Problem 98 - Fundamentals of Mathematics

Question

For the following problems, find the solutions. Subtract 37 from the difference between 263 and \(175 .\)

Step-by-Step Solution

Answer

The final result is 51.

Step 1: Find the Difference

Begin by calculating the difference between the two numbers specified in the exercise: 263 and 175. This involves performing the subtraction \(263 - 175\).

Step 2: Perform the Subtraction

Subtract 175 from 263. \(263 - 175 = 88\). So, the difference between 263 and 175 is 88.

Step 3: Subtract 37 from the Result

Now use the result from Step 2, which is 88, and subtract 37 from it to find the final solution. \(88 - 37 = 51\).

Step 4: Verify the Calculation

Check the calculations for accuracy by adding the second numbers back to the results. Adding 37 to 51 gives back 88, and adding 175 to this should give 263, confirming the calculation is correct.

Key Concepts

Subtraction

Subtraction is one of the basic operations in arithmetic. It allows us to find the difference between two numbers. In simple terms, if you have a certain quantity and you take some away, subtraction gives you the remaining quantity. For instance, if you start with 263 apples and give away 175, you perform the subtraction operation to find out how many apples are left: 263 minus 175.

When dealing with subtraction, it's crucial to line up the numbers correctly by their rightmost digits — this is helpful especially when working with larger numbers. Doing so ensures we subtract units from units, tens from tens, and so on. This process involves borrowing from neighboring digits when the top digit is smaller than the bottom one in any given column. Mastering subtraction is key to performing other arithmetic operations, making it an essential skill in mathematics.

Verification of Results

Verifying the result of a subtraction problem is important because it confirms the accuracy of your calculations. Generally, you can verify subtraction by performing an inverse operation, which is addition. When you add the result of the subtraction back to the number you subtracted, you should arrive at the original number.

Let's use our previous example: after subtracting 175 from 263, we got 88. Then, we further subtracted 37 from 88 to reach 51. To verify, simply add 37 back to 51 — you should end up with 88 again. This confirms that the initial subtraction was done correctly. Repeating this verification process with subsequent subtractions or using different methods can help pinpoint errors and ensure accuracy in your math work.

Number Operations

Number operations such as addition, subtraction, multiplication, and division form the foundation of mathematics. In any arithmetic problem, understanding when and how to use these operations is crucial. Subtraction, in particular, is about finding what remains when one quantity is taken away from another.

The problem we examined is a good example of how multiple number operations often work together. We first performed a subtraction: 263 minus 175. Then, we used that result for another subtraction: 88 minus 37. Such stepwise operations are common, and it's important to follow each carefully. Understanding these operations helps unlock more complex math problems and builds a strong foundation in mathematical reasoning.

• Do each operation one step at a time.
• Double-check your math for each operation.
• Understand the relation between different operations.

A firm grasp of number operations not only aids in problem-solving but also boosts your overall confidence in math.