Problem 8 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 48 \\ +36 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 84.

Step 1: Add the Units

First, add the numbers in the units place. We have 8 (from 48) and 6 (from 36). When we add these together, we get 14. Write down the 4 in the units place and carry over 1 to the tens place.

Step 2: Add the Tens with Carryover

Now, add the numbers in the tens place: 4 (from 48) and 3 (from 36), giving us 7. Don't forget to add the carryover 1 from the previous step, resulting in 8. Write down 8 in the tens place.

Step 3: Write the Final Sum

Combining both results from Step 1 and Step 2, the final answer is 84.

Key Concepts

Place Value

Place value is a fundamental concept in arithmetic addition. Understanding it helps perform operations like summing numbers efficiently. Place value tells us the value of a digit based on its position in a number. Each position in a number has a value that is a power of ten. For example, in the number 48, the digit 4 is in the 'tens place' and represents the value 40 (4 times 10). Similarly, the digit 8 is in the 'units place', and it represents the value 8 (8 times 1).

Knowing place value helps us in aligning numbers correctly for addition operations, ensuring digits of the same place line up vertically. This alignment is crucial because it allows us to add values correctly from the rightmost digit (units) to the left (hundreds, thousands, etc.). When we add numbers, each digit's place determines its contribution to the final sum.

Carryover

Carryover is a concept that occurs during addition when the sum of digits in a particular place exceeds 9. It involves transferring the extra value to the next highest place value.

Imagine you're adding 8 (from 48) and 6 (from 36) in the units place. Their sum is 14, which is greater than 9. In this situation:

• We write down the 4 in the current place (units).
• The 1, is carried over to the tens place.

This process of moving that extra digit to the next higher place is called 'carryover'. It ensures that numbers are correctly accounted for their value and align with place value rules.

Carryover maintains the integrity of the addition operation across different place values and is a legitimate step when summing digits from columns.

Units and Tens Place

The units and tens places are the two rightmost positions in a multi-digit number and play a crucial role during arithmetic addition.

• Units Place: This is the first digit from the right in any given number. It represents the value's single digit. For the number 48, the 8 is in the units place, which is added to the 6 from the same place in 36 to initially get 14.
• Tens Place: This is the second digit from the right, representing ten times the value. In 48, the 4 is in the tens place. When adding the tens digit from two numbers (like 4 with 3 from 36), we also add any carryover from the units place. The small carryover change can affect the tens place sum as seen when we add 1 from the carryover, forming the 8 in the tens place result.

By understanding these places, students can successfully add numbers between these places, ensuring sums are correct. Knowing this simplifies the whole addition process, reducing the likelihood of errors.