Solving A Complex Mathematical Expression Step-by-Step
Hey guys! Today, we are diving into a fascinating mathematical problem that might look intimidating at first glance, but don't worry, we'll break it down step by step. Our mission is to solve the expression: 16 2/3 * 3/125 - 18/39 ÷ 1 11/13 + 13/20. This problem combines multiplication, division, addition, and subtraction of fractions and mixed numbers, offering a comprehensive workout for our math skills. So, let's put on our thinking caps and get started!
Understanding the Order of Operations
Before we even touch the numbers, it's crucial to understand the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is the golden rule that ensures we solve the problem correctly. If we mess up the order, we'll end up with the wrong answer, and nobody wants that, right? So, keep PEMDAS/BODMAS in your mind like your favorite song!
In our expression, we have multiplication, division, subtraction, and addition. According to PEMDAS/BODMAS, we'll tackle multiplication and division first, working from left to right, and then handle addition and subtraction, again from left to right. This systematic approach is our best friend in solving complex math problems. It's like having a roadmap that guides us through the maze of numbers and operations.
Converting Mixed Numbers and Simplifying Fractions
Now, let's get our hands dirty with the numbers! The first thing we see is a mixed number: 16 2/3. Mixed numbers are a mix of a whole number and a fraction, and they can be a bit tricky to work with directly. So, our first move is to convert this mixed number into an improper fraction. To do this, we multiply the whole number (16) by the denominator of the fraction (3) and then add the numerator (2). This gives us (16 * 3) + 2 = 50. We then put this result over the original denominator, giving us 50/3. See? Not so scary after all!
Next, let's convert the mixed number 1 11/13 into an improper fraction. We do the same thing: multiply the whole number (1) by the denominator (13) and add the numerator (11). This gives us (1 * 13) + 11 = 24. Place this over the original denominator, and we get 24/13. Now we have all our numbers in fraction form, which makes them much easier to handle in calculations. It's like translating a foreign language into your native tongue – suddenly everything makes sense!
Performing Multiplication and Division
With our fractions ready, let's tackle the multiplication and division. Our expression now looks like this: 50/3 * 3/125 - 18/39 ÷ 24/13 + 13/20. We'll start with the multiplication: 50/3 * 3/125. To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, 50 * 3 = 150, and 3 * 125 = 375. This gives us the fraction 150/375.
But hold on! We can simplify this fraction before moving on. Both 150 and 375 are divisible by 75. Dividing both the numerator and the denominator by 75, we get 2/5. Simplifying fractions whenever possible makes our calculations easier and keeps the numbers manageable. It's like decluttering your workspace before starting a big project – it just makes everything smoother.
Now, let's move on to the division: 18/39 ÷ 24/13. Dividing fractions might seem tricky, but there's a neat trick: we flip the second fraction and multiply. So, 18/39 ÷ 24/13 becomes 18/39 * 13/24. Let's multiply: 18 * 13 = 234, and 39 * 24 = 936. This gives us the fraction 234/936.
Again, we can simplify this fraction. Both 234 and 936 are divisible by 234. Dividing both the numerator and the denominator by 234, we get 1/4. Simplifying fractions is like finding a shortcut in a maze – it saves us time and effort.
Adding and Subtracting Fractions
Our expression is now significantly simpler: 2/5 - 1/4 + 13/20. We've conquered the multiplication and division, and now it's time for addition and subtraction. But there's a catch: we can only add or subtract fractions that have the same denominator. So, we need to find a common denominator for 5, 4, and 20. The least common multiple (LCM) of these numbers is 20.
To get each fraction to have a denominator of 20, we need to multiply the numerator and denominator of each fraction by the appropriate number. For 2/5, we multiply both by 4, giving us 8/20. For 1/4, we multiply both by 5, giving us 5/20. The fraction 13/20 already has the correct denominator, so we can leave it as is.
Now our expression looks like this: 8/20 - 5/20 + 13/20. We can now perform the subtraction and addition. First, 8/20 - 5/20 = 3/20. Then, 3/20 + 13/20 = 16/20.
Final Simplification
We're almost there! We have the fraction 16/20. But as good mathematicians, we always want to simplify our answers as much as possible. Both 16 and 20 are divisible by 4. Dividing both the numerator and the denominator by 4, we get 4/5.
Conclusion
And there you have it! The solution to the complex mathematical expression 16 2/3 * 3/125 - 18/39 ÷ 1 11/13 + 13/20 is 4/5. We did it! We took a seemingly daunting problem and broke it down into manageable steps, using the order of operations, converting mixed numbers, simplifying fractions, and finding common denominators. Math can be challenging, but with a systematic approach and a bit of practice, we can conquer any problem.
Remember, guys, the key to mastering math is to practice regularly and break down complex problems into smaller, more manageable steps. Don't be afraid to make mistakes – they're part of the learning process. Keep practicing, and you'll become a math whiz in no time! And if you ever get stuck, just remember PEMDAS/BODMAS and take it one step at a time. You've got this!