Problem 60 - Fundamentals of Mathematics

Question

Add and subtract as in dicated. Subtract the difference of 8,412 and 576 from the sum of 22,140 and 8,617 .

Step-by-Step Solution

Answer

The result is 22,921.

Step 1: Understand the Problem

We need to subtract the result of one operation (the difference between 8,412 and 576) from another operation (the sum of 22,140 and 8,617). This is a two-part problem where we first perform addition and then subtraction.

Step 2: Calculate the Sum

First, we'll find the sum of 22,140 and 8,617.Calculate:\[22,140 + 8,617 = 30,757\]This is the first part of our problem.

Step 3: Calculate the Difference

Next, we need to find the difference between 8,412 and 576.Calculate:\[8,412 - 576 = 7,836\]This result will be subtracted from the sum obtained in Step 2.

Step 4: Subtract the Difference from the Sum

Now, we use the results from the previous steps. Subtract the difference (7,836) from the sum (30,757).Calculate:\[30,757 - 7,836 = 22,921\]This is the final result of the entire operation.

Key Concepts

Understanding Addition

Addition is one of the basic operations in arithmetic. It involves combining two or more numbers to get a total or sum. When we talk about adding numbers, we're essentially putting together groups of objects to see how many we have overall.

Here's what you need to remember about addition:

• The numbers being added are called addends; the result is called the sum.
• Order doesn't affect the sum, meaning that 3 + 5 is the same as 5 + 3.
• Addition is cumulative and associative, which means that grouping of numbers doesn't change the sum. For example, (2 + 3) + 4 is the same as 2 + (3 + 4).

In the exercise, we calculated the sum of 22,140 and 8,617. The operation looked like this:\[ 22,140 + 8,617 = 30,757 \]It's important to align numbers by their place value when adding to avoid mistakes and ensure an accurate result.

Exploring Subtraction

Subtraction is another fundamental arithmetic operation. It is essentially the process of taking one number away from another, which gives us the difference. While adding combines groups, subtracting takes away from a group.

Key points for subtraction include:

• The number you start with is the minuend, and the number you subtract is the subtrahend. The result is the difference.
• Unlike addition, subtraction is not commutative or associative. The order of the numbers is important. In subtraction, 10 - 3 is not the same as 3 - 10.
• When the minuend is smaller than the subtrahend, the result will be negative.

In our exercise, we subtracted 576 from 8,412:\[ 8,412 - 576 = 7,836 \]This difference was used in a later step to be subtracted from another sum. Paying attention to the order is crucial in subtraction, just as seen here.

Effective Problem-Solving

Problem-solving in arithmetic often involves breaking down a complex expression into simpler, manageable parts. This is what we saw in our exercise, which combined addition and subtraction.

To solve problems efficiently, consider these strategies:

• Read the question carefully to understand what is being asked.
• Identify the operations required, and plan the order in which to solve them.
• Perform each operation step by step, checking your work as you go.
• Reassess your results within the context of the problem to ensure they make sense.

In this case, we calculated a sum and a difference separately and then subtracted the latter from the former:First, the sum: \[ 22,140 + 8,617 = 30,757 \] Next, the difference: \[ 8,412 - 576 = 7,836 \] Finally, we solved the problem by subtracting: \[ 30,757 - 7,836 = 22,921 \]By following a structured approach, you can tackle complex arithmetic problems step-by-step with confidence.