Dividing Decimals And Fractions: Step-by-Step Solutions

Hey guys! Today, we're diving deep into the world of division, specifically focusing on decimals and fractions. This might seem tricky at first, but don't worry! We'll break it down step-by-step, so you'll be a pro in no time. Let's tackle these problems together and make math a little less intimidating. Remember, practice makes perfect, so let's get started!

Understanding Decimal Division

When we talk about decimal division, it's essential to grasp the core concept: dividing a number that has a decimal point by another number, which could also have a decimal point. Mastering this skill is crucial because decimals are everywhere in our daily lives, from calculating grocery bills to measuring ingredients for a recipe. So, understanding how to divide decimals accurately is super practical. To make things crystal clear, we'll walk through several examples, breaking down each step. This way, you'll not only learn the mechanics but also understand the logic behind each calculation. By the end of this section, you'll feel much more confident in tackling any decimal division problem that comes your way. Remember, the key is to practice and take it one step at a time!

Problem A: 6.4 ÷ 3.2

Alright, let's kick things off with our first problem: 6.4 ÷ 3.2. The key here is to eliminate the decimals to make the division easier. To do this, we'll multiply both numbers by 10. Why 10? Because it moves the decimal point one place to the right. So, 6.4 becomes 64, and 3.2 becomes 32. Now, our problem looks a lot simpler: 64 ÷ 32. This is a straightforward division problem, and we know that 64 divided by 32 is 2. And there you have it! The answer to 6.4 ÷ 3.2 is 2. This method works because we're essentially scaling up both numbers by the same factor, which doesn't change the final result of the division. Keep this trick in your back pocket – it's a lifesaver for decimal divisions!

Problem B: 1.25 ÷ (-7.25)

Next up, we have 1.25 ÷ (-7.25). Notice that we're dealing with a negative number here, so remember that the result will be negative. Just like before, let's get rid of those decimals. This time, we need to multiply both numbers by 100 because we have two decimal places. So, 1.25 becomes 125, and -7.25 becomes -725. Now our problem is 125 ÷ (-725). This might look intimidating, but we can simplify it. We can rewrite this division as a fraction: 125/-725. Now, let's find the greatest common divisor (GCD) to simplify the fraction. Both numbers are divisible by 25, so let's divide both the numerator and the denominator by 25. 125 ÷ 25 = 5, and -725 ÷ 25 = -29. So, our simplified fraction is 5/-29, which can also be written as -5/29. Therefore, 1.25 ÷ (-7.25) = -5/29. Remember, when dividing a positive number by a negative number, the result is always negative!

Problem C: (-6.5) ÷ 5

Moving on, let's tackle (-6.5) ÷ 5. This time, we have a negative decimal being divided by a whole number. First things first, let's deal with that decimal. We'll multiply -6.5 by 10 to get -65. Now, we need to remember to divide our final answer by 10 later to account for this multiplication. So, for now, we have -65 ÷ 5. This is a more manageable division. We know that 65 divided by 5 is 13, and since we have a negative number, the result is -13. Now, don't forget that we multiplied by 10 earlier, so we need to divide -13 by 10 to get our final answer. -13 ÷ 10 = -1.3. Therefore, (-6.5) ÷ 5 = -1.3. Always keep track of those decimal places!

Problem D: (-2.56) ÷ (-8)

Now, let's dive into (-2.56) ÷ (-8). This one's interesting because we have a negative decimal being divided by a negative whole number. Remember, a negative divided by a negative gives us a positive result, so our answer will be positive. First, let's deal with the decimal in -2.56. We'll multiply it by 100 to get -256. This means we'll need to divide our final answer by 100 later. Now we have -256 ÷ (-8). We know that 256 divided by 8 is 32. Since both numbers are negative, the result is positive 32. Now, let's not forget that we multiplied by 100 earlier, so we need to divide 32 by 100 to get our final answer. 32 ÷ 100 = 0.32. Therefore, (-2.56) ÷ (-8) = 0.32. Great job – we're on a roll!

