What Is 35 Of 75

What is 35 of 75? Understanding Percentages, Fractions, and Ratios

This article explores the question "What is 35 of 75?" in depth. We'll delve beyond a simple numerical answer, examining the underlying mathematical concepts of percentages, fractions, and ratios, and illustrating how these concepts apply in various real-world scenarios. This comprehensive guide is designed for learners of all levels, from those needing a basic refresher to those seeking a deeper understanding of proportional relationships.

Understanding the Problem: Percentages, Fractions, and Ratios

The question "What is 35 of 75?" essentially asks us to determine what percentage 35 represents of the total value 75. This involves understanding the interconnectedness of percentages, fractions, and ratios.

Percentage: A percentage represents a fraction of 100. For example, 50% means 50 out of 100, or 50/100.
Fraction: A fraction represents a part of a whole. It is expressed as a numerator (the top number) over a denominator (the bottom number). For instance, 1/2 represents one part out of two equal parts.
Ratio: A ratio compares two quantities. It can be expressed as a fraction or using a colon (:). For example, the ratio of apples to oranges could be 2:3 or 2/3.

In our problem, we need to find the percentage, fraction, and ratio that 35 represents when compared to 75. These three concepts are fundamentally linked and offer different ways to express the same relationship.

Calculating the Percentage

The most straightforward way to answer "What is 35 of 75?" is to calculate the percentage. We can do this using the following formula:

(Part / Whole) * 100% = Percentage

In our case:

(35 / 75) * 100% = 46.67%

Therefore, 35 is 46.67% of 75. This means that 35 represents approximately 46.67 out of every 100 parts of 75.

Expressing the Relationship as a Fraction

We can also express the relationship between 35 and 75 as a fraction:

35/75

This fraction can be simplified by finding the greatest common divisor (GCD) of 35 and 75, which is 5. Dividing both the numerator and denominator by 5, we get:

7/15

Therefore, 35 is 7/15 of 75. This simplified fraction represents the same proportional relationship as the percentage and decimal value.

Understanding the Ratio

The ratio of 35 to 75 can be expressed as:

35:75

Like the fraction, this ratio can be simplified by dividing both numbers by their GCD (5):

7:15

This simplified ratio indicates that for every 7 parts of 35, there are 15 parts in the total of 75.

Real-World Applications

Understanding percentages, fractions, and ratios is crucial in various real-world scenarios. Consider these examples:

Sales and Discounts: If a store offers a 20% discount on an item priced at $75, you would calculate the discount as (20/100) * $75 = $15. The final price would be $75 - $15 = $60.
Test Scores: If you answered 35 questions correctly out of a total of 75, your score would be 46.67%. This helps you understand your performance relative to the total number of questions.
Survey Results: If a survey of 75 people shows that 35 prefer a certain product, you can express this as 46.67% or 7/15, showing the proportion of people who favor that product.
Recipe Scaling: If a recipe calls for 7 parts of ingredient A and 15 parts of ingredient B, and you want to make a larger batch, you can scale the recipe proportionally using the ratio 7:15.
Financial Analysis: Ratios are fundamental in financial analysis. For example, the current ratio compares current assets to current liabilities, providing insights into a company's liquidity.

Beyond the Basics: Working with Proportions

The core concept underlying this problem is proportion. A proportion states that two ratios are equal. We can use proportions to solve various problems involving percentages, fractions, and ratios.

For example, let's say we want to find out what number is 46.67% of 100. We can set up a proportion:

35/75 = x/100

To solve for x, we cross-multiply:

35 * 100 = 75 * x

3500 = 75x

x = 3500/75 = 46.67

This confirms that 46.67 is 46.67% of 100. This illustrates the power of proportions in solving problems related to percentages, fractions, and ratios.

Further Exploration: Decimal Representation

The result of 35/75 can also be expressed as a decimal. Dividing 35 by 75 gives:

0.466666...

This is a recurring decimal, meaning the digit 6 repeats infinitely. For practical purposes, it's usually rounded to a certain number of decimal places, such as 0.4667, which is equivalent to 46.67%.

Frequently Asked Questions (FAQ)

Q: What is the simplest form of the fraction 35/75?

A: The simplest form is 7/15, obtained by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

Q: Can I use a calculator to solve this problem?

A: Yes, a calculator can be used to divide 35 by 75 and then multiply the result by 100 to find the percentage.

Q: Why are percentages, fractions, and ratios important?

A: These concepts are essential for understanding and working with proportions in various fields, including mathematics, science, finance, and everyday life.

Q: How can I improve my understanding of these concepts?

A: Practice solving various problems involving percentages, fractions, and ratios. You can find many practice exercises online or in textbooks.

Conclusion

In conclusion, 35 is 46.67% of 75. This seemingly simple question unlocks a deeper understanding of the interconnectedness of percentages, fractions, and ratios – fundamental mathematical concepts with wide-ranging real-world applications. Mastering these concepts is crucial for success in various fields and for navigating everyday situations requiring proportional reasoning. By understanding the underlying principles and practicing their application, you can confidently tackle similar problems and develop a strong foundation in mathematical literacy. Remember to utilize the formulas, techniques, and examples provided in this article to solidify your understanding and enhance your problem-solving skills.