Problem 9 - Fundamentals of Mathematics

Question

For the next three problems, find the sums. $$ \begin{array}{r} 847 \\ 825 \\ 796 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 2468.

Step 1: Understand the Problem

We are asked to find the sum of three numbers: 847, 825, and 796.

Step 2: Align the Numbers

Align the numbers vertically so that each digit is in the correct column. This means aligning them by their hundreds, tens, and ones place: \[\begin{array}{r}847 \825 \796 \\end{array}\]

Step 3: Add the Units Place

Start by adding the digits in the rightmost column (the units): 7 + 5 + 6 = 18. Write down 8 and carry over 1 to the tens column.

Step 4: Add the Tens Place

Next, add the digits in the tens column along with the carried over 1: 4 + 2 + 9 + 1 = 16. Write down 6 and carry over 1 to the hundreds column.

Step 5: Add the Hundreds Place

Finally, add the digits in the hundreds column with the carried over 1: 8 + 8 + 7 + 1 = 24. Since we have no more places to carry over to, write down the entire result, 24.

Step 6: Solution Summary

The resulting sum by combining all the columns' totals and carrying over is: 2468.

Key Concepts

Aligning numbers

When you are adding multiple numbers together, it's crucial to align them properly. This means placing each number's digits directly under the corresponding place value of the other numbers: units under units, tens under tens, and hundreds under hundreds. Aligning numbers helps avoid mistakes and ensures that you add the correct digits together. If you misalign numbers, you might accidentally add a tens digit as if it were a units digit, giving you a completely incorrect sum. Consider the three numbers from the exercise: 847, 825, and 796. You would write these vertically like this:

• 847
• 825
• 796

This stacks the units, tens, and hundreds together, much like how bricks are carefully aligned when building a wall. You wouldn’t want a brick (or a digit) sticking out of line, causing the structure to be unstable.

Carrying over

When adding numbers, sometimes the sum of a column is more than 9. When this happens, you must use the concept of carrying over. Carrying over means taking the tens component of your sum and adding it to the next left place value column. For example, in the units column from our exercise: 7 + 5 + 6 equals 18. You write down the 8 in the units place and carry over the 1 (from the 18) to the tens place. Then you repeat the process for each column, always remembering to include any carried over numbers. For instance, in the tens column, we add 4 + 2 + 9 and the carried over 1, giving us 16. Write down the 6 and carry over the 1 to the hundreds place. Carrying over is a useful technique because it allows you to handle larger sums without losing any value, ensuring that each digit occupies the correct place in the total number.

Place value

The concept of place value is essential for understanding addition as well as carrying over. Each digit in a number has a specific place value that determines its actual value based on its position. In the number 847:

• 8 is in the hundreds place, meaning it represents 800.
• 4 is in the tens place, representing 40.
• 7 is in the units place, representing just 7.

When adding numbers like 847, 825, and 796, it is this place value that guides the alignment process and helps us decide when and how to carry over. Every time you move a digit one place to the left, its value increases tenfold. For instance, moving from the units to the tens place makes a number ten times bigger. Understanding this makes addition more intuitive, helping you grasp why we line up numbers and carry digits from one column to the next.