Problem E: (-9) ÷ (-1.8)

Let's move on to (-9) ÷ (-1.8). Here, we're dividing a negative whole number by a negative decimal. Remember, a negative divided by a negative equals a positive, so our answer will be positive. To make things easier, let's get rid of the decimal in -1.8 by multiplying it by 10. This gives us -18. We also need to multiply -9 by 10, which gives us -90. Now, our problem is -90 ÷ (-18). Both numbers are negative, so we know the answer will be positive. 90 divided by 18 is 5. So, (-9) ÷ (-1.8) = 5. See how simplifying the problem makes it much easier to solve?

Problem F: 7 ÷ (-2.8)

Next up, we have 7 ÷ (-2.8). We're dividing a positive number by a negative decimal, so our result will be negative. Let's get rid of the decimal in -2.8 by multiplying it by 10, which gives us -28. We also need to multiply 7 by 10, which gives us 70. Now our problem is 70 ÷ (-28). We can simplify this by dividing both numbers by their greatest common divisor, which is 14. 70 ÷ 14 = 5, and -28 ÷ 14 = -2. So, we now have 5 ÷ (-2), which can be written as -5/2. This can also be expressed as a decimal: -2.5. Therefore, 7 ÷ (-2.8) = -2.5. Keep practicing, and you'll become a pro at these!

Mastering Fraction Division

Okay, guys, now let's switch gears and talk about fraction division. Dividing fractions might seem a bit different from dividing whole numbers or decimals, but once you understand the trick, it's actually pretty straightforward. The key thing to remember is that dividing by a fraction is the same as multiplying by its reciprocal. Yep, you heard that right! We'll break down what that means and how to do it, step by step. Fractions are a fundamental part of math, and they pop up everywhere, so mastering fraction division is super useful. We'll go through several examples together to make sure you've got a solid grasp of the concept. By the end of this section, you'll be dividing fractions like a total boss!

Problem G: 28 ÷ (-66)

Let's start with 28 ÷ (-66). This one looks a bit different because it's written as a whole number divided by another whole number, but we can easily turn these into fractions. We can rewrite this as the fraction 28/-66. Now, just like before, we want to simplify this fraction as much as possible. To do this, we need to find the greatest common divisor (GCD) of 28 and 66. The GCD is 2. So, let's divide both the numerator and the denominator by 2. 28 ÷ 2 = 14, and -66 ÷ 2 = -33. This gives us the simplified fraction 14/-33, which can also be written as -14/33. So, 28 ÷ (-66) = -14/33. Remember, simplifying fractions makes them much easier to work with!

Problem H: 20/27 ÷ 56/81

Alright, let's dive into our final problem: 20/27 ÷ 56/81. This is a classic fraction division problem. Remember the trick we talked about earlier? Dividing by a fraction is the same as multiplying by its reciprocal. So, we need to flip the second fraction (56/81) and multiply. The reciprocal of 56/81 is 81/56. Now our problem looks like this: 20/27 * 81/56. Before we multiply, let's see if we can simplify anything. We can simplify diagonally: 20 and 56 have a common factor of 4, and 27 and 81 have a common factor of 27. 20 ÷ 4 = 5, and 56 ÷ 4 = 14. 27 ÷ 27 = 1, and 81 ÷ 27 = 3. So, our problem now looks like this: 5/1 * 3/14. Now we can multiply the numerators and the denominators: 5 * 3 = 15, and 1 * 14 = 14. So, the result is 15/14. Therefore, 20/27 ÷ 56/81 = 15/14. You nailed it!

Wrapping Up Division Problems

Awesome job, everyone! We've covered a lot of ground today, from decimal division to fraction division. Remember, the key to mastering these concepts is practice. The more you work through these types of problems, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're a part of the learning process. And remember, there are tons of resources out there to help you, so keep exploring and keep learning. You've got this! Math might seem tough sometimes, but with a little effort and the right approach, you can conquer any challenge. Keep up the great work, and I'll see you in the next math adventure!