Evaluating Expressions: A/10 + A/15 For Given Values

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate the expression a/10 + a/15 for different values of a. This means we'll be substituting the given values of a (which are 1, 2, 5, 7, 4, 5, and 6) into the expression and calculating the result. It might sound a bit complicated, but trust me, it's super straightforward once you get the hang of it! So, grab your calculators (or your mental math skills!) and let's jump right in!

Understanding the Expression

Before we start plugging in numbers, let's take a closer look at the expression a/10 + a/15. This is a simple algebraic expression involving fractions. The variable a represents a number that can change, and in this case, we have a list of numbers that a will take on. The expression tells us to do two things: first, divide a by 10, and then divide a by 15. Finally, we need to add the results of these two divisions together. Understanding this order of operations is crucial for getting the correct answer. Think of it like a recipe – you need to follow the steps in the right order to get the delicious final product! This is the same concept with math; understanding the structure helps significantly.

To make things even easier, we can actually simplify the expression before we start substituting the values of a. This involves finding a common denominator for the fractions 1/10 and 1/15. The least common multiple of 10 and 15 is 30. So, we can rewrite the fractions as follows:

• a/10 = (3a)/30
• a/15 = (2a)/30

Now, our expression becomes:

(3a)/30 + (2a)/30

Since the fractions now have the same denominator, we can simply add the numerators:

(3a + 2a) / 30 = (5a) / 30

And we can simplify further by dividing both the numerator and the denominator by 5:

(5a) / 30 = a / 6

So, the simplified expression is a/6. This means that instead of doing two divisions and an addition, we can simply divide a by 6. This will save us some time and effort in the long run. This simplification step highlights the importance of understanding algebraic manipulation to make problem-solving more efficient. Simplifying complex problems into smaller components makes them easier to manage and solve accurately. Now that we've simplified the expression, it's much easier to evaluate for different values of 'a'.

Evaluating for Different Values of 'a'

Now comes the fun part – evaluating the simplified expression a/6 for each given value of a. We have the following values: 1, 2, 5, 7, 4, 5, and 6. For each value, we'll simply substitute it into the expression and calculate the result. Let's go through each one step-by-step:

1. a = 1: Substitute 1 for a in the expression a/6: 1/6. So, when a is 1, the value of the expression is 1/6.
2. a = 2: Substitute 2 for a in the expression a/6: 2/6. We can simplify this fraction by dividing both the numerator and denominator by 2, giving us 1/3. So, when a is 2, the value of the expression is 1/3.
3. a = 5: Substitute 5 for a in the expression a/6: 5/6. This fraction is already in its simplest form. So, when a is 5, the value of the expression is 5/6.
4. a = 7: Substitute 7 for a in the expression a/6: 7/6. This is an improper fraction (the numerator is greater than the denominator), which means it's greater than 1. We can also express it as a mixed number: 1 1/6. So, when a is 7, the value of the expression is 7/6 or 1 1/6.
5. a = 4: Substitute 4 for a in the expression a/6: 4/6. We can simplify this fraction by dividing both the numerator and denominator by 2, giving us 2/3. So, when a is 4, the value of the expression is 2/3.
6. a = 5: We already calculated this one! When a is 5, the value of the expression is 5/6.
7. a = 6: Substitute 6 for a in the expression a/6: 6/6. This simplifies to 1. So, when a is 6, the value of the expression is 1.

And there you have it! We've evaluated the expression a/6 for all the given values of a. We've seen how substituting different values into an expression changes the result. This is a fundamental concept in algebra and is used extensively in various mathematical and scientific fields. Understanding how to evaluate expressions like this is essential for building a strong foundation in mathematics.

Results Summary

To make it super clear, let's summarize our results in a neat little table. This will help us see all the values at a glance and reinforce our understanding. A table is a great way to organize information and make it easy to compare different results.

As you can see, the value of the expression changes depending on the value of a. When a is small, the value of the expression is also small. As a increases, the value of the expression also increases. This shows the direct relationship between a and the expression a/6. This relationship is a key concept in understanding functions and how variables affect outcomes in mathematical expressions.

We've successfully evaluated the expression for all the given values, and we've organized our findings in a clear and concise table. This process not only gives us the answers but also helps us understand the underlying concepts and the relationship between variables and expressions. Keep practicing these types of problems, and you'll become a pro at evaluating expressions in no time!

Importance of Order of Operations

Before we wrap up, let's quickly touch on the importance of the order of operations. Remember way back at the beginning, we talked about how crucial it is to perform operations in the correct sequence? This is a fundamental principle in mathematics, and if we ignore it, we'll end up with the wrong answers. The order of operations is often remembered by the acronym PEMDAS (or BODMAS, depending on where you learned it), which stands for:

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

In our problem, we had a simple expression involving division and addition. But in more complex expressions, following the order of operations is essential. For example, if we had an expression like (2 + 3) * 4, we would first perform the addition inside the parentheses (2 + 3 = 5), and then multiply the result by 4 (5 * 4 = 20). If we ignored the parentheses and performed the multiplication first, we would get 3 * 4 = 12, and then add 2, giving us 14, which is incorrect.

Understanding and applying the order of operations is crucial for accuracy in mathematics. It's like the grammar of math – it ensures that everyone interprets an expression in the same way. So, always remember PEMDAS (or BODMAS) when you're dealing with complex expressions. This ensures consistency and accuracy in your mathematical calculations.

Conclusion

Alright, guys, we've reached the end of our journey into evaluating expressions! We took a seemingly complex problem and broke it down into manageable steps. We learned how to simplify an expression, substitute values for a variable, and calculate the result. We also highlighted the importance of the order of operations and how it ensures accuracy in our calculations.

By working through this problem, we've not only found the answers but also reinforced some fundamental mathematical concepts. Understanding how to evaluate expressions is a key skill in algebra and beyond. It's the foundation for solving equations, graphing functions, and tackling more advanced mathematical problems.

So, keep practicing, keep exploring, and keep having fun with math! Remember, math isn't just about numbers and formulas; it's about problem-solving, critical thinking, and understanding the world around us. And who knows, maybe one day you'll be using these skills to solve real-world problems and make a positive impact on the world. Until next time, keep those brains engaged and those calculators handy! You got this!