Solving Mathematical Expressions A Step-by-Step Guide

Hey guys! Today, we're going to break down some mathematical expressions and solve them together. Math can seem intimidating, but trust me, with a step-by-step approach, it becomes much easier. We'll tackle expressions involving multiplication, subtraction, addition, and division. So, grab your calculators (or your brains!) and let’s dive in!

a) +0.75 * 1

When we talk about basic multiplication, especially involving decimals, it's crucial to understand the underlying principles. The expression +0.75 * 1 might seem simple, but it highlights a fundamental rule in mathematics: any number multiplied by 1 remains the same. This is because 1 is the multiplicative identity, meaning it doesn't change the value of the number it's multiplied with.

Let's break this down further. The number +0.75 represents seventy-five hundredths. When you multiply this by 1, you're essentially asking, "What is one group of 0.75?" The answer, of course, is 0.75. There's no change in the quantity because we're only taking one whole unit of it. This concept is vital not just for simple calculations but also for more complex algebraic manipulations where understanding the properties of numbers can simplify equations.

Now, why is this important in a broader mathematical context? Well, imagine you're dealing with fractions or percentages. Multiplying by 1 (or its equivalent forms like 2/2, 100/100, etc.) allows you to change the appearance of a number without altering its value. This is particularly useful when you need to find a common denominator or convert a decimal to a percentage. For instance, if you want to express 0.75 as a percentage, you can multiply it by 100/1 (which is essentially multiplying by 1) to get 75%. Understanding this simple multiplication rule is a cornerstone for many mathematical operations and problem-solving techniques. So, remember, guys, when you see a number multiplied by 1, the answer is just the original number itself!

Answer: +0.75 * 1 = 0.75

b) 3.7 - 1.2

Okay, let's jump into some subtraction with decimals. The expression 3.7 - 1.2 might look straightforward, but it’s a great example of how decimal subtraction works. Subtraction, at its core, is about finding the difference between two numbers. In this case, we're figuring out what's left when we take 1.2 away from 3.7.

To make things super clear, let’s think about place values. The number 3.7 has a 3 in the ones place and a 7 in the tenths place. Similarly, 1.2 has a 1 in the ones place and a 2 in the tenths place. When subtracting decimals, it’s super important to line up the decimal points. This ensures that you're subtracting tenths from tenths, ones from ones, and so on. If the decimal points aren’t aligned, you might end up subtracting the wrong place values, leading to an incorrect answer.

So, let's line them up:

  3.  7
- 1.  2
-------

Now, we subtract the tenths first: 7 tenths minus 2 tenths equals 5 tenths. Then, we subtract the ones: 3 ones minus 1 one equals 2 ones. Put it all together, and you get 2.5. See? It’s like subtracting whole numbers, but you just have to keep that decimal point in line!

Why is this skill important? Well, decimals pop up everywhere in real life, from measuring ingredients in a recipe to calculating the change you get at the store. Being comfortable with decimal subtraction means you can handle everyday situations with confidence. Plus, mastering this basic operation is a building block for more complex math, like algebra and calculus. So, keep practicing, guys, and you’ll become subtraction superstars in no time!

Answer: 3.7 - 1.2 = 2.5

c) 5 - 0.3 + 3

Let's tackle an expression that mixes subtraction and addition – 5 - 0.3 + 3. This type of problem highlights the importance of following the order of operations, which, in this case, is pretty straightforward: we perform the operations from left to right. This rule ensures that we all get the same answer, no matter who's doing the math.

First up, we've got 5 - 0.3. This is a subtraction problem involving a whole number and a decimal. To make things crystal clear, we can think of 5 as 5.0. Now, we're subtracting 0.3 from 5.0. When subtracting decimals, remember to line up those decimal points! This ensures that you're subtracting tenths from tenths, ones from ones, and so on.

So, let’s do the subtraction: 5.0 - 0.3. You might need to borrow from the ones place to subtract the tenths. If you do that correctly, you’ll find that 5.0 - 0.3 equals 4.7. Great! We've completed the first part of the expression.

Now, we move on to the addition part: + 3. We're adding 3 to the result we just got, which is 4.7. This is another decimal addition. We can think of 3 as 3.0 to keep the decimal places aligned. So, we have 4.7 + 3.0. Adding these two together is pretty simple: 7 tenths plus 0 tenths is 7 tenths, and 4 ones plus 3 ones is 7 ones. Combine them, and you get 7.7.

Why is understanding the order of operations so crucial? Well, imagine if we did the addition before the subtraction – we'd get a completely different answer! Following the correct order ensures consistency and accuracy in all mathematical calculations. Plus, this skill is essential as you move on to more complex equations and algebraic expressions. So, remember guys, left to right is the way to go when you're dealing with addition and subtraction in the same expression!

Answer: 5 - 0.3 + 3 = 7.7

d) 1.2 / -1.4

Alright, let's dive into some division with decimals and negative numbers! The expression 1.2 / -1.4 brings together a couple of important mathematical concepts. First, we're dealing with decimals, and second, we're dividing by a negative number. Understanding how these two things interact is key to getting the right answer.

Division, at its heart, is about splitting a quantity into equal parts or groups. When we see 1.2 / -1.4, we’re asking, "How many times does -1.4 fit into 1.2?" Because we're dividing by a negative number, we know that the result will be negative. This is a crucial rule in math: a positive number divided by a negative number gives you a negative result.

Now, let’s tackle the decimal part. Dividing decimals can seem tricky, but we can simplify it by getting rid of the decimal points. To do this, we can multiply both the dividend (1.2) and the divisor (-1.4) by 10. This shifts the decimal point one place to the right in both numbers, turning our expression into 12 / -14. Multiplying both numbers by the same value doesn't change the overall result of the division, but it makes the calculation easier to handle.

So, we now have 12 / -14. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 6 / -7. So, the final simplified fraction is -6/7. To get a decimal approximation, you can divide 6 by 7 using a calculator or long division. The result is approximately -0.857.

Why is it important to understand division with decimals and negative numbers? Well, these concepts pop up in all sorts of real-world situations, from calculating unit prices at the store to understanding financial ratios. Mastering these skills gives you a solid foundation for more advanced math and helps you make sense of the numbers around you. So, keep practicing, guys, and you'll become division dynamos in no time!

Answer: 1.2 / -1.4 ≈ -0.857

So, there you have it, guys! We've tackled several mathematical expressions involving different operations. Remember, the key to mastering math is practice and a solid understanding of the basic rules. Keep working at it, and you’ll be solving complex problems in no time!