Problem 16 - Fundamentals of Mathematics

Question

Find the sums and differences. $$520-216$$

Step-by-Step Solution

Answer

The difference is 304.

Step 1: Set Up the Expression

Start by writing down the subtraction problem as it is given: \(520 - 216\).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Key Concepts

Subtraction

Subtraction is the mathematical process of taking one quantity away from another. It is one of the basic arithmetic operations and is usually represented by the minus sign (-). In this operation, the number from which another number is to be subtracted is called the minuend, while the number that is subtracted is called the subtrahend. The result of subtraction is called the difference.

When performing subtraction, it is important to align the numbers correctly, especially when dealing with multi-digit numbers. Let's look at how you subtract 216 from 520.

• Write the minuend (520) above the subtrahend (216), aligning them by their rightmost digits.
• Begin subtracting from the rightmost digit (or units column). If the top digit is smaller than the bottom digit, you'll need to borrow from the column to the left.

For example: \[\begin{array}{c} 520 \- 216 \\hline\,\end{array}\]1. Start from the units place: 0 is less than 6, so you must borrow 1 from the tens place, making it 10 in the units place.2. 10 - 6 equals 4.3. In the tens place, after borrowing, you have a 1 in the tens column of 520 and 1 in 216, now subtract 1 from 1 leaving 0.4. In the hundreds place, subtract 2 from 5, resulting in 3.

Therefore, the difference between 520 and 216 is 304.

Addition

Addition is another fundamental arithmetic operation that combines quantities. Represented by the plus sign (+), addition is used to find the total or sum when combining numbers. Even though the original exercise focused only on subtraction, understanding addition is crucial for grasping overall arithmetic fluency.

Addition involves adding numbers together to find a sum. Here’s how you would generally perform addition:

• Align numbers by their rightmost columns, similar to subtraction.
• Begin with the rightmost digit and move leftward, adding digits column by column.
• If the sum of a column exceeds 9, carry over the excess to the next left column.

For instance, let's add 316 and 204:\[\begin{array}{c} 316 \+ 204 \\hline\ \end{array}\]1. Start with the rightmost digits: 6 + 4 = 10. Write 0 and carry over 1.2. Move to the next column from the right: 1 + 0 + 1 = 2.3. In the hundreds column, 3 + 2 = 5.

The sum of 316 and 204 is 520. This illustrates how understanding addition can help reverse-engineer a subtraction problem, adding pairs to check the results.

Step-by-Step Solution

Breaking an arithmetic problem down into step-by-step solutions aids in understanding the logic behind each action. This method provides clarity, especially when dealing with complex calculations.

Here is a breakdown of solving the subtraction problem \(520 - 216\):

• Step 1: Write down the subtraction problem: \(520 - 216\).
• Step 2: Align the numbers by their place values (units, tens, and hundreds).
• Step 3: Begin from the units place. You may need to borrow if the upper digit is smaller than the lower digit.
• Step 4: Subtract each column moving leftward.
• Step 5: Record the result after subtraction.

Using a step-by-step method:
1. From our example, borrow 1 from the tens place as 0 is smaller than 6.2. Subtract 6 from 10 to get 4.3. In the tens place, 1 from borrowing subtracts 1 in 216, giving 0.4. In the hundreds place, 5 minus 2 results in 3.

Presenting solutions step-by-step not only enhances understanding but also builds confidence in problem-solving skills. Following these structured steps helps in reducing errors and preparing for more advanced math problems.