Evaluating The Expression: $8 imes rac{15}{5}-(5+9)$

Let's break down how to evaluate the expression $8 \times \frac{15}{5}-(5+9)$. In this comprehensive guide, we'll walk through each step, ensuring you understand the order of operations and the reasoning behind every calculation. So, buckle up, math enthusiasts, and let's get started!

Understanding the Order of Operations

When we're faced with a mathematical expression like this, we can't just go from left to right. We need a set of rules to ensure everyone gets the same answer. That's where the order of operations comes in. It's like a mathematical GPS, guiding us through the expression. The most common mnemonic to remember the order is PEMDAS, which stands for:

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

Why is Order Important?

Imagine if we didn't have an order. Some people might add before multiplying, leading to completely different results. The order of operations ensures consistency and clarity in mathematics. Think of it as the grammar of math – it provides structure and meaning.

Let's Apply PEMDAS to Our Problem

Now, let's apply this to our expression: $8 \times \frac{15}{5}-(5+9)$. We'll go through each step of PEMDAS one by one to ensure we solve this correctly. Remember, math is a journey, not a race! Take your time and understand each step.

Step 1: Parentheses

Our expression has a set of parentheses: $(5+9)$. According to PEMDAS, we need to tackle this first. It's like clearing the first hurdle in a race. We need to simplify what's inside the parentheses before we can move on to the rest of the expression.

Performing the Addition

Inside the parentheses, we have a simple addition problem: $5 + 9$. Guys, this is pretty straightforward! 5 plus 9 equals 14. So, we can replace $(5+9)$ with $14$ in our expression.

Our Updated Expression

Now our expression looks like this: $8 \times \frac{15}{5} - 14$. See how we've simplified things already? We've knocked out the parentheses, and now we're ready to move on to the next step.

Step 2: Multiplication and Division

Next up, we have multiplication and division. Remember, PEMDAS tells us to perform these operations from left to right. It's like reading a sentence – we work through it in the order we see it.

Tackling Multiplication and Division from Left to Right

Looking at our expression, $8 \times \frac{15}{5} - 14$, we see multiplication first: $8 \times \frac{15}{5}$. To make this easier, let's remember that multiplying by a fraction is the same as multiplying by the numerator and then dividing by the denominator. So, we're essentially doing $8 \times 15$ and then dividing by $5$.

Performing the Multiplication: $8 \times 15$

Let's calculate $8 \times 15$. You can do this in your head, on paper, or with a calculator. The result is 120. So now we have $120 \div 5$.

Performing the Division: $120 \div 5$

Now we divide 120 by 5. This gives us 24. So, $8 \times \frac{15}{5}$ simplifies to 24. This is a huge step forward! We've conquered the multiplication and division part of the expression.

Our Even Simpler Expression

Our expression is now down to: $24 - 14$. We're in the home stretch, guys! Just one operation left.

Step 3: Addition and Subtraction

Finally, we have addition and subtraction. Just like with multiplication and division, we perform these operations from left to right. In our case, we only have subtraction left, so this is easy peasy!

Performing the Subtraction: $24 - 14$

Let's subtract 14 from 24. This is a straightforward subtraction problem. 24 minus 14 equals 10. So, the final answer is 10!

The Grand Finale

We've done it! We've successfully evaluated the expression $8 \times \frac{15}{5}-(5+9)$. By following the order of operations, we've arrived at the solution: 10.

Let's Recap the Steps

To make sure we've got it all down, let's quickly recap the steps we took:

1. Parentheses: We started by simplifying the expression inside the parentheses: $(5+9) = 14$.
2. Multiplication and Division: We then performed the multiplication and division from left to right: $8 \times \frac{15}{5} = 24$.
3. Addition and Subtraction: Finally, we performed the subtraction: $24 - 14 = 10$.

Common Mistakes to Avoid

It's easy to make mistakes in math, but knowing the common pitfalls can help you avoid them. Here are a few things to watch out for when evaluating expressions:

• Forgetting PEMDAS: The most common mistake is not following the order of operations. Always remember PEMDAS to guide you.
• Incorrectly Performing Multiplication/Division or Addition/Subtraction: Make sure you're performing these operations in the correct order from left to right.
• Simple Arithmetic Errors: Double-check your calculations to avoid simple mistakes. Even the smallest error can throw off your final answer.

Practice Makes Perfect

The best way to master evaluating expressions is to practice. The more you practice, the more comfortable you'll become with the order of operations and the different types of problems you might encounter.

Try These Practice Problems

Here are a few practice problems you can try on your own:

1. $12 + 3 \times 4 - 6 \div 2$
2. $(10 - 2) \div 4 + 5 \times 2$
3. $15 \div (3 + 2) \times 4 - 1$

Work through these problems step-by-step, using PEMDAS as your guide. Check your answers with a calculator or an online solver to make sure you're on the right track.

Real-World Applications of Order of Operations

The order of operations isn't just some abstract mathematical concept. It's used in many real-world applications, from programming to finance. Whenever you're dealing with complex calculations, you'll need to understand and apply the order of operations to get the correct result.

Programming

In programming, expressions are used to perform calculations and make decisions. The order of operations is crucial for ensuring that these expressions are evaluated correctly. If you write code that doesn't follow the order of operations, you'll likely get unexpected results.

Finance

In finance, calculations often involve multiple operations, such as addition, subtraction, multiplication, and division. For example, calculating compound interest requires understanding the order of operations to ensure that the interest is calculated correctly.

Conclusion

Evaluating expressions using the order of operations is a fundamental skill in mathematics. By understanding PEMDAS and practicing regularly, you can become confident in your ability to solve even the most complex expressions. Remember, math is like a puzzle – each step builds on the previous one. So, keep practicing, stay curious, and you'll master it in no time! Guys, keep up the awesome work, and happy calculating!