Solving The Math Expression: Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that might look a bit intimidating at first, but trust me, we'll break it down together. We're going to solve the expression: $\frac{1}{2}(\frac{1}{10}-\frac{3}{5}) + -1.9$. Don't worry, we'll go through each step nice and slow so everyone can follow along. Let's get started!

Understanding the Expression

Before we jump into the calculations, let's make sure we understand what we're looking at. The expression $\frac{1}{2}(\frac{1}{10}-\frac{3}{5}) + -1.9$ involves a combination of fractions, subtraction, multiplication, and adding a negative decimal. The key to solving any mathematical expression like this is to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This helps us tackle the problem in the correct sequence, ensuring we arrive at the right answer. Understanding each component—the fractions, the decimal, and the operations—is the first step toward demystifying the problem and making it more approachable. So, before we even pick up our calculators or pencils, let's take a moment to appreciate the structure of this expression and mentally prepare ourselves for the journey ahead. Remember, math isn't about rushing to the solution; it's about understanding the path.

Breaking Down the Components

Let's break down each part: We have fractions inside parentheses $(\frac{1}{10} - \frac{3}{5})$, which means we'll deal with those first. Then, we'll multiply the result by $\frac{1}{2}$, and finally, we'll add -1.9 to the result. Each of these steps is a building block towards the final solution. Recognizing these individual components makes the overall expression less daunting and allows us to focus on each operation one at a time. Think of it like assembling a puzzle – each piece has its place, and once connected correctly, the whole picture becomes clear. By understanding the role of each number and operation, we set ourselves up for success in solving the problem. It's all about taking it step-by-step and making sure we're solid on the basics before moving on.

Step 1: Solving Inside the Parentheses

The first thing we need to do according to PEMDAS is to solve the operation inside the parentheses: $(\frac{1}{10} - \frac{3}{5})$. To subtract these fractions, we need a common denominator. Finding the common denominator is crucial in fraction arithmetic. The smallest common denominator for 10 and 5 is 10. So, we need to convert $\frac{3}{5}$ to have a denominator of 10. We can do this by multiplying both the numerator and the denominator of $\frac{3}{5}$ by 2. This gives us $\frac{3 imes 2}{5 imes 2} = \frac{6}{10}$. Now we have the expression: $(\frac{1}{10} - \frac{6}{10})$. Subtracting the numerators gives us: $\frac{1 - 6}{10} = \frac{-5}{10}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This simplifies $\frac{-5}{10}$ to $\frac{-1}{2}$. So, the expression inside the parentheses simplifies to $-\frac{1}{2}$.

Converting to a Common Denominator

As we just saw, getting a common denominator is super important when we're trying to add or subtract fractions. Think of it like trying to compare apples and oranges – they're different! But if we convert them into something comparable, like fruit slices, it becomes much easier. In our case, we needed to make the denominators (the bottom numbers) of the fractions the same so we could actually subtract them. This process of finding a common denominator allows us to express fractions in equivalent forms, making the arithmetic operations straightforward. By ensuring that the fractions are on the same “playing field,” we avoid mixing apples and oranges and get an accurate result. It's a fundamental step in fraction manipulation, and mastering it opens the door to tackling more complex math problems with confidence.

Simplifying the Fraction

After performing the subtraction inside the parentheses, we ended up with $\frac{-5}{10}$. But we didn't stop there! We simplified it to $\frac{-1}{2}$. Why? Because simplifying fractions makes them easier to work with in further calculations and gives us the most reduced form of the fraction. Simplifying a fraction means dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common factor (GCF). In our case, the GCF of -5 and 10 is 5. Dividing both by 5 gives us the simplified fraction. This step is not just about neatness; it's about making the numbers as manageable as possible. Simpler numbers mean simpler calculations, which reduce the chances of making mistakes down the line. So, always take a moment to see if you can simplify a fraction – your future self will thank you!

Step 2: Multiplication

Now that we've solved the parentheses, we move on to the multiplication: $\frac{1}{2} imes (-\frac{1}{2})$. To multiply fractions, we simply multiply the numerators together and the denominators together. So, we have: $\frac{1 imes -1}{2 imes 2} = \frac{-1}{4}$. Multiplying fractions is pretty straightforward once you've got the hang of it. Remember, a positive number multiplied by a negative number gives a negative result, which is why our answer is negative. This step transforms the fraction multiplication into a single, simplified fraction that we can use in the next part of our calculation. Keep in mind, the simplicity of this operation is a testament to the elegance of fractions in mathematics – direct and efficient.

Multiplying Numerators and Denominators

When we multiply fractions, we're not just performing a mechanical operation; we're actually combining parts of wholes. The rule of thumb is to multiply the numerators (the top numbers) together to get the new numerator, and multiply the denominators (the bottom numbers) together to get the new denominator. This method works because fractions represent parts of a whole, and multiplying them is like taking a fraction of a fraction. It’s a fundamental concept in arithmetic and serves as a building block for more advanced mathematical operations. Understanding this principle allows us to confidently tackle multiplication problems involving fractions and appreciate the inherent logic behind this mathematical operation.

Handling Negative Signs

In our problem, we multiplied a positive fraction (rac{1}{2}) by a negative fraction ($-\frac{1}{2}$). It's crucial to remember the rules of signs in multiplication: a positive times a positive is positive, a negative times a negative is also positive, but a positive times a negative (or vice versa) is negative. Keeping track of negative signs is essential for getting the correct answer in mathematical calculations. In our case, the negative sign in $-\frac{1}{2}$ carries through to the product, making our result $\frac{-1}{4}$. Paying attention to these details ensures that our calculations are accurate and that we're building a strong foundation for more complex mathematical operations.

Step 3: Addition

Finally, we need to add the result from the multiplication to -1.9: $\frac{-1}{4} + -1.9$. To do this, it's easiest to convert both numbers to decimals. We know that $\frac{-1}{4}$ is equal to -0.25. So, our expression becomes: -0.25 + -1.9. Adding these two negative numbers is straightforward. When adding negative numbers, we simply add their absolute values and keep the negative sign. The absolute value of -0.25 is 0.25, and the absolute value of -1.9 is 1.9. Adding these gives us 0.25 + 1.9 = 2.15. Since both numbers were negative, our final answer is -2.15.

Converting Fractions to Decimals

In this step, we converted the fraction $\frac{-1}{4}$ to a decimal, which is -0.25. Converting fractions to decimals can make addition and subtraction easier, especially when dealing with numbers that are already in decimal form. To convert a fraction to a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). In our case, -1 divided by 4 equals -0.25. This conversion allows us to perform the addition using decimals, which many people find more intuitive than working with fractions directly. It’s a handy skill to have in your mathematical toolkit, and it often simplifies calculations.

Adding Negative Numbers

When we add negative numbers, it's like moving further into the negative side of the number line. The process of adding negative numbers involves adding their absolute values and then applying the negative sign to the result. In our problem, we added -0.25 and -1.9. We found the absolute values (0.25 and 1.9), added them together (2.15), and then applied the negative sign, giving us -2.15. This rule is fundamental in arithmetic and is essential for accurately handling negative numbers in mathematical expressions. Understanding this principle ensures that we can confidently add negative numbers and avoid common pitfalls.

Final Answer

So, after carefully working through each step, we've found that the solution to the expression $\frac{1}{2}(\frac{1}{10}-\frac{3}{5}) + -1.9$ is -2.15. Great job, guys! We tackled fractions, multiplication, and negative numbers. Remember, the key is to take it one step at a time and follow the order of operations. Keep practicing, and you'll become math pros in no time!