1}{24}\right)$ from start to finish. By meticulously following the order of operations and applying the rules of fraction arithmetic, we arrived at the simplified answer of $\frac{409}{576}$. Each step, from finding common denominators to squaring a fraction and performing subtraction, was crucial. This problem highlights the importance of precision and patience in mathematics. Whether you're a student learning these concepts for the first time or a seasoned mathematician, breaking down complex problems into smaller, manageable parts is always a sound strategy. We hope this detailed walkthrough has been both informative and encouraging. Remember, with practice, even the most intimidating mathematical expressions can be conquered. Keep exploring, keep practicing, and most importantly, keep enjoying the fascinating world of mathematics!

For further exploration into fraction arithmetic and algebraic simplification, you might find resources at Khan Academy or Math is Fun incredibly helpful. These platforms offer extensive explanations, practice problems, and engaging tutorials that can deepen your understanding.