Problem 44 - Fundamentals of Mathematics

Question

Combine the numbers as indicated. $$3,704+2,344+429+10,374+74$$

Step-by-Step Solution

Answer

The sum is 16,925.

Step 1: Align the numbers for addition

To add the numbers together, write each number under one another making sure the digits of the same place value are aligned. This will allow us to add each column easily.

Step 2: Add the ones column

Start by adding the digits in the ones place (rightmost column): \(4 + 4 + 9 + 4 = 21\). Write down \(1\) and carry over \(2\) to the tens place.

Step 3: Add the tens column

Add the digits in the tens place including the carried \(2\): \(0 + 4 + 2 + 7 + 7 = 20\). Write down \(0\) and carry over \(2\) to the hundreds place.

Step 4: Add the hundreds column

Add the digits in the hundreds place including the carried \(2\): \(7 + 3 + 4 + 3 = 17\). Write down \(7\) and carry over \(1\) to the thousands place.

Step 5: Add the thousands column

Add the digits in the thousands place including the carried \(1\): \(3 + 2 + 0 + 0 + 1 = 6\). Write down the \(6\), and there is no more carry since the maximum value is \(6\).

Step 6: Add the ten-thousands column if applicable

In this exercise, only the number \(10,374\) affects the ten-thousands column. Thus, simply write down the \(1\) from this column.

Step 7: Conclusion: Write down the final sum

Combine all the results from previous steps to form the complete number. So, the final sum of all the numbers is \(16,925\).

Key Concepts

Aligned Addition

Aligned addition means lining up numbers in such a way that their digits match by place value. This arrangement helps you add numbers efficiently and accurately by columns.
For example, when adding numbers such as 3,704 and 2,344, ensure each number is written so that:

• The ones digit of one number is directly below the ones digit of the other number.
• The tens digits are below each other, and so on with hundreds, thousands, etc.

This alignment allows you to clearly see which digits should be added together. It reduces confusion and ensures that you don’t mistakenly add digits of different place values together.

Place Value

Place value is the value of a digit based on its position in a number. Understanding place value helps in performing operations like addition accurately.
Every digit in a number has a place value, such as:

• Ones
• Tens
• Hundreds
• Thousands

For instance, consider the number 3,704:

• '4' is in the ones place, so it represents 4.
• '0' is in the tens place, representing 0 tens.
• '7' is in the hundreds place, representing 700.
• '3' is in the thousands place, representing 3,000.

Knowing how each digit contributes to the overall number is crucial when you are adding numbers together. This understanding helps ensure that you add the correct digits in the right column.

Addition with Carrying

Addition with carrying, also known as "regrouping," occurs when the sum of digits in a column exceeds 9. This means you cannot write the entire sum in the same column and must "carry" part of it to the next higher place value.
This happens frequently and is an important part of multi-digit addition.

• For example, when adding the ones place as in 4 + 4 + 9 + 4 = 21, you write down the 1 and carry the 2 to the tens column.
• Similarly, with 0 + 4 + 2 + 7 + 7 = 20 in the tens place, you write down the 0 and carry the 2 to the hundreds place.

Carrying helps correctly compute the sum by redistributing values across place values, ensuring that each column accurately reflects its potential contribution to the total number.

Column Addition

Column addition involves adding numbers by summing each corresponding column individually, starting from the rightmost column (usually the ones place) moving left.
This method breaks down the larger task of adding multi-digit numbers into simpler parts.

• Start with the ones column and add all numbers there.
• Move to the tens column, then the hundreds column, and so forth.

By focusing on one column at a time, you simplify the process and avoid overwhelming yourself by managing smaller pieces of the addition process.
Each column's addition might involve carrying, as seen in multiple-digit sums. After processing each column, make sure to write down results appropriately before moving to the next column, ensuring an accurate final result.