What Is 15 Of 90

What is 15 of 90? Unpacking Fractions, Percentages, and Ratios

This article delves into the seemingly simple question, "What is 15 of 90?" While the immediate answer might appear straightforward, exploring this question allows us to unpack fundamental mathematical concepts like fractions, percentages, and ratios, highlighting their interconnectedness and practical applications. Understanding these concepts is crucial for various aspects of life, from everyday budgeting to complex scientific calculations. This guide will provide a comprehensive explanation, accessible to all levels of mathematical understanding.

Understanding the Core Concepts

Before diving into the solution, let's clarify the three essential mathematical ideas: fractions, percentages, and ratios.

1. Fractions: A fraction represents a part of a whole. It's expressed as a ratio of two numbers, the numerator (top number) and the denominator (bottom number). The numerator indicates the number of parts you have, while the denominator shows the total number of parts in the whole. In our example, "15 of 90" can be represented as the fraction 15/90.

2. Percentages: A percentage is a way of expressing a fraction as a proportion of 100. It indicates how many parts out of 100 represent a given quantity. The symbol "%" represents "per cent" or "out of 100". To convert a fraction to a percentage, you divide the numerator by the denominator and multiply the result by 100.

3. Ratios: A ratio compares two or more quantities. It shows the relative sizes of the quantities. Like fractions, ratios can be simplified by dividing both parts by their greatest common divisor (GCD). The ratio of 15 to 90 can be expressed as 15:90.

Solving "What is 15 of 90?"

Now, let's address the primary question. "What is 15 of 90?" essentially asks us to determine what proportion 15 represents of the total 90. We can solve this using fractions, percentages, or ratios.

Method 1: Using Fractions

The simplest approach is to express the problem as a fraction: 15/90. This fraction represents the portion of 90 that is 15. To simplify this fraction, we find the greatest common divisor (GCD) of 15 and 90. The GCD of 15 and 90 is 15. Dividing both the numerator and the denominator by 15, we get:

15/90 = (15 ÷ 15) / (90 ÷ 15) = 1/6

Therefore, 15 is one-sixth (1/6) of 90.

Method 2: Using Percentages

To express 15 as a percentage of 90, we use the formula:

(Part / Whole) x 100%

Substituting our values:

(15 / 90) x 100% = 0.1667 x 100% = 16.67%

Therefore, 15 is 16.67% of 90. Note that this is an approximation; the exact percentage is a recurring decimal.

Method 3: Using Ratios

The ratio of 15 to 90 is written as 15:90. To simplify this ratio, we find the GCD of 15 and 90, which is 15. Dividing both sides by 15, we get:

15:90 = 1:6

This simplified ratio shows that the relationship between 15 and 90 is the same as the relationship between 1 and 6. For every 1 part, there are 6 parts in total.

Practical Applications and Real-World Examples

Understanding fractions, percentages, and ratios is essential for navigating numerous real-world scenarios. Here are some examples:

• Shopping: Calculating discounts, comparing prices, and determining the best deals often involves using percentages. For instance, a 20% discount on a $90 item would be (20/100) x $90 = $18.

• Cooking: Recipes often require scaling up or down. Understanding ratios allows you to adjust ingredient quantities proportionally. If a recipe calls for a 1:2 ratio of sugar to flour, you can maintain the same flavor profile by increasing both quantities while preserving the ratio.

• Finance: Calculating interest rates, understanding loan repayments, and budgeting personal finances heavily rely on percentages and fractions. For instance, calculating the interest earned on a savings account requires applying a percentage to the principal amount.

• Data Analysis: In data analysis and statistics, percentages and ratios are frequently used to represent data proportions, trends, and comparisons. For instance, if a survey shows that 15 out of 90 respondents prefer a specific product, the percentage representation is useful to show this preference concisely.

• Science and Engineering: Many scientific and engineering calculations require the manipulation of fractions, ratios, and percentages for precise measurements and calculations. Examples include calculating concentrations of solutions, efficiency of machines, or analyzing experimental results.

Further Exploration: Working with Larger Numbers and Decimals

The principles demonstrated above can be applied to any numbers, including larger numbers and decimals. Let's consider an example using decimals:

What is 15.5 of 90.7?

We can use the same approach:

1. Fraction: 15.5 / 90.7 ≈ 0.1707

2. Percentage: (15.5 / 90.7) x 100% ≈ 17.07%

3. Ratio: 15.5 : 90.7 (This ratio cannot be simplified further using integers).

Notice that even with decimals, the principles remain the same. We express the part as a fraction or ratio of the whole and then convert it to a percentage if needed. The only difference is that calculations might involve decimal arithmetic.

Frequently Asked Questions (FAQ)

Q1: What is the simplest form of the fraction 15/90?

A1: The simplest form is 1/6. We achieved this by dividing both the numerator and the denominator by their greatest common divisor, which is 15.

Q2: How do I convert a fraction to a percentage?

A2: To convert a fraction to a percentage, divide the numerator by the denominator and multiply the result by 100%. For example, 1/6 = (1 ÷ 6) x 100% ≈ 16.67%

Q3: Can I use a calculator to solve these