Problem 47 - Fundamentals of Mathematics

Question

For the following problems, perform the additions. If you can, check each sum with a calculator. $$41+61+85+62$$

Step-by-Step Solution

Answer

The sum is 249.

Step 1: Arrange the Numbers

Start by writing all the numbers vertically, one below the other, aligning them by place value. This will help in systematically adding numbers from right to left.

Step 2: Add the Units Column

Add the digits in the units column: \(1 + 1 + 5 + 2 = 9\). Write 9 in the units place of the sum.

Step 3: Add the Tens Column

Next, add the digits in the tens column: \(4 + 6 + 8 + 6 = 24\). Since this results in a two-digit number, place the 4 in the tens place of the sum and carry over the 2 to the hundreds place.

Step 4: Combine Results

Combine the carried over value with the result from the tens column: the carried over 2 becomes the hundreds place digit, while the calculated tens and units become 4 and 9, respectively. So, the total sum is \(249\).

Step 5: Verify with a Calculator

Use a calculator to add \(41 + 61 + 85 + 62\). Input the numbers and check that the result matches the manual calculation, which should also confirm \(249\).

Key Concepts

Place Value

Place value is a fundamental concept in mathematics that helps us understand the significance of a digit based on its position within a number. Each digit in a number has a different value depending on where it is placed. Understanding place value allows us to effectively perform arithmetic operations like addition.

When working with place values, each column in a number represents a different power of ten. Starting from the right, the first column is the units column, followed by the tens column, hundreds column, and so on. This means that moving left, each column represents a ten times greater value than the previous one.

• The units column: Represents single units (1-9).
• The tens column: Represents tens (10-90).
• Hundreds and thousands follow in similar patterns, based on multiplying each subsequent place by ten.

By arranging numbers vertically and aligning them by their place values, it becomes easier to perform addition and ensure the accuracy of the result.

Units Column

The units column is the first column we consider when performing addition, starting from the right. It consists of the digits that represent quantities less than ten. Understanding how to add numbers in this column is crucial, as it forms the foundation of our addition process.

In the exercise, the numbers were:

• 41
• 61
• 85
• 62

By focusing on the units column, we added the digits:

• 1 (from 41)
• 1 (from 61)
• 5 (from 85)
• 2 (from 62)

Summing them up as: \(1 + 1 + 5 + 2 = 9\).

This result was placed in the units position of the final sum, setting the stage for us to confidently move on to the next column, the tens column.

Tens Column

The tens column is the second column from the right and involves digits that represent multiples of ten. In our exercise, after adding the units column, we move on to this column to ensure every number is accounted for. When summing numbers in the tens column, it's important to note any total exceeding 9, as this results in a carry over.

To add the tens column digits:

• 4 (from 41)
• 6 (from 61)
• 8 (from 85)
• 6 (from 62)

Their sum is calculated as:\(4 + 6 + 8 + 6 = 24\).

Since the result is a two-digit number, with 24, we put the 4 in the tens position and prepare for a carry over with the 2 moving to the hundreds column. This demonstrates how addition in the tens column influences the overall result.

Carry Over

The concept of "carry over" comes into play when a sum of digits in any column results in a number greater than nine. It requires transferring the extra value to the next leftmost column. This ensures that each digit maintains its proper place value. Carrying over is especially crucial in preserving the integrity of mathematical operations and results.

From our exercise, when the tens column added up to 24, the 2 had to be carried over to the hundreds column. This was done because 24 consists of 2 tens and 4 units, meaning it can't be fully represented in the tens column alone.

• The 4 from 24 was kept in the tens place.
• The 2 was carried over to the hundreds column, where it was added to any present values.

Thus, the carry over mechanism is an essential part of addition, ensuring the sum reflects each digit's place value correctly, resulting in an accurate total of 249 in this case.