Math Problem: Step-by-Step Guide & Solution

Hey guys! Let's dive into a classic math problem: $1 + \frac{3}{2} \times ( \frac{7}{4} - \frac{5}{3} )$. This might look a little intimidating at first, but trust me, we'll break it down step-by-step to make it super clear. We're gonna use the order of operations (PEMDAS/BODMAS) to get to the right answer. This ensures we do things in the correct order – parentheses/brackets first, then exponents/orders, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Knowing this rule is key to solving a lot of math problems, not just this one! Understanding the problem is the first step. The expression involves a mix of addition, multiplication, and subtraction with fractions. We'll need to remember how to handle fractions – finding common denominators and performing the arithmetic. Let's start with what's inside the parentheses. This is usually the trickiest part, but we'll tackle it together. We'll simplify the fraction part first, then the multiplication and then finally the addition. Let's make sure we have the correct answer. Now, let's get started!

Step 1: Solving the Parentheses

Alright, let's focus on the part inside the parentheses: $(\frac{7}{4} - \frac{5}{3})$. To subtract these fractions, we need a common denominator. The smallest number that both 4 and 3 divide into evenly is 12. So, we'll rewrite both fractions with a denominator of 12. To change $\frac{7}{4}$ to have a denominator of 12, we multiply both the numerator and denominator by 3: $\frac{7 \times 3}{4 \times 3} = \frac{21}{12}$. For $\frac{5}{3}$, we multiply both the numerator and denominator by 4: $\frac{5 \times 4}{3 \times 4} = \frac{20}{12}$. Now we can subtract the fractions: $\frac{21}{12} - \frac{20}{12} = \frac{1}{12}$. Great job, guys! We've simplified the expression inside the parentheses to $\frac{1}{12}$. Remember, the core of this step is to find that common denominator. Always look for the least common multiple (LCM) of the denominators to keep the numbers manageable. Always double-check your calculations, especially when dealing with fractions. A small mistake here can throw off the whole problem. We are off to a great start, and we've already done the first and the most difficult part. Let’s keep going!

Step 2: Multiplication

Next, we need to handle the multiplication: $\frac{3}{2} \times \frac{1}{12}$. Remember, to multiply fractions, you simply multiply the numerators together and the denominators together. So, we have: $\frac{3 \times 1}{2 \times 12} = \frac{3}{24}$. Now, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us: $\frac{3 \div 3}{24 \div 3} = \frac{1}{8}$. Awesome! We've successfully completed the multiplication part. Notice how we simplified the resulting fraction. This makes the next step, addition, much easier. Also, try to simplify fractions before multiplying if possible. For instance, in our original multiplication problem, we could have simplified $\frac{3}{2} \times \frac{1}{12}$ by dividing the 3 in the numerator of the first fraction and the 12 in the denominator of the second fraction by 3. This would have given us $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$, the same result, but with smaller numbers to work with from the start. This approach can save you a little time and effort. We're almost there! Just one more step to go and we'll have solved the problem. You guys are doing amazing!

Step 3: Addition

Finally, we're at the last step: $1 + \frac{1}{8}$. To add a whole number to a fraction, it's helpful to think of the whole number as a fraction with a denominator of 1. So, we can rewrite 1 as $\frac{8}{8}$. Now the problem becomes: $\frac{8}{8} + \frac{1}{8}$. Since the fractions now have a common denominator, we can simply add the numerators: $\frac{8 + 1}{8} = \frac{9}{8}$. And there you have it! Our final answer is $\frac{9}{8}$. You can also express this as a mixed number: $1 \frac{1}{8}$. Congrats, guys! You successfully solved the math problem! Let’s celebrate! This problem is a great example of how you can break down a complex-looking math expression into simpler steps. We've used key math concepts like the order of operations, finding common denominators, and simplifying fractions. These skills are fundamental in math and will be useful in a lot of problems you encounter in the future. Remember, practice makes perfect. The more you work with fractions and the order of operations, the easier they will become. Don't be afraid to try different problems and to ask for help if you get stuck. Keep up the amazing work! Don't worry, even if you made mistakes, the important thing is that you learned something new! Math can be fun! If you understood all these steps, you are well on your way to becoming a math whiz. You can try similar problems to practice your newly acquired skills.

Conclusion: Summary and Key Takeaways

We did it, guys! We successfully navigated the math problem: $1 + \frac{3}{2} \times ( \frac{7}{4} - \frac{5}{3} )$. We've seen how important it is to follow the order of operations (PEMDAS/BODMAS) to solve the problem correctly. We started by tackling the parentheses, where we had to subtract fractions. This involved finding a common denominator (12 in our case) and rewriting the fractions before subtracting. Then, we moved on to the multiplication, where we multiplied the fraction $\frac{3}{2}$ by the result inside the parentheses, $\frac{1}{12}$. We simplified the resulting fraction to $\frac{1}{8}$. Finally, we added 1 to the fraction $\frac{1}{8}$, converting 1 to the fraction $\frac{8}{8}$ and adding the numerators, which gave us the final answer of $\frac{9}{8}$ or $1 \frac{1}{8}$. This whole process highlights the core mathematical concepts and also the value of breaking down complex problems into smaller, manageable steps. Remember the key takeaways: the order of operations is your best friend. Always solve the parentheses first, then exponents (if any), then multiplication and division (from left to right), and finally addition and subtraction (also from left to right). When working with fractions, always make sure to find a common denominator before adding or subtracting. Simplify your fractions whenever possible to make calculations easier. This problem provides a solid foundation for more complex mathematical concepts and problem-solving. Keep practicing, keep learning, and don't be afraid to challenge yourselves! You guys are awesome!