Volume Conversions And Equation Solutions

Hey math enthusiasts! Let's dive into some volume conversions and equation solving, tackling the problems step by step. We'll be working with cubic meters (m³) and cubic decimeters (dm³), so get ready to flex those conversion muscles! We'll also be looking for a special number, a root that's divisible by 9. Buckle up, guys, it's going to be fun!

Part 1: Volume Addition

Understanding Volume Units

First things first, let's get comfortable with our units. Remember that 1 cubic meter (1 m³) is equal to 1000 cubic decimeters (1000 dm³). Think of it like this: a meter is a decent-sized unit, and a decimeter is smaller (a tenth of a meter). So, when we cube them (multiply the length, width, and height by themselves), the relationship changes too. This is super important to keep in mind as we move through the problems. Now let's get started.

Problem d) $7 ext{ m}^3 42 ext{ dm}^3 + 567 ext{ dm}^3$

For this problem, we need to add two volumes together. However, they're expressed in different units: cubic meters and cubic decimeters. Before we can add them, we need to convert everything to the same unit. A smart move is to convert everything to cubic decimeters (dm³), since that's the smaller unit. We know that $1 ext{ m}^3 = 1000 ext{ dm}^3$, so, let's handle the first part of the expression, $7 ext{ m}^3 42 ext{ dm}^3$. We know that $7 ext{ m}^3$ is equal to $7 * 1000 ext{ dm}^3 = 7000 ext{ dm}^3$. Now, let's add the existing $42 ext{ dm}^3$ from the problem. This means that, $7 ext{ m}^3 42 ext{ dm}^3 = 7000 ext{ dm}^3 + 42 ext{ dm}^3 = 7042 ext{ dm}^3$. Next, we need to add $567 ext{ dm}^3$. So our equation becomes, $7042 ext{ dm}^3 + 567 ext{ dm}^3 = 7609 ext{ dm}^3$. Thus, the result of the operation is $7609 ext{ dm}^3$. Pretty straightforward, right? Just remember the conversion factor, and you'll be golden! This also shows that when dealing with volume, understanding the relationships between different units is key.

Part 2: Volume Subtraction

Working with Cubic Meters and Decimeters

Now, let's move on to subtraction, which is the second part of our volume problems. Keep in mind the same principle of unit conversion; we must have matching units before performing our operations. This time, we will be subtracting a volume expressed in cubic decimeters from a volume expressed in cubic meters. Let's convert everything to cubic decimeters. The problem is written as follows:

Problem e) $21 ext{ m}^3 - 1283 ext{ dm}^3$

In the given problem, we have $21 ext{ m}^3$. We need to convert this to cubic decimeters. We know that $1 ext{ m}^3 = 1000 ext{ dm}^3$. Therefore, $21 ext{ m}^3 = 21 * 1000 ext{ dm}^3 = 21000 ext{ dm}^3$. Now we can substitute this result into the equation, $21000 ext{ dm}^3 - 1283 ext{ dm}^3 = 19717 ext{ dm}^3$. That means our answer is $19717 ext{ dm}^3$. Keep practicing, and you'll become a master of these volume conversions. Remember to always pay attention to the units; it's the most critical part.

Part 3: Solving Equations and Identifying Divisibility

Equation Solving Fundamentals

This is where the fun really begins! We need to solve an equation. The exact equation isn't provided in the problem description. But, let's assume we have some equation to solve for 'x', for example, $x + 5 = 14$. We need to isolate 'x' on one side of the equation. To do this, we subtract 5 from both sides. So, we get, $x + 5 - 5 = 14 - 5$, which simplifies to $x = 9$. So, the solution for x in this case is 9. After solving the equation, we would examine the solution (the 'root' of the equation, in mathematical terms) to see if it is divisible by 9. Divisibility means whether the solution, when divided by 9, results in a whole number, with no remainder. For instance, if the root is 18, then 18 / 9 = 2, so 18 is divisible by 9. If, however, the root is 20, then 20 / 9 = 2 with a remainder of 2; therefore, 20 is not divisible by 9.

Finding the Root and the Divisibility Check

Let's work with an example. Let's say that after solving the equation, we found that the root is 27. Now we have to determine if 27 is divisible by 9. We can divide 27 by 9: $27 / 9 = 3$. Since the result is a whole number (3), we can confidently say that 27 is indeed divisible by 9. Another example: let's say we solved another equation and found that the root is 35. Now we check if 35 is divisible by 9. We divide 35 by 9: $35 / 9 = 3$ with a remainder of 8. Because we have a remainder, we can say that 35 is not divisible by 9. The key here is to solve for the variable, find the value of the root, and then perform a simple division to check divisibility. This is not always the case, but the number 9 has a rule that can be used to tell if a number is divisible by 9. The rule says that if you add up the digits in the number and the result is divisible by 9, then the entire number is divisible by 9. For instance, using the prior example of the number 27, we add the digits, $2 + 7 = 9$. Since 9 is divisible by 9, the entire number is divisible by 9. Now for the number 35, we add the digits $3 + 5 = 8$. Since 8 is not divisible by 9, the entire number is not divisible by 9. This little trick can be very helpful in finding the answer. Make sure you're comfortable with these steps: solve the equation, get the root, and then check if the root is divisible by 9. You've got this!

Conclusion

Great job, guys! We've covered volume conversions and how to check for divisibility. Remember, math is all about practice. Keep working through problems, and don't be afraid to ask for help if you get stuck. You’re building a strong foundation, and with each problem you solve, you’re becoming more confident in your mathematical abilities. Keep up the great work, and happy calculating!