Problem 31 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 1,007 \\ +\quad 331 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 1,338.

Step 1: Align the Numbers

Start by writing the numbers in a column format so that the digits are lined up according to their place value. Here, write \( 1,007 \) above \( 331 \), ensuring the ones, tens, and hundreds places are aligned.

Step 2: Add the Ones Place

Add the numbers in the ones place: \( 7 + 1 = 8 \). Write 8 under the line in the ones column.

Step 3: Add the Tens Place

Add the numbers in the tens place: \( 0 + 3 = 3 \). Write 3 under the line in the tens column.

Step 4: Add the Hundreds Place

Add the numbers in the hundreds place: \( 0 + 3 = 3 \). Write 3 under the line in the hundreds column.

Step 5: Add the Thousands Place

In the thousands place, there is only one number to consider from the first number: \( 1 \). Write 1 under the line in the thousands column as there is no corresponding digit to add from the second number.

Step 6: Write the Final Answer

Combine the results from each step: in the thousands place you have 1, hundreds place you have 3, tens place you have 3, and ones place you have 8. Therefore, the sum is \( 1,338 \).

Key Concepts

Place Value

Understanding place value is like having a guide to navigate through the world of numbers. It helps us see how each digit in a number has a specific value depending on where it is located. For instance, in the number 1,007, the '1' is not just one; it represents 1,000 because it is in the thousands place.

• Ones Place: Represents a single unit. In our example, it is the '7'.
• Tens Place: Groups of ten. Here, it’s a '0'.
• Hundreds Place: Groups of a hundred. Another '0' here.
• Thousands Place: Groups of a thousand. The number '1' gives this value.

Place value provides structure to our number system. It’s the framework that allows us to perform operations like addition and subtraction.

Adding Multi-Digit Numbers

Adding multi-digit numbers is simply about bringing together all the place values in a controlled manner. Each place value column must be added, beginning from the ones and moving leftward to higher values.
Consider adding 1,007 and 331. We start from the rightmost digits in the ones place where 7 and 1 reside. This gives us 8.

• Tens Place: Adding 0 and 3 gives 3.
• Hundreds Place: Adding 0 and 3 gives 3 as well.
• Thousands Place: Here, it’s just 1, since there’s no corresponding digit in 331.

Adding multi-digit numbers gets more complex when we need to carry over numbers, but this particular problem doesn't involve carrying. This makes it a perfect starting point for beginners.

Column Addition

Column addition is a straightforward technique to add numbers by writing them one below the other. This allows you to easily line up their digits according to their place values—ones below ones, tens below tens, and so on.
By positioning the numbers this way, you ensure that each addition is performed in its proper place, minimizing errors.
With 1,007 and 331:

• Align the numbers vertically.
• Work from right to left.
• Add each column starting from the ones place to the thousands.

Column addition keeps you organized, ensuring that each digit is considered correctly, leading to accurate results.