Problem 33 - Fundamentals of Mathematics

Question

For the following problems, perform the additions. If you can, check each sum with a calculator. $$36+48$$

Step-by-Step Solution

Answer

The sum of 36 and 48 is 84.

Step 1: Understanding the Problem

We need to add two numbers: 36 and 48. Our goal is to find the sum of these two numbers.

Step 2: Align the Numbers

Align the numbers vertically by their place values, ensuring that the digits in the same column represent the same place value (units on top of units, tens on top of tens). This helps in performing the addition systematically:\[ \begin{array}{c} \;36 \ +48 \ \hline \end{array}\]

Step 3: Add the Units Place

Start by adding the digits in the units place (6 and 8): 6 + 8 = 14. Since 14 is greater than 9, place the 4 in the units place and carry over the 1 to the tens place.

Step 4: Add the Tens Place

Now add the digits in the tens place along with the carried over 1: 3 + 4 + 1 = 8. Write 8 in the tens place below the line.

Step 5: Write the Final Answer

Combine the results from the tens and units place to get the final sum: 84. Thus,\[ \begin{array}{c} \;36 \ +48 \ \hline 84 \end{array}\]

Step 6: Verification Using a Calculator

Use a calculator to confirm the addition: enter 36 + 48. The result should be 84, verifying the manual calculation.

Key Concepts

Aligning Numbers

Aligning numbers correctly is a crucial step in ensuring that the addition process is smooth and accurate. When you align numbers, you're arranging them in a way that visually helps keep track of what you're adding. It's simple:

- Write each number so that the digits in the same place value are directly above or below one another.
- In our example, 36 and 48 are aligned in two columns, with the units (ones) and tens in their respective columns.

This alignment aids in avoiding confusion, especially in cases where there's a carryover involved. When numbers are lined up properly, you can clearly see which digits need to be added together, helping to reduce mistakes. Take your time to align them properly, and you will find that the process becomes a lot more intuitive.

Place Value

The concept of place value is fundamental in understanding how numbers work in addition. Each digit in a number has a specific value based on its position. For example, in the number 36:

• The '6' is in the units place, meaning it stands for six ones.
• The '3' is in the tens place, indicating it stands for three tens or thirty.

When adding two numbers, you should add the digits in the same place value. This is why aligning the numbers becomes helpful. In our example, both 36 and 48 need their 'units' digits added first, and then the 'tens'.

Understanding place value means recognizing that the tens place digit represents a higher overall value than one in the units place. This understanding is key, particularly when dealing with carry over, as you need to correctly adjust not just the single ones but carry over to the tens.

Carry Over

Carry over is the term used when the sum of two digits in one place value exceeds what a single digit can represent, typically more than 9. For instance, when adding the units digits of 36 and 48:

• You start with the units place: 6 + 8 = 14.
• The digit '4' stays in the units place.
• The '1' is carried over to the tens place because 14 is over 9.

This step is called carrying over, as you're carrying the additional value to the next higher place value.

Why is it important?

This step ensures all sums are accurately reflected as part of the larger number. Skipping the carry over may lead to incorrect results. It's essential to add both the digits in that place value and any carry over amounts for accurate results. In our addition, after carrying over, we add the tens digits and the carried over '1', making sure the final sum is accurate.