Problem 18 - Fundamentals of Mathematics

Question

Find the sums and differences. Subtract 425 from 816 .

Step-by-Step Solution

Answer

The result of subtracting 425 from 816 is 391.

Step 1: Identify the Numbers

We are given two numbers: 816 and 425. We need to subtract 425 from 816.

Step 2: Set Up the Subtraction Problem

Write the numbers in a column, placing the larger number (816) on top of the smaller number (425). Ensure the digits are aligned according to their place value:\[ \begin{array}{c} 816 \ -425 \ \hline \end{array}\]

Step 3: Subtract the Digits in Each Column

Start subtracting from the rightmost column (units):1. Units column: 6 - 5 = 1.2. Tens column: 1 - 2. Since 1 is smaller than 2, borrow 1 from the hundreds column: 11 - 2 = 9.3. Hundreds column: 7 - 4 = 3.After borrowing, our column looks like this:\[ \begin{array}{c} 7 \ -4 \ \hline 3 \ \end{array} \quad \begin{array}{c} 11 \ - 2 \ \hline 9 \ \end{array} \quad \begin{array}{c} 6 \ -5 \ \hline 1 \ \end{array}\]

Step 4: Write Down the Result

Combine the subtraction results from each column to find the final answer:\[ \begin{array}{c} 391 \ \end{array}\]

Key Concepts

Place Value

Understanding place value is a crucial part of performing subtraction and other arithmetic operations. Place value refers to the value of a digit based on its position within a number. This concept helps us break down numbers into their respective constituent parts, such as units, tens, hundreds, and so on. For example, in the number 816, the digit 8 represents 800 because it is in the hundreds place, the digit 1 represents 10 in the tens place, and the digit 6 represents 6 in the units place.
When setting up a subtraction problem, aligning the digits according to their place values is essential. This ensures accurate subtraction. If the digits are mixed up, the result will not reflect the true difference between the numbers. Always make sure each digit is directly beneath the corresponding place value of the numbers being subtracted.

Borrowing in Subtraction

Borrowing in subtraction is a technique used when a digit in a particular place value is too small to allow subtraction from it. For instance, in the problem of subtracting 425 from 816, we run into a situation where the digit

• in the tens place of 816, which is 1, is smaller than the digit in the tens place of 425, which is 2.

In such cases, we "borrow" 1 from the next higher place value to make subtraction possible. When borrowing, we reduce the digit in the next column to the left by 1 and increase the current column's digit by 10. In our example, we borrow 1 from the hundreds place, making the digit in the tens place equal to 11. We then perform the subtraction as 11 - 2 = 9.
Borrowing is a handy tool that helps maintain the place values' integrity during subtraction, making it easier to find the correct difference.

Columnar Subtraction

Columnar subtraction refers to the method of subtracting numbers by writing them one below the other, such that their digits are aligned according to their respective place values. This setup makes it easier to visualize the subtraction process and minimizes errors. To perform columnar subtraction:

• Write the larger number on top.
• Write the smaller number below it, ensuring that the digits are aligned according to place value.
• Subtract beginning from the rightmost column, which represents the units.

In the example of subtracting 425 from 816, after setting up our numbers in columns, we start subtracting from the units column and move leftwards. This systematic approach helps in handling more complex problems with multiple digits, ensuring that we correctly execute borrowing if necessary and achieve the proper result.
Columnar subtraction is a tried and tested method that simplifies the subtraction of multi-digit numbers, making arithmetic operations manageable and less prone to mistake.