Mastering Arithmetic Operations A Step-by-Step Guide

JU07/15/2025 08, 2025 by THE IDEN 53 views

Mastering Basic Arithmetic Operations A Comprehensive Guide

Arithmetic operations are the bedrock of mathematics, forming the foundation for more complex concepts. This comprehensive guide will delve into various arithmetic problems, providing step-by-step solutions and explanations to enhance your understanding. From basic addition and subtraction to multiplication, division, fractions, decimals, and mixed numbers, we will cover a wide range of topics to solidify your arithmetic skills. Whether you are a student looking to improve your grades or an adult seeking to refresh your knowledge, this guide is designed to be your go-to resource for mastering basic arithmetic operations.

(i) Addition: 92053 + 21152

Addition, one of the fundamental operations in arithmetic, involves combining two or more numbers to find their sum. In this section, we will explore the addition of whole numbers, specifically the problem 92053 + 21152. Understanding addition is crucial as it forms the basis for more complex mathematical operations. Let's break down the process step by step to ensure clarity and accuracy.

Step-by-Step Solution

To solve 92053 + 21152, we align the numbers vertically, ensuring that the digits in the same place value (ones, tens, hundreds, etc.) are aligned. This makes it easier to add each column separately.

  92053
+ 21152
-------

Add the ones column: 3 + 2 = 5. Write down 5 in the ones place.
Add the tens column: 5 + 5 = 10. Write down 0 in the tens place and carry over 1 to the hundreds column.
Add the hundreds column: 0 + 1 + 1 (carry-over) = 2. Write down 2 in the hundreds place.
Add the thousands column: 2 + 1 = 3. Write down 3 in the thousands place.
Add the ten-thousands column: 9 + 2 = 11. Write down 11 as there are no more columns to add.

Combining these results, we get:

  92053
+ 21152
-------
 113205

Final Answer

Therefore, 92053 + 21152 = 113205. This straightforward addition problem illustrates the basic principles of adding large numbers, emphasizing the importance of aligning place values and carrying over when necessary. Mastering this skill is essential for tackling more complex arithmetic challenges.

(ii) Subtraction: 62525 - 38744

Subtraction is another fundamental arithmetic operation that involves finding the difference between two numbers. In this section, we will tackle the subtraction problem 62525 - 38744. Subtraction is vital for various real-world applications, from managing finances to calculating distances. A clear understanding of the subtraction process ensures accuracy and efficiency in problem-solving. Let's delve into the step-by-step solution.

Step-by-Step Solution

To solve 62525 - 38744, we align the numbers vertically, similar to addition, ensuring that digits in the same place value are aligned. This method simplifies the subtraction process by allowing us to subtract column by column.

  62525
- 38744
-------

Subtract the ones column: 5 - 4 = 1. Write down 1 in the ones place.
Subtract the tens column: 2 - 4. Since 2 is less than 4, we need to borrow 1 from the hundreds column. The 5 in the hundreds place becomes 4, and the 2 in the tens place becomes 12. Now, 12 - 4 = 8. Write down 8 in the tens place.
Subtract the hundreds column: 4 - 7. Again, 4 is less than 7, so we borrow 1 from the thousands column. The 2 in the thousands place becomes 1, and the 4 in the hundreds place becomes 14. Now, 14 - 7 = 7. Write down 7 in the hundreds place.
Subtract the thousands column: 1 - 8. Since 1 is less than 8, we borrow 1 from the ten-thousands column. The 6 in the ten-thousands place becomes 5, and the 1 in the thousands place becomes 11. Now, 11 - 8 = 3. Write down 3 in the thousands place.
Subtract the ten-thousands column: 5 - 3 = 2. Write down 2 in the ten-thousands place.

Combining these results, we get:

  62525
- 38744
-------
  23781

Final Answer

Therefore, 62525 - 38744 = 23781. This subtraction problem highlights the importance of borrowing from higher place values when the digit being subtracted is larger than the digit it is being subtracted from. Mastering this technique is essential for accurate subtraction of larger numbers.

(iii) Multiplication: 1910 × 60

Multiplication is a fundamental arithmetic operation that involves repeated addition. In this section, we will explore the multiplication problem 1910 × 60. Multiplication is essential for various calculations, such as determining areas, volumes, and quantities. A solid understanding of multiplication techniques ensures accurate and efficient problem-solving. Let’s break down the solution step by step.

