Problem 40 - Fundamentals of Mathematics

Question

Combine the numbers as indicated. $$2,957+9,006$$

Step-by-Step Solution

Answer

The sum of 2,957 and 9,006 is 11,963.

Step 1: Set up the Addition

Write down the numbers vertically, aligning the digits according to their place value (ones, tens, hundreds, thousands). Write 2,957 below 9,006, ensuring that the ones, tens, hundreds, and thousands places are aligned.

Step 2: Add the Ones Place

Start with the rightmost digit (the ones place). Add 7 (from 2,957) to 6 (from 9,006), which equals 13. Place 3 in the ones column of the result and carry over the 1 to the tens column.

Step 3: Add the Tens Place

Move to the tens column. Add 5 (from 2,957) to 0 (from 9,006), plus the 1 carried over. This equals 6. Write 6 in the tens place of the result.

Step 4: Add the Hundreds Place

Next, add the digits in the hundreds place. Add 9 (from 2,957) to 0 (from 9,006). This equals 9, and write it in the hundreds place of the result.

Step 5: Add the Thousands Place

Finally, add the digits in the thousands place. Add 2 (from 2,957) to 9 (from 9,006). This equals 11. Place 1 in the thousands column of the result and carry over 1 to the ten-thousands place.

Step 6: Finalize the Sum

In the ten-thousands place, write the carried over 1. So, the final result is 11,963.

Key Concepts

Place Value

Understanding place value is essential when performing arithmetic addition, especially with larger numbers. Each digit in a number has a specific place value which determines its value in the number's structure. For instance, in the number 2,957:

• The digit '7' is in the ones place, having the value of 7.
• The digit '5' is in the tens place, signifying 50.
• The digit '9' occupies the hundreds place, meaning 900.
• The digit '2' is in the thousands place, representing 2,000.

Each digit must be aligned according to its place value when performing arithmetic operations. By lining up digits, like in 2,957 and 9,006, each figure maintains its true value based on its position.

Carrying Over

The concept of carrying over is a critical aspect of addition, especially when adding multi-digit numbers. It occurs when the sum of digits in a column exceeds 9, necessitating an adjustment to the next highest column. For example:

• In adding the ones place digits of 2,957 and 9,006, you get 13. Here, 3 is placed in the ones column, and 1 is carried over to the tens place.
• In the thousands place, adding 2 and 9 gives 11. Then, 1 is placed in thousands, and another 1 is carried over to the ten-thousands place.

Carrying over ensures that each column does not exceed a single digit, preserving the accuracy of addition across place values.

Column Addition

Column addition is a systematic method of adding numbers by aligning their place values in columns. This technique simplifies visualizing and calculating sums, especially with multi-digit numbers. Here's a simple guide: 1. **Align the numbers**: Write the numbers vertically so each place value (ones, tens, hundreds, etc.) lines up straight. 2. **Start with the smallest value**: Always begin adding from the rightmost column, which is the ones place. 3. **Move left with carryovers**: After calculating each column, move left, adding any digits carried over from the previous column. Column addition helps manage large numbers by breaking down the addition into manageable steps, ensuring each digit is correctly accounted for in its respective column.