Problem 21 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 131,294 \\ +\quad 9,087 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 140,381.

Step 1: Align the numbers

First, write the numbers vertically, aligning them by their rightmost digits. This alignment ensures that each digit is in its correct place value.

Step 2: Add the digits in the rightmost column

Add the digits in the ones place: 4 + 7 = 11. Write down the 1 in the ones place and carry over the 1 to the tens column.

Step 3: Add the digits in the tens column

Now, add the digits in the tens place including the carryover: 9 + 8 + 1 = 18. Write down the 8 in the tens place and carry over 1 to the hundreds column.

Step 4: Add the digits in the hundreds column

Add the digits in the hundreds place and the carryover: 2 + 0 + 1 = 3. Write down the 3 in the hundreds place.

Step 5: Add the digits in the thousands column

Add the digits in the thousands place: 1 + 9 = 10. Write down 0 in the thousands place and carry over 1 to the ten-thousands column.

Step 6: Add the digits in the ten-thousands column

Add the digits in the ten-thousands place with the carryover: 3 + 1 = 4. Write down this result in the ten-thousands digit.

Step 7: Add the digits in the hundred-thousands column

Only the top number has a digit in the hundred-thousands place. Bring down the 1 to complete the sum.

Key Concepts

Place Value

Understanding place value is crucial when adding whole numbers. Each digit in a number has a different value based on its position. For instance, in the number 131,294, the digit '1' in the hundred-thousands place represents 100,000, while the '4' in the ones place represents just 4.

It's helpful to split the number into columns, each representing a specific place value:

• Ones
• Tens
• Hundreds
• Thousands
• Ten-Thousands
• Hundred-Thousands

By aligning numbers according to their place values during addition, we ensure that each digit is accurately contributed to the final sum. Always start aligning from the rightmost column, the ones place, and proceed leftwards.

Carry Over in Addition

The concept of 'carry over' in addition helps manage those situations where a column's sum exceed 9. For example, in our current task, adding the ones place digits results in 11. Since this value includes a tens unit, we keep the '1' in the ones column and 'carry over' the remaining 1 to the next higher column, the tens column.

This carry over continues through each affected column. Consider the sum in the tens column: 9 + 8 + 1 (the carried over 1) equals 18. Here, '8' stays in the tens place, and another '1' is carried over to the hundreds column. When each digit is correctly added along with any carried over digits, the carry over system ensures accuracy, even in long sequences of numbers.

Column Addition

Column addition involves aligning numbers vertically and adding digits column by column. Each column corresponds to a place value and should be added separately.

For example, the column for the ones place in our example involves adding 4 and 7, providing us with the sum 11. We continue this approach for each subsequent column:

• Ones column: 4 + 7 → Write 1, carry over 1
• Tens column: 9 + 8 + 1 → Write 8, carry over 1
• Hundreds column: 2 + 0 + 1 → Write 3
• Thousands column: 1 + 9 → Write 0, carry over 1
• Ten-Thousands column: 3 + 1 → Write 4
• Hundred-Thousands column: 1 → Write 1

Each row reflects a complete part of the overall sum, handled step by step.

Addition Steps

Large calculations can seem daunting, but by dividing them into steps, they become manageable. These steps typically involve:

• Aligning numbers by digits, rightmost first
• Adding digits starting from the right (the ones place)
• Carrying over any surplus exceeding 9 to the next column
• Continuing the addition and carry over sequentially to the left
• Writing the final sum when all columns are complete

Following these systematic steps, as demonstrated with the numbers 131,294 and 9,087, ensures accuracy and clarity in solving addition problems. Breaking down the process makes the task proceed smoothly and helps develop strong foundational skills in arithmetic.