Step-by-Step Solution

To solve 1910 × 60, we can use the standard multiplication method. This involves multiplying each digit of one number by each digit of the other number and then adding the results. Since we are multiplying by 60, we can simplify the process by first multiplying 1910 by 6 and then multiplying the result by 10 (which is equivalent to adding a zero at the end).

Multiply 1910 by 6:
- 6 × 0 = 0
- 6 × 1 = 6
- 6 × 9 = 54. Write down 4 and carry over 5.
- 6 × 1 = 6. Add the carry-over 5 to get 11. Write down 11.
```
   1910
×    6
-------
  11460
```
Multiply the result by 10 (add a zero at the end):
```
 11460 × 10 = 114600
```

Alternatively, we can multiply directly by 60:

       1910
    ×   60
    -------
       0000  (1910 × 0)
    11460   (1910 × 6, shifted one place to the left)
    -------
    114600

Final Answer

Therefore, 1910 × 60 = 114600. This multiplication problem demonstrates the standard method for multiplying large numbers, highlighting the importance of place value and carrying over. Proficiency in multiplication is crucial for tackling more advanced mathematical problems.

(iv) Division: 144360 ÷ 22

Division is an arithmetic operation that involves splitting a number into equal parts or groups. In this section, we will solve the division problem 144360 ÷ 22. Division is crucial for various applications, such as sharing quantities, determining rates, and solving equations. A clear understanding of the division process ensures accurate and efficient problem-solving. Let's break down the solution step by step.

Step-by-Step Solution

To solve 144360 ÷ 22, we use the long division method. This involves dividing the dividend (144360) by the divisor (22) step by step.

Set up the long division:
```
        ______
22 | 144360
```
Divide the first few digits of the dividend (144) by the divisor (22):
- 22 goes into 144 six times (6 × 22 = 132).
```
     6_____
22 | 144360
   - 132
   ------
     12
```

Bring down the next digit (3) from the dividend:

     6_____
22 | 144360
   - 132
   ------
     123

Divide 123 by 22:

22 goes into 123 five times (5 × 22 = 110).

     65____
22 | 144360
   - 132
   ------
     123
   - 110
   ------
      13

Bring down the next digit (6) from the dividend:

     65____
22 | 144360
   - 132
   ------
     123
   - 110
   ------
      136

Divide 136 by 22:

22 goes into 136 six times (6 × 22 = 132).

     656___
22 | 144360
   - 132
   ------
     123
   - 110
   ------
      136
   - 132
   ------
       4

Bring down the last digit (0) from the dividend:

     656___
22 | 144360
   - 132
   ------
     123
   - 110
   ------
      136
   - 132
   ------
       40

Divide 40 by 22:

22 goes into 40 once (1 × 22 = 22).

     6561
22 | 144360
   - 132
   ------
     123
   - 110
   ------
      136
   - 132
   ------
       40
   -  22
   ------
       18

The remainder is 18.

Final Answer

Therefore, 144360 ÷ 22 = 6561 with a remainder of 18, or approximately 6561.82 when expressed as a decimal. This division problem demonstrates the long division method, highlighting the steps of dividing, multiplying, subtracting, and bringing down digits. Proficiency in division is crucial for solving various mathematical problems and real-world applications.

(v) Addition of Mixed Numbers: 2 1/2 + 3 1/2

Mixed numbers combine whole numbers and fractions, and adding them requires a specific approach. In this section, we will solve the addition problem 2 1/2 + 3 1/2. Understanding how to add mixed numbers is crucial for various applications, such as cooking, measuring, and construction. Let's break down the solution step by step.

Step-by-Step Solution

To add mixed numbers, we can add the whole numbers and the fractions separately. The problem is 2 1/2 + 3 1/2.

Add the whole numbers:
- 2 + 3 = 5
Add the fractions:
- 1/2 + 1/2 = 2/2
Simplify the fraction:
- 2/2 = 1
Combine the results:
- 5 (from the whole numbers) + 1 (from the fractions) = 6

Alternatively, we can convert the mixed numbers to improper fractions first and then add them.

Convert mixed numbers to improper fractions:
- 2 1/2 = (2 × 2 + 1) / 2 = 5/2
- 3 1/2 = (3 × 2 + 1) / 2 = 7/2
Add the improper fractions:
- 5/2 + 7/2 = 12/2
Simplify the improper fraction:
- 12/2 = 6

Final Answer

Therefore, 2 1/2 + 3 1/2 = 6. This problem demonstrates two methods for adding mixed numbers: adding whole numbers and fractions separately, and converting mixed numbers to improper fractions before adding. Proficiency in both methods enhances problem-solving flexibility.

