Solving 115 17 1/3 ÷ 2 2/2 + 0.68 × 2.5 A Step-by-Step Algebraic Solution

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Introduction

In this comprehensive exploration, we will meticulously dissect the algebraic expression 115 17 1/3 : 2 2/2 + 0.68 × 2.5, delving into the intricacies of its components and the order of operations required to arrive at the correct solution. Algebra, a fundamental branch of mathematics, serves as the bedrock for numerous scientific and technological disciplines. Mastering algebraic expressions is crucial for problem-solving, critical thinking, and analytical reasoning. This article aims to provide a step-by-step guide, ensuring clarity and fostering a deep understanding of the underlying principles.

Breaking Down the Expression

To effectively tackle the expression, let's first break it down into its constituent parts. We have a combination of whole numbers, fractions, decimals, and arithmetic operations, namely division, addition, and multiplication. The expression can be visually represented as follows:

  • 115 17 1/3: This represents a mixed number, a combination of a whole number (115) and a fraction (17 1/3).
  • : (Division): The division operator indicates that the preceding quantity will be divided by the subsequent one.
  • 2 2/2: Another mixed number, comprising a whole number (2) and a fraction (2/2).
  • + (Addition): The addition operator signifies that the results of the division and multiplication operations will be added together.
  • 0.68: A decimal number.
  • × (Multiplication): The multiplication operator denotes that the two adjacent quantities will be multiplied.
  • 2.5: Another decimal number.

Converting Mixed Numbers to Improper Fractions

Before we can perform any arithmetic operations, it's essential to convert the mixed numbers into improper fractions. This simplifies the calculations and allows for seamless application of mathematical rules.

115 17/3 can be converted as follows:

  1. Multiply the whole number (115) by the denominator of the fraction (3): 115 × 3 = 345
  2. Add the numerator of the fraction (17) to the result: 345 + 17 = 362
  3. Place the sum (362) over the original denominator (3): 362/3

Therefore, 115 17/3 is equivalent to 362/3.

Similarly, 2 2/2 can be converted:

  1. Multiply the whole number (2) by the denominator of the fraction (2): 2 × 2 = 4
  2. Add the numerator of the fraction (2) to the result: 4 + 2 = 6
  3. Place the sum (6) over the original denominator (2): 6/2

Simplifying 6/2, we get 3.

Rewriting the Expression

Now that we've converted the mixed numbers into improper fractions, we can rewrite the original expression as:

362/3 : 3 + 0.68 × 2.5

Applying the Order of Operations (PEMDAS/BODMAS)

To ensure we arrive at the correct solution, we must adhere to the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms provide a hierarchical guide for performing calculations.

In our expression, we have division, addition, and multiplication. According to PEMDAS/BODMAS, we must perform division and multiplication before addition.

Performing Division

We begin by dividing 362/3 by 3. Dividing by a whole number is the same as multiplying by its reciprocal. The reciprocal of 3 is 1/3. Therefore:

(362/3) : 3 = (362/3) × (1/3)

Multiplying the fractions, we get:

(362 × 1) / (3 × 3) = 362/9

Performing Multiplication

Next, we multiply 0.68 by 2.5:

0. 68 × 2.5 = 1.7

Rewriting the Expression Again

Now we can rewrite the expression with the results of the division and multiplication operations:

362/9 + 1.7

Performing Addition

To add 362/9 and 1.7, we need to express both terms with a common denominator. We can convert 1.7 into a fraction by placing it over 1 (1.7/1) and then multiplying the numerator and denominator by 10 to eliminate the decimal (17/10). Now we need to find a common denominator for 9 and 10, which is 90.

Convert 362/9 to an equivalent fraction with a denominator of 90:

(362/9) × (10/10) = 3620/90

Convert 17/10 to an equivalent fraction with a denominator of 90:

(17/10) × (9/9) = 153/90

Now we can add the fractions:

3620/90 + 153/90 = 3773/90

Converting Improper Fraction to Mixed Number (Optional)

We can convert the improper fraction 3773/90 back into a mixed number for easier interpretation:

Divide 3773 by 90:

  • 3773 ÷ 90 = 41 with a remainder of 83

Therefore, 3773/90 is equivalent to 41 83/90.

Expressing as Decimal (Optional)

Alternatively, we can express the result as a decimal by dividing 3773 by 90:

3773 ÷ 90 ≈ 41.922 (rounded to three decimal places)

Final Answer

The solution to the algebraic expression 115 17 1/3 : 2 2/2 + 0.68 × 2.5 is:

  • 3773/90 (Improper Fraction)
  • 41 83/90 (Mixed Number)
  • 41.922 (Decimal, rounded to three decimal places)

Conclusion

Through a meticulous step-by-step approach, we have successfully解剖 and solved the algebraic expression 115 17 1/3 : 2 2/2 + 0.68 × 2.5. We began by converting mixed numbers into improper fractions, then meticulously applied the order of operations (PEMDAS/BODMAS) to ensure accuracy. The result can be expressed in various forms – as an improper fraction, a mixed number, or a decimal – each providing a different perspective on the same numerical value.

The mastery of algebraic expressions is not merely an academic exercise; it is a fundamental skill that empowers individuals to navigate complex problems in diverse fields. By understanding the underlying principles and diligently applying the rules of algebra, we can unlock a world of mathematical possibilities.

This detailed exploration serves as a testament to the power of methodical problem-solving and the importance of a solid foundation in algebraic concepts. As we continue our journey through the realm of mathematics, let us embrace the challenges and celebrate the triumphs that come with unraveling the intricacies of numerical expressions.

Further Practice

To solidify your understanding of algebraic expressions, consider tackling additional problems that involve mixed numbers, decimals, and various arithmetic operations. Seek out resources such as textbooks, online tutorials, and practice worksheets. The more you practice, the more confident and proficient you will become in your algebraic abilities.

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Solving 115 17 1/3 ÷ 2 2/2 + 0.68 × 2.5 A Step-by-Step Algebraic Solution