Problem 302 - Prealgebra

Question

In the following exercises, simplify. $$3,816+8,184$$

Step-by-Step Solution

Answer

12,000

Step 1: Align the numbers

Write the numbers one under the other, aligning the digits according to their place values. 3,816 + 8,184

Step 2: Add the ones place

Add the digits in the ones place: 6 + 4 = 10. Write down the 0 and carry over the 1.

Step 3: Add the tens place

Add the digits in the tens place, including the carryover: 1 + 8 + 1 = 10. Write down the 0 and carry over the 1.

Step 4: Add the hundreds place

Add the digits in the hundreds place, including the carryover: 8 + 1 + 1 = 10. Write down the 0 and carry over the 1.

Step 5: Add the thousands place

Add the digits in the thousands place, including the carryover: 3 + 8 + 1 = 12. Write down the 2 and carry over the 1, adding it as a new ten thousand place.

Step 6: Write the final sum

The final sum after including all carryovers is 12,000.

Key Concepts

Place Value

Place value is a fundamental concept in math that helps us understand the value of each digit in a number based on its position. For example, in the number 3,816, each digit has a specific place value:
• 3 is in the thousands place
• 8 is in the hundreds place
• 1 is in the tens place
• 6 is in the ones place
Each digit's place value is ten times greater than the digit to its right. This structure allows us to add numbers systematically by aligning the digits according to their place values. By aligning 3,816 and 8,184, it ensures that we are adding digits of the same place value together: ones with ones, tens with tens, and so on.

Carry Over

The carryover method is used when the sum of digits in a particular place value is more than 9. This is because any number greater than 9 needs an additional place to hold the extra value. For instance, when we add 6 + 4 in the ones place, we get 10.
We write down the 0 in the ones place and carry over 1 to the tens place. This process repeats for higher place values. Every time a column adds up to 10 or more, we write down the last digit and carry the remaining value over to the next higher place value.
In our exercise, each place value addition resulted in 10 or more, necessitating a carryover each time. By understanding this method, students can tackle more complex additions confidently.

Sum

The sum is the result of adding two or more numbers together. In the exercise, the task was to find the sum of 3,816 and 8,184. By following the steps of aligning the numbers, adding each place value, and carefully carrying over as needed, we finally arrive at the sum. Here, we found:
• Ones place: 6 + 4 = 10 (write 0, carry over 1)
• Tens place: 1 + 8 + 1 = 10 (write 0, carry over 1)
• Hundreds place: 8 + 1 + 1 = 10 (write 0, carry over 1)
• Thousands place: 3 + 8 + 1 = 12 (write 2, carry over 1)
Adding the final carryover gives us 12,000 as the sum. Each step builds on the previous one, showing how place value and carryover work together seamlessly to achieve the final result.