(vi) Subtraction of Mixed Numbers: 3 1/2 - 2 5/6

Subtracting mixed numbers requires careful attention to both the whole numbers and the fractional parts. In this section, we will solve the subtraction problem 3 1/2 - 2 5/6. Understanding how to subtract mixed numbers is essential for various applications, such as cooking, carpentry, and financial calculations. Let's break down the solution step by step.

Step-by-Step Solution

To solve 3 1/2 - 2 5/6, we first need to find a common denominator for the fractions. The least common denominator (LCD) for 2 and 6 is 6. We then convert the fractions to equivalent fractions with the common denominator.

Convert the fractions to have a common denominator:
- 1/2 = (1 × 3) / (2 × 3) = 3/6
So, the problem becomes 3 3/6 - 2 5/6.
Subtract the whole numbers and fractions separately:
- Subtract the whole numbers: 3 - 2 = 1
- Subtract the fractions: 3/6 - 5/6. Since 3/6 is less than 5/6, we need to borrow 1 from the whole number.
Borrow 1 from the whole number and convert it to a fraction:
- Borrow 1 from 3, making it 2.
- Convert the borrowed 1 to 6/6 (since the denominator is 6).
- Add the borrowed fraction to the existing fraction: 3/6 + 6/6 = 9/6
Now, the problem becomes 2 9/6 - 2 5/6.
Subtract the fractions:
- 9/6 - 5/6 = 4/6
Subtract the whole numbers:
- 2 - 2 = 0
Combine the results:
- 0 + 4/6 = 4/6
Simplify the fraction:
- 4/6 = 2/3

Alternatively, we can convert the mixed numbers to improper fractions first and then subtract them.

Convert mixed numbers to improper fractions:
- 3 1/2 = (3 × 2 + 1) / 2 = 7/2
- 2 5/6 = (2 × 6 + 5) / 6 = 17/6
Find a common denominator (LCD) for 2 and 6, which is 6.
Convert the fractions to equivalent fractions with the common denominator:
- 7/2 = (7 × 3) / (2 × 3) = 21/6
Subtract the improper fractions:
- 21/6 - 17/6 = 4/6
Simplify the fraction:
- 4/6 = 2/3

Final Answer

Therefore, 3 1/2 - 2 5/6 = 2/3. This subtraction problem demonstrates the importance of finding a common denominator and borrowing from the whole number when necessary. Proficiency in subtracting mixed numbers is essential for various practical applications.

(vii) Addition of Fractions: 1/3 + 3/5

Adding fractions requires a common denominator to ensure accurate results. In this section, we will solve the addition problem 1/3 + 3/5. Understanding how to add fractions is fundamental for various mathematical concepts and real-world applications, such as cooking, measuring, and dividing resources. Let's break down the solution step by step.

Step-by-Step Solution

To add 1/3 + 3/5, we first need to find a common denominator. The least common denominator (LCD) for 3 and 5 is 15. We then convert each fraction to an equivalent fraction with the common denominator.

Find the least common denominator (LCD):
- The LCD of 3 and 5 is 15.
Convert the fractions to equivalent fractions with the LCD:
- 1/3 = (1 × 5) / (3 × 5) = 5/15
- 3/5 = (3 × 3) / (5 × 3) = 9/15
Add the fractions:
- 5/15 + 9/15 = (5 + 9) / 15 = 14/15
Simplify the fraction (if possible):
- 14/15 is already in its simplest form as 14 and 15 have no common factors other than 1.

Final Answer

Therefore, 1/3 + 3/5 = 14/15. This problem illustrates the importance of finding a common denominator before adding fractions. Mastering this skill is crucial for various mathematical and practical applications.

(viii) Subtraction of Mixed Numbers: 3 1/3 - 2 2/3

Subtracting mixed numbers, especially when the fractional part of the subtrahend is larger, requires careful handling. In this section, we will solve the subtraction problem 3 1/3 - 2 2/3. Proficiency in subtracting mixed numbers is essential for various real-world applications, such as carpentry, cooking, and financial planning. Let's break down the solution step by step.

Step-by-Step Solution

To solve 3 1/3 - 2 2/3, we can first convert the mixed numbers to improper fractions. This method simplifies the subtraction process, especially when the fraction being subtracted is larger than the fraction it is being subtracted from.

