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Decoding the Enigma: Understanding 75 out of 90 (75/90) and its Implications

The fraction 75/90, or its equivalent decimal representation 0.8333..., might seem like a simple mathematical concept at first glance. However, understanding its implications goes beyond basic arithmetic and delves into the realms of percentages, ratios, proportions, and even statistical interpretations. This article will explore the multifaceted nature of 75/90, providing a comprehensive understanding for learners of all levels, from elementary school students to advanced mathematics enthusiasts. We'll unpack its meaning, explore practical applications, and address common misconceptions.

Introduction: The Significance of Fractions

Fractions represent parts of a whole. They are fundamental to numerous aspects of our lives, from cooking and construction to finance and scientific research. Understanding fractions is crucial for developing a strong foundation in mathematics and its applications in various fields. 75/90, in particular, provides a valuable case study for understanding fraction simplification, percentage calculations, and their interpretations in real-world contexts.

Simplifying the Fraction: Finding the Greatest Common Divisor (GCD)

Before delving into further analysis, it's crucial to simplify the fraction 75/90. Simplification involves reducing the fraction to its lowest terms by dividing both the numerator (75) and the denominator (90) by their greatest common divisor (GCD). The GCD is the largest number that divides both 75 and 90 without leaving a remainder.

To find the GCD, we can use various methods, including prime factorization.

• Prime Factorization:
  • 75 = 3 x 5 x 5 = 3 x 5²
  • 90 = 2 x 3 x 3 x 5 = 2 x 3² x 5

The common factors are 3 and 5. Therefore, the GCD is 3 x 5 = 15.

• Dividing by the GCD:

Dividing both the numerator and the denominator by 15:

75 ÷ 15 = 5 90 ÷ 15 = 6

Therefore, the simplified fraction is 5/6. This simplified form is crucial because it represents the same value as 75/90 but is easier to work with in calculations and comparisons.

Converting to Percentage: Expressing the Fraction as a Proportion

Converting a fraction to a percentage allows us to express the fraction as a proportion out of 100. This is a common and useful way to represent fractions, particularly in contexts like grades, discounts, and statistical data.

To convert 5/6 to a percentage, we can perform the following calculation:

(5/6) x 100% = 83.333...%

This means that 75/90 represents approximately 83.33%. The recurring decimal indicates that the percentage is not exact but a close approximation.

Practical Applications: Real-World Scenarios

The fraction 75/90, and its simplified form 5/6, has numerous practical applications in everyday life. Here are some examples:

• Academic Performance: If a student answered 75 questions correctly out of 90 on a test, their score is 75/90 or 83.33%. This score provides a clear indication of their performance relative to the total number of questions.

• Completion Rates: In project management, if 75 tasks out of 90 planned tasks are completed, the completion rate is 83.33%. This allows for easy monitoring of progress and identification of potential delays.

• Survey Results: If 75 out of 90 respondents answered "yes" to a particular survey question, the percentage of "yes" responses is 83.33%. This provides valuable insights into public opinion or consumer preferences.

• Manufacturing: In a manufacturing process, if 75 out of 90 products pass quality control, the yield rate is 83.33%. This metric is essential for monitoring production efficiency and identifying areas for improvement.

Ratio and Proportion: Understanding the Relationship

75/90 can also be interpreted as a ratio, expressing the relationship between two quantities. The ratio of correct answers to total questions (75:90) simplifies to 5:6, indicating that for every 5 correct answers, there are 6 total questions. This ratio is useful for comparing performances across different tests or datasets. Proportionality is crucial here, as we can use this ratio to predict the number of correct answers if the number of questions changes. For example, if there were 120 questions, we would expect approximately 100 correct answers (5/6 * 120 = 100).

Beyond the Basics: Exploring Statistical Interpretations

In statistics, fractions like 75/90 can represent sample proportions. Imagine a sample of 90 individuals, where 75 exhibit a particular characteristic. The fraction 75/90 (or 5/6) represents the sample proportion exhibiting that characteristic. This sample proportion can be used to estimate the population proportion, but it's important to consider the margin of error and confidence intervals to account for the uncertainty associated with using a sample to infer about a larger population.

Addressing Common Misconceptions

Several common misconceptions surround fractions:

• Confusing Simplification with Changing Value: Simplifying a fraction does not change its value. 75/90 and 5/6 are equivalent and represent the same proportion.

• Improper Simplification: Students might incorrectly simplify a fraction by dividing the numerator and denominator by different numbers, leading to an incorrect result. Always use the GCD for accurate simplification.

• Incorrect Percentage Conversion: Mistakes can occur during the conversion from fraction to percentage, often due to incorrect calculation or misunderstanding of decimal places.

Frequently Asked Questions (FAQ)

• Q: Is 75/90 a proper or improper fraction?

  • A: 75/90 is a proper fraction because the numerator (75) is smaller than the denominator (90).

• Q: What is the simplest form of 75/90?

  • A: The simplest form of 75/90 is 5/6.

• Q: How do I convert 75/90 to a decimal?

  • A: Divide the numerator (75) by the denominator (90). The result is 0.8333...

• Q: Can I use a calculator to simplify fractions?

  • A: Yes, many calculators have functions for simplifying fractions. Alternatively, you can use online fraction calculators.

Conclusion: The Power of Understanding Fractions

The fraction 75/90, seemingly simple, offers a rich learning opportunity. By understanding its simplification, percentage conversion, ratio interpretation, and statistical implications, we gain valuable insights into the broader world of mathematics and its applications. Mastering fractions is fundamental to success in many academic and professional fields, highlighting the importance of developing a strong foundation in this essential mathematical concept. This exploration of 75/90 serves as a valuable stepping stone for further exploration of more complex fractional concepts and their practical uses. Remember, a solid grasp of fractions is not just about numbers; it's about understanding proportions, relationships, and the power of mathematical representation in the world around us.