Solving Numerical Expressions: Step-by-Step Calculation

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Hey guys! Let's break down how to solve the numerical expression (3/4 + 1/2) × (2/3 - 1/6). It might look a little intimidating at first, but don't worry, we'll go through it step by step. We'll cover the exact steps you need to take and, of course, find the final answer. So, grab your pencils and let's get started!

Understanding the Order of Operations

Before we dive into the actual calculation, it's super important to remember the order of operations. You might have heard of it as PEMDAS or BODMAS. It’s a set of rules that tells us the order in which we should perform mathematical operations. So, let’s break this down:

  • Parentheses (or Brackets)
  • Exponents (or Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Keeping this order in mind is crucial. If we don’t follow it, we might end up with the wrong answer. In our expression, we have parentheses, addition, subtraction, and multiplication, so we'll tackle them in the correct order.

Step 1: Solving Inside the Parentheses

The first thing we need to do is handle what's inside the parentheses. We've got two sets of parentheses here: (3/4 + 1/2) and (2/3 - 1/6). Let’s take them one at a time.

Solving (3/4 + 1/2)

To add these fractions, we need a common denominator. The smallest common denominator for 4 and 2 is 4. So, we’ll convert 1/2 to have a denominator of 4. To do this, we multiply both the numerator and the denominator of 1/2 by 2:

1/2 = (1 × 2) / (2 × 2) = 2/4

Now we can add the fractions:

3/4 + 2/4 = 5/4

So, the first set of parentheses simplifies to 5/4. Awesome! We’re making progress.

Solving (2/3 - 1/6)

Next up, we tackle the second set of parentheses: (2/3 - 1/6). Again, we need a common denominator to subtract these fractions. The smallest common denominator for 3 and 6 is 6. So, we’ll convert 2/3 to have a denominator of 6. We multiply both the numerator and the denominator of 2/3 by 2:

2/3 = (2 × 2) / (3 × 2) = 4/6

Now we can subtract the fractions:

4/6 - 1/6 = 3/6

We can simplify 3/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

3/6 = (3 ÷ 3) / (6 ÷ 3) = 1/2

So, the second set of parentheses simplifies to 1/2. Perfect! We've handled the parentheses.

Step 2: Multiplication

Now that we’ve simplified the expressions inside the parentheses, we’re left with:

(5/4) × (1/2)

This means we need to multiply these two fractions together. Multiplying fractions is pretty straightforward – we multiply the numerators together and the denominators together:

(5/4) × (1/2) = (5 × 1) / (4 × 2) = 5/8

And that’s it! We’ve performed the multiplication. The result is 5/8.

Final Result and the Answer

So, after following all the steps, we've found that the final result of the expression (3/4 + 1/2) × (2/3 - 1/6) is 5/8.

Now, let’s compare this to the options given:

A) 1 B) 2/3 C) 5/6 D) 1/2

None of these options match our result of 5/8. It seems there might be a mistake in the provided options, or perhaps we need to present our answer in a different format. However, based on our calculations, 5/8 is the correct answer.

Common Mistakes and How to Avoid Them

When solving numerical expressions, it’s easy to make mistakes if you’re not careful. Here are a few common pitfalls and tips on how to avoid them:

  1. Forgetting the Order of Operations: This is the most common mistake. Always remember PEMDAS/BODMAS. If you skip a step or do operations in the wrong order, you'll likely get the wrong answer. Always double-check that you're following the correct order.
  2. Incorrectly Finding Common Denominators: When adding or subtracting fractions, finding the correct common denominator is crucial. A wrong denominator means a wrong answer. Make sure you’re finding the least common multiple (LCM) of the denominators.
  3. Math Errors: Simple arithmetic errors can happen, especially when you’re working quickly. Take your time and double-check your calculations. Use a calculator if you need to.
  4. Simplifying Fractions Too Late: Sometimes, simplifying fractions earlier in the process can make the calculations easier. Look for opportunities to simplify before you multiply or divide, especially.
  5. Not Showing Your Work: It might seem faster to do calculations in your head, but writing down each step helps you catch mistakes and makes it easier to review your work. Always show your work – it’s a lifesaver!

Tips for Accuracy

To ensure you’re getting the correct answer every time, try these tips:

  • Write Neatly: Messy handwriting can lead to misreading your own numbers. Keep your work organized and easy to read.
  • Double-Check Each Step: Before moving on to the next operation, take a quick look back to make sure you haven’t made any errors.
  • Use a Calculator: For complex calculations, a calculator can be a great tool. Just be sure you’re entering the numbers and operations correctly.
  • Practice Regularly: The more you practice, the more comfortable you’ll become with these types of problems. Try different expressions and challenge yourself!

Real-World Applications of Numerical Expressions

You might be wondering, why do we even need to know how to solve these numerical expressions? Well, they’re actually used in lots of real-world situations! Here are a few examples:

  • Cooking: When you’re doubling or halving a recipe, you need to work with fractions and perform calculations to get the right amounts of ingredients. Numerical expressions come in handy here!
  • Finance: Calculating interest, figuring out discounts, and managing budgets all involve numerical expressions. Whether you’re saving money or planning a big purchase, these skills are essential.
  • Construction and Engineering: Measuring materials, calculating areas and volumes, and designing structures require a solid understanding of numerical expressions. It’s crucial for accuracy and safety.
  • Science: Many scientific calculations, like measuring chemical reactions or analyzing data, involve numerical expressions. If you’re into science, you’ll definitely use these skills.
  • Everyday Shopping: Figuring out the best deals, calculating sale prices, and splitting costs with friends all involve working with numbers and expressions. It helps you make smart choices.

Practicing with More Examples

To really nail this, let’s look at a couple more examples. The best way to get comfortable with numerical expressions is to practice, practice, practice!

Example 1: 2 × (1/3 + 1/4)

  1. Parentheses: First, we solve the expression inside the parentheses. We need a common denominator for 1/3 and 1/4, which is 12.

    • 1/3 = 4/12
    • 1/4 = 3/12
    • 4/12 + 3/12 = 7/12

  2. Multiplication: Now we multiply 2 by 7/12.

    • 2 × (7/12) = (2 × 7) / 12 = 14/12

  3. Simplify: We can simplify 14/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

    • 14/12 = 7/6

So, the final answer is 7/6.

Example 2: (1/2 - 1/5) × (3/4 + 1/8)

  1. Parentheses: Let’s tackle the first set of parentheses.

    • Common denominator for 2 and 5 is 10.
    • 1/2 = 5/10
    • 1/5 = 2/10
    • 5/10 - 2/10 = 3/10

  2. Parentheses: Now, the second set.

    • Common denominator for 4 and 8 is 8.
    • 3/4 = 6/8
    • 6/8 + 1/8 = 7/8

  3. Multiplication: Multiply the results.

    • (3/10) × (7/8) = (3 × 7) / (10 × 8) = 21/80

So, the final answer is 21/80.

Conclusion

Solving numerical expressions might seem tricky at first, but by remembering the order of operations and taking it one step at a time, you can master them! The key is to understand the rules, practice regularly, and double-check your work. Don't rush, and you’ll be solving these expressions like a pro in no time. We've walked through the steps, looked at common mistakes, and even explored real-world applications. Now it's your turn to practice and build your skills. You got this!