Convert the mixed numbers to improper fractions:
- 3 1/3 = (3 × 3 + 1) / 3 = 10/3
- 2 2/3 = (2 × 3 + 2) / 3 = 8/3
Subtract the improper fractions:
- 10/3 - 8/3 = (10 - 8) / 3 = 2/3

Alternatively, we can subtract the whole numbers and fractions separately, borrowing if necessary.

Subtract the whole numbers and fractions separately:
- Subtract the whole numbers: 3 - 2 = 1
- Subtract the fractions: 1/3 - 2/3. Since 1/3 is less than 2/3, we need to borrow 1 from the whole number.
Borrow 1 from the whole number and convert it to a fraction:
- Borrow 1 from 3, making it 2.
- Convert the borrowed 1 to 3/3 (since the denominator is 3).
- Add the borrowed fraction to the existing fraction: 1/3 + 3/3 = 4/3
Now, the problem becomes 2 4/3 - 2 2/3.
Subtract the fractions:
- 4/3 - 2/3 = 2/3
Subtract the whole numbers:
- 2 - 2 = 0
Combine the results:
- 0 + 2/3 = 2/3

Final Answer

Therefore, 3 1/3 - 2 2/3 = 2/3. This problem demonstrates the process of subtracting mixed numbers, including converting to improper fractions and borrowing when necessary. Proficiency in both methods provides flexibility in problem-solving.

(ix) Addition of Decimals: 0.29 + 0.738

Adding decimals requires aligning the decimal points to ensure accurate results. In this section, we will solve the addition problem 0.29 + 0.738. Understanding how to add decimals is essential for various real-world applications, such as financial calculations, measurements, and scientific computations. Let's break down the solution step by step.

Step-by-Step Solution

To add 0.29 and 0.738, we align the decimal points vertically. This ensures that we are adding digits with the same place value (tenths, hundredths, thousandths, etc.).

Align the decimal points:
```
 0.29
```

0.738

Add zeros as placeholders if needed:
```
 0.290
```

0.738

Add the numbers column by column, starting from the rightmost column (thousandths):
- 0 + 8 = 8. Write down 8 in the thousandths place.
- 9 + 3 = 12. Write down 2 in the hundredths place and carry over 1 to the tenths column.
- 2 + 7 + 1 (carry-over) = 10. Write down 0 in the tenths place and carry over 1 to the ones column.
- 0 + 0 + 1 (carry-over) = 1. Write down 1 in the ones place.
```
 0.290
```

0.738
1.028

Final Answer

Therefore, 0.29 + 0.738 = 1.028. This problem illustrates the importance of aligning decimal points when adding decimals. Accurate alignment ensures that digits with the same place value are added together, leading to the correct result. Mastering this skill is crucial for various mathematical and practical applications.

(x) Subtraction of Decimals: 4.221 - 3.98

Subtracting decimals requires aligning the decimal points, similar to addition, to ensure accurate results. In this section, we will solve the subtraction problem 4.221 - 3.98. Understanding how to subtract decimals is essential for various practical applications, such as financial calculations, measurements, and scientific computations. Let's break down the solution step by step.

Step-by-Step Solution

To subtract 3.98 from 4.221, we align the decimal points vertically. This ensures that we are subtracting digits with the same place value (tenths, hundredths, thousandths, etc.).

Align the decimal points:
```
 4.221
```

3.98

Add zeros as placeholders if needed:
```
 4.221
```

3.980

Subtract the numbers column by column, starting from the rightmost column (thousandths):
- 1 - 0 = 1. Write down 1 in the thousandths place.
- 2 - 8. Since 2 is less than 8, we need to borrow 1 from the tenths place. The 2 in the tenths place becomes 1, and the 2 in the hundredths place becomes 12. Now, 12 - 8 = 4. Write down 4 in the hundredths place.
- 1 - 9. Since 1 is less than 9, we need to borrow 1 from the ones place. The 4 in the ones place becomes 3, and the 1 in the tenths place becomes 11. Now, 11 - 9 = 2. Write down 2 in the tenths place.
- 3 - 3 = 0. Write down 0 in the ones place.
```
 4.221
```

3.980
0.241

Final Answer

Therefore, 4.221 - 3.98 = 0.241. This problem illustrates the importance of aligning decimal points and borrowing when necessary during subtraction of decimals. Accurate alignment and borrowing ensure that digits with the same place value are subtracted correctly, leading to the correct result.

(xi) Multiplication of Decimals: 5.27 × 3.0

Multiplying decimals involves multiplying the numbers as if they were whole numbers and then placing the decimal point in the correct position. In this section, we will solve the multiplication problem 5.27 × 3.0. Understanding how to multiply decimals is essential for various real-world applications, such as calculating costs, measurements, and scientific computations. Let's break down the solution step by step.

Step-by-Step Solution

To multiply 5.27 by 3.0, we can treat the numbers as whole numbers initially and then place the decimal point in the final result.

Multiply the numbers as if they were whole numbers:
- 527 × 3 = 1581
Count the total number of decimal places in the original numbers:
- 5.27 has 2 decimal places.
- 3.0 has 1 decimal place.
- Total decimal places: 2 + 1 = 3
Place the decimal point in the result so that there are 3 decimal places:
- 1581 becomes 15.81

Alternatively, we can use the standard multiplication method:

     5.27
  ×  3.0
    -------
     000   (5.27 × 0)
  1581    (5.27 × 3, shifted one place to the left)
    -------
  15.810

Final Answer

Therefore, 5.27 × 3.0 = 15.81. This problem demonstrates the process of multiplying decimals, including treating the numbers as whole numbers initially and placing the decimal point in the correct position based on the total number of decimal places in the original numbers. Proficiency in multiplying decimals is crucial for various practical applications.

(xii) Subtraction of Decimals: 6.084 - 0.254

Subtracting decimals requires aligning the decimal points to ensure accurate results, similar to addition. In this section, we will solve the subtraction problem 6.084 - 0.254. Understanding how to subtract decimals is essential for various real-world applications, such as financial calculations, measurements, and scientific computations. Let's break down the solution step by step.

Step-by-Step Solution

To subtract 0.254 from 6.084, we align the decimal points vertically. This ensures that we are subtracting digits with the same place value (tenths, hundredths, thousandths, etc.).

Align the decimal points:
```
 6.084
```

0.254

Subtract the numbers column by column, starting from the rightmost column (thousandths):
- 4 - 4 = 0. Write down 0 in the thousandths place.
- 8 - 5 = 3. Write down 3 in the hundredths place.
- 0 - 2. Since 0 is less than 2, we need to borrow 1 from the ones place. The 6 in the ones place becomes 5, and the 0 in the tenths place becomes 10. Now, 10 - 2 = 8. Write down 8 in the tenths place.
- 5 - 0 = 5. Write down 5 in the ones place.
```
 6.084
```

0.254
5.830

Simplify the result by removing the trailing zero (if any):
- 5.830 becomes 5.83

Final Answer

Therefore, 6.084 - 0.254 = 5.83. This problem illustrates the importance of aligning decimal points and borrowing when necessary during subtraction of decimals. Accurate alignment and borrowing ensure that digits with the same place value are subtracted correctly, leading to the correct result. Proficiency in subtracting decimals is crucial for various mathematical and practical applications.

This comprehensive guide has covered a range of basic arithmetic operations, from addition and subtraction to multiplication, division, fractions, decimals, and mixed numbers. Each section provided a step-by-step solution, emphasizing the importance of understanding the underlying principles. By mastering these fundamental arithmetic skills, you can build a strong foundation for more advanced mathematical concepts and real-world applications.

(i) Addition: 92053 + 21152

Step-by-Step Solution

Final Answer

(ii) Subtraction: 62525 - 38744

Step-by-Step Solution

Final Answer

(iii) Multiplication: 1910 × 60

Step-by-Step Solution

Final Answer

(iv) Division: 144360 ÷ 22

Step-by-Step Solution

Final Answer

(v) Addition of Mixed Numbers: 2 1/2 + 3 1/2

Step-by-Step Solution

Final Answer

(vi) Subtraction of Mixed Numbers: 3 1/2 - 2 5/6

Step-by-Step Solution

Final Answer

(vii) Addition of Fractions: 1/3 + 3/5

Step-by-Step Solution

Final Answer

(viii) Subtraction of Mixed Numbers: 3 1/3 - 2 2/3

Step-by-Step Solution

Final Answer

(ix) Addition of Decimals: 0.29 + 0.738

Step-by-Step Solution

0.738

0.738

0.738

Final Answer

(x) Subtraction of Decimals: 4.221 - 3.98

Step-by-Step Solution

3.98

3.980

3.980

Final Answer

(xi) Multiplication of Decimals: 5.27 × 3.0

Step-by-Step Solution

Final Answer

(xii) Subtraction of Decimals: 6.084 - 0.254

Step-by-Step Solution

0.254

0.254

Final Answer