Mastering Arithmetic Operations A Step-by-Step Guide

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In this comprehensive guide, we will delve into various arithmetic operations, providing a detailed explanation and step-by-step solutions for each. This guide aims to enhance your understanding of fundamental mathematical concepts and improve your problem-solving skills. We will cover addition, subtraction, multiplication, and division, using specific examples to illustrate each operation. Let's embark on this mathematical journey together!

1. Addition: 876.8 + 42 + 6.862

Addition, one of the fundamental arithmetic operations, involves combining two or more numbers to find their total sum. In this section, we will explore the addition of three numbers: 876.8, 42, and 6.862. To ensure accuracy and clarity, we will break down the process into manageable steps.

First, let’s align the numbers vertically, paying close attention to the decimal points. This alignment ensures that we add the corresponding place values correctly. We have:

  876.800
   42.000
+  6.862
--------

Notice that we have added zeros to the end of 876.8 and 42 to make the decimal places consistent. This step is crucial for accurate addition, especially when dealing with numbers that have different decimal lengths.

Now, let’s add the numbers column by column, starting from the rightmost column (thousandths place):

  • Thousandths place: 0 + 0 + 2 = 2
  • Hundredths place: 0 + 0 + 6 = 6
  • Tenths place: 8 + 0 + 8 = 16. Write down 6 and carry over 1 to the ones place.
  • Ones place: 6 + 2 + 6 + 1 (carried over) = 15. Write down 5 and carry over 1 to the tens place.
  • Tens place: 7 + 4 + 0 + 1 (carried over) = 12. Write down 2 and carry over 1 to the hundreds place.
  • Hundreds place: 8 + 0 + 0 + 1 (carried over) = 9

So, the sum is:

  876.800
   42.000
+  6.862
--------
  925.662

Therefore, 876.8 + 42 + 6.862 = 925.662. This detailed step-by-step approach ensures that every aspect of the addition process is clear, reducing the chances of errors. Mastering addition is a fundamental skill in mathematics, and understanding the process thoroughly is essential for tackling more complex problems.

2. Subtraction: 900.5 - 678.963

Subtraction is another core arithmetic operation that involves finding the difference between two numbers. In this section, we will focus on subtracting 678.963 from 900.5. Similar to addition, aligning the numbers correctly is vital for accurate subtraction. Let’s break down the process:

First, align the numbers vertically, ensuring that the decimal points are in the same column. Add zeros as placeholders where necessary:

  900.500
- 678.963
--------

Now, subtract column by column, starting from the rightmost column (thousandths place). This often requires borrowing from the digits to the left.

  • Thousandths place: 0 - 3. We need to borrow. Borrow 1 from the hundredths place, making it 9 in the hundredths place and changing the thousandths place to 10. Now, 10 - 3 = 7.
  • Hundredths place: 9 - 6 = 3.
  • Tenths place: 4 - 9 (since we borrowed 1 earlier). We need to borrow again. Borrow 1 from the ones place, making it 9 in the ones place and changing the tenths place to 14. Now, 14 - 9 = 5.
  • Ones place: 9 - 8 = 1 (since we borrowed 1 earlier).
  • Tens place: 9 - 7 = 2.
  • Hundreds place: 8 - 6 = 2.

So, the result is:

  900.500
- 678.963
--------
  221.537

Therefore, 900.5 - 678.963 = 221.537. This detailed process of subtraction, including borrowing, is critical for accurate calculations. Consistent practice will solidify your understanding and speed up your problem-solving ability. Understanding how to borrow correctly is particularly important in subtraction, and this example illustrates the process clearly.

3. Multiplication: 5.24 x 24.34

Multiplication is a fundamental arithmetic operation that involves repeated addition. In this section, we will multiply 5.24 by 24.34. Multiplication of decimal numbers requires careful attention to the placement of the decimal point in the final answer. Let's break down the process into steps.

First, ignore the decimal points and multiply the numbers as if they were whole numbers:

   524
 x 2434
-------
  2096
 15720
209600
1048000
--------

Now, add the results:

    2096
   15720
  209600
+1048000
--------
 1275416

Next, count the total number of decimal places in the original numbers. In 5.24, there are two decimal places, and in 24.34, there are also two decimal places. So, the total number of decimal places is 2 + 2 = 4.

Now, place the decimal point in the result by counting four places from the right:

1275.416

Therefore, 5.24 x 24.34 = 127.5416. This step-by-step explanation of multiplication ensures that you understand each part of the process, from multiplying the numbers without decimals to placing the decimal point correctly in the final answer. Consistent practice with different numbers will build confidence and improve accuracy.

4. Division: 9.95 ÷ 21.5

Division is an arithmetic operation that involves splitting a number into equal parts. In this section, we will divide 9.95 by 21.5. Dividing decimal numbers requires a few extra steps to ensure accuracy. Let’s break down the process.

First, we need to eliminate the decimal from the divisor (21.5). To do this, we multiply both the dividend (9.95) and the divisor (21.5) by 10:

  1. 95 x 10 = 99.5
  2. 5 x 10 = 215

Now, we perform the division: 99.5 ÷ 215. Set up the long division:

      0.462
215 | 99.500
      86 0
      -----
      13 50
      12 90
      ------
       600
       430
       ----
       170

Steps:

  1. 215 does not go into 99, so we look at 995. 215 goes into 995 four times (4 x 215 = 860).
  2. Subtract 860 from 995 to get 135. Bring down the next digit (0) to make 1350.
  3. 215 goes into 1350 six times (6 x 215 = 1290).
  4. Subtract 1290 from 1350 to get 60. Bring down the next digit (0) to make 600.
  5. 215 goes into 600 two times (2 x 215 = 430).
  6. The division can continue, but for this example, we will stop at three decimal places.

So, 9.95 ÷ 21.5 ≈ 0.462. This step-by-step process for division with decimals includes clearing the decimal from the divisor and performing long division. Each step is critical for arriving at the correct answer. Practice with different division problems to build confidence and accuracy.

By mastering these fundamental arithmetic operations – addition, subtraction, multiplication, and division – you lay a solid foundation for tackling more advanced mathematical concepts. Each operation has its nuances, and understanding the step-by-step processes, including techniques like borrowing in subtraction and placing the decimal point in multiplication and division, is key to success in mathematics.