Step-by-Step Guide To Solve 0.75 + 2/5 × 1 3/4
Hey guys! Math can sometimes seem like a puzzle, but don't worry, we're here to break it down step by step. Today, we're tackling the problem 0.75 + 2/5 × 1 3/4. This might look a little intimidating at first glance, but with the right approach, it's totally manageable. We'll go through each operation methodically, making sure you understand exactly what's happening and why. So, grab your pencils and let's dive in!
Understanding the Order of Operations
Before we jump into the nitty-gritty of solving this mathematical problem, it's super important to understand the order of operations. You might have heard of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Think of it as a roadmap for solving mathematical expressions – it tells us which operations to tackle first to avoid any confusion and get to the right answer.
In our case, the expression is 0.75 + 2/5 × 1 3/4. According to PEMDAS, we need to handle the multiplication part (2/5 × 1 3/4) before we even think about the addition. Ignoring this rule can lead to a completely wrong answer, and we definitely don't want that! By following PEMDAS, we ensure that we're solving the equation in a logical and consistent way. This isn't just about getting the right answer this time; it's about building a solid foundation for tackling more complex math problems in the future. So, remember PEMDAS: it's your best friend in the world of math!
Step 1: Converting Mixed Fractions to Improper Fractions
Okay, so the first thing we need to do to solve this equation properly is to deal with that mixed fraction: 1 3/4. Mixed fractions can be a little tricky to work with directly, especially when we're multiplying or dividing. So, the best thing to do is convert it into an improper fraction. An improper fraction is simply one where the numerator (the top number) is greater than or equal to the denominator (the bottom number).
To convert 1 3/4 into an improper fraction, we follow a simple process. First, we multiply the whole number part (1) by the denominator (4). That gives us 1 × 4 = 4. Next, we add the numerator (3) to this result: 4 + 3 = 7. This new number, 7, becomes our new numerator. The denominator stays the same, which is 4. So, the improper fraction equivalent of 1 3/4 is 7/4. Now, we've transformed our mixed fraction into a form that's much easier to work with in calculations. This conversion is a crucial step because it allows us to perform multiplication and division with fractions without any hiccups. Trust me, it makes the whole process much smoother!
Step 2: Performing the Multiplication
Now that we've converted the mixed fraction into an improper fraction, we can focus on the multiplication part of our equation: 2/5 × 7/4. Multiplying fractions might seem daunting, but it's actually pretty straightforward. The rule is simple: you multiply the numerators (the top numbers) together, and then you multiply the denominators (the bottom numbers) together. That's it!
So, let's apply this to our problem. The numerators are 2 and 7, and their product is 2 × 7 = 14. The denominators are 5 and 4, and their product is 5 × 4 = 20. Therefore, 2/5 × 7/4 equals 14/20. But hold on, we're not quite done yet! It's always a good practice to simplify fractions whenever possible. In this case, both the numerator (14) and the denominator (20) are divisible by 2. Dividing both by 2 gives us 7/10. So, the simplified result of the multiplication is 7/10. This step of simplifying is important because it makes the fraction easier to understand and work with in future calculations. Plus, it's just good mathematical etiquette to present your answer in its simplest form!
Step 3: Converting Decimals to Fractions
Alright, let's tackle the next part of our equation. We have a decimal, 0.75, and we need to add it to a fraction. To do this easily, it's best to convert the decimal into a fraction. This way, we'll be working with the same type of numbers, and the addition will be much smoother.
The decimal 0.75 represents seventy-five hundredths. Think of it this way: the 7 is in the tenths place, and the 5 is in the hundredths place. So, we can write 0.75 as the fraction 75/100. Makes sense, right? Now, just like with the previous fraction, we want to simplify this as much as possible. Both 75 and 100 are divisible by 5, so let's divide both by 5. This gives us 15/20. We can simplify further! Both 15 and 20 are also divisible by 5. Dividing again, we get 3/4. So, the simplified fraction equivalent of 0.75 is 3/4. This conversion is super handy because now we can add fractions together without any decimal drama. Remember, converting decimals to fractions (and vice versa) is a key skill in math, so it's worth practicing until it feels like second nature.
Step 4: Performing the Addition
Okay, we're in the home stretch now! We've simplified everything, and we're ready to perform the addition. Our equation now looks like this: 3/4 + 7/10. To add fractions, they need to have the same denominator, which is called the common denominator. Right now, our denominators are 4 and 10, so we need to find a common denominator for these two numbers.
The easiest way to find a common denominator is to look for the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators can divide into evenly. For 4 and 10, the LCM is 20. So, we want to convert both fractions so that they have a denominator of 20.
To convert 3/4 to have a denominator of 20, we need to multiply both the numerator and the denominator by 5 (20 ÷ 4 = 5). This gives us (3 × 5) / (4 × 5) = 15/20. To convert 7/10 to have a denominator of 20, we need to multiply both the numerator and the denominator by 2 (20 ÷ 10 = 2). This gives us (7 × 2) / (10 × 2) = 14/20. Now we can rewrite our equation as 15/20 + 14/20. Adding these fractions is easy: we simply add the numerators and keep the denominator the same. So, 15/20 + 14/20 = 29/20. And there we have it! The sum of the fractions is 29/20.
Step 5: Expressing the Answer in Simplest Form
We've done all the hard work, but there's one final step: expressing our answer in its simplest form. We've arrived at the fraction 29/20, which is an improper fraction because the numerator (29) is larger than the denominator (20). While improper fractions are perfectly valid, it's often more helpful to express them as mixed numbers. A mixed number combines a whole number and a proper fraction (where the numerator is smaller than the denominator).
To convert 29/20 into a mixed number, we need to figure out how many times 20 goes into 29. It goes in once, with a remainder. So, we have a whole number of 1. The remainder is 29 - 20 = 9. This remainder becomes the numerator of our fractional part, and the denominator stays the same (20). Therefore, 29/20 is equal to the mixed number 1 9/20. Now, we should check if the fractional part (9/20) can be simplified further. In this case, 9 and 20 don't have any common factors other than 1, so 9/20 is already in its simplest form. This means that 1 9/20 is the simplest form of our answer. Expressing the answer in its simplest form not only makes it easier to understand but also demonstrates a good grasp of mathematical principles. Great job, guys! We've solved the problem!
Conclusion
So, there you have it! We've successfully navigated through the problem 0.75 + 2/5 × 1 3/4 step by step. Remember, the key to solving math problems like this is to break them down into smaller, more manageable parts. We started by understanding the order of operations (PEMDAS), then we converted mixed fractions to improper fractions, decimals to fractions, performed the multiplication and addition, and finally, expressed our answer in the simplest form. Each step is a building block that leads us closer to the solution.
Don't be afraid to tackle these kinds of problems. With practice, you'll become more confident and comfortable with each step. Math is like any other skill – the more you practice, the better you get. So, keep challenging yourselves, and remember that every problem you solve is a step forward. And hey, if you ever get stuck, just remember the steps we've covered today. You've got this!
