Solving Arithmetic Expressions: A Step-by-Step Guide

Hey guys! Today, we're going to break down a complex arithmetic problem step by step. So, grab your calculators (or your mental math skills!) and let's dive into solving this expression: 880 ÷ 20 × 143 - 168 ÷ 56 - 148 ÷ 37 × 44 × 5.

Understanding the Order of Operations

Before we get started, it's super important to remember the order of operations, often remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order ensures we all get to the same correct answer. Without it, we would have a very confusing and inconsistent mathematical landscape.

When we talk about order of operations, we're basically setting the rules for how we simplify math expressions. It's like the grammar of mathematics! Think of it this way: just as you need grammar to make a sentence understandable, you need the order of operations to make a mathematical expression solvable and universally understood. For instance, consider the simple expression 2 + 3 * 4. If we just went from left to right, we might do 2 + 3 first to get 5, and then multiply by 4 to get 20. But, according to PEMDAS, multiplication comes before addition. So, we should do 3 * 4 first to get 12, and then add 2 to get 14. See how different the answers are? That's why the order of operations is so crucial.

The beautiful thing about the order of operations is that it brings consistency to mathematics. Whether you're in elementary school learning basic arithmetic or working on advanced calculus, the rules remain the same. This consistency allows mathematicians, scientists, engineers, and anyone else using math to communicate their ideas clearly and effectively. It's a universal language that transcends borders and disciplines.

Step-by-Step Solution

Let's apply PEMDAS to our problem:

880 ÷ 20 × 143 - 168 ÷ 56 - 148 ÷ 37 × 44 × 5

Step 1: Division (Left to Right)

First, we handle the divisions from left to right.

• 880 ÷ 20 = 44
• 168 ÷ 56 = 3
• 148 ÷ 37 = 4

Now our expression looks like this:

44 × 143 - 3 - 4 × 44 × 5

Step 2: Multiplication (Left to Right)

Next up, we tackle the multiplications from left to right.

• 44 × 143 = 6292
• 4 × 44 = 176
• 176 × 5 = 880

So the expression becomes:

6292 - 3 - 880

Step 3: Subtraction (Left to Right)

Finally, we perform the subtractions from left to right.

• 6292 - 3 = 6289
• 6289 - 880 = 5409

Therefore, the final answer is:

5409

Breaking Down Each Operation

Let's dive a bit deeper into each of the operations to make sure we understand exactly what's going on.

Division: 880 ÷ 20

Division is the operation of splitting a quantity into equal groups. In our case, we're dividing 880 by 20. This tells us how many times 20 fits into 880. Thinking about it, we're essentially asking: "If I have 880 items and I want to put them into 20 equal groups, how many items will be in each group?" The answer, as we calculated, is 44. Each group would have 44 items. Division is the inverse operation of multiplication, meaning that if 880 ÷ 20 = 44, then 20 × 44 = 880.

Division: 168 ÷ 56

Here, we're dividing 168 by 56. This is asking, "How many times does 56 fit into 168?" The answer is 3. You can think of this as splitting 168 into groups of 56. You'd get exactly 3 groups. Again, division is the inverse of multiplication, so 168 ÷ 56 = 3 means 56 × 3 = 168. Understanding this relationship helps in verifying your division calculations.

Division: 148 ÷ 37

We are dividing 148 by 37. This means, "How many times does 37 fit into 148?" The answer is 4. Therefore, 148 ÷ 37 = 4, which also means 37 × 4 = 148. This operation highlights the idea of division as the process of finding how many equal-sized groups can be made from a total quantity.

Multiplication: 44 × 143

Multiplication is a shortcut for repeated addition. When we multiply 44 by 143, we're essentially adding 143 to itself 44 times. This can be a bit tedious to do manually, which is why we often use calculators or multiplication tables. The result, 6292, is the total when you add 143 to itself 44 times. In real-world terms, if you had 44 stacks of items, with each stack containing 143 items, you would have a total of 6292 items.

Multiplication: 4 × 44

Multiplying 4 by 44 is like adding 44 to itself 4 times. This is a simpler multiplication, resulting in 176. Think of it as having 4 groups of 44 items each. The total number of items would be 176. Multiplication is commutative, meaning the order doesn't matter: 4 × 44 is the same as 44 × 4. However, in the context of the order of operations, it's essential to perform multiplications and divisions from left to right.

Multiplication: 176 × 5

Multiplying 176 by 5 is like adding 176 to itself 5 times. This gives us 880. So, if you had 5 piles of items, with each pile containing 176 items, you would have a total of 880 items. This multiplication step is crucial in our problem as it combines the results of previous operations to move closer to the final answer.

Subtraction: 6292 - 3

Subtraction is the operation of taking away one quantity from another. When we subtract 3 from 6292, we're essentially removing 3 units from 6292. This results in 6289. Subtraction is the inverse operation of addition. If 6292 - 3 = 6289, then 6289 + 3 = 6292. Understanding this relationship helps in checking your subtraction calculations.

Subtraction: 6289 - 880

Finally, we subtract 880 from 6289. This means we're taking 880 units away from 6289. The result is 5409. This is the final subtraction in our problem, giving us the ultimate result. Just like with the previous subtraction, we can check this calculation by adding: 5409 + 880 = 6289.

Tips for Solving Arithmetic Problems

• Double-Check Your Work: It's easy to make a small mistake, so always double-check each step.
• Use a Calculator: If you're allowed to, use a calculator to avoid errors.
• Break Down the Problem: Break the problem into smaller, more manageable steps.
• Practice Regularly: The more you practice, the better you'll become at solving these types of problems.

Conclusion

So there you have it! By following the order of operations and breaking down the problem into smaller steps, we were able to solve the expression and arrive at the answer: 5409. Keep practicing, and you'll become a math whiz in no time! Remember, math can be fun when you approach it step by step. Keep at it, and you will definitely see improvements. You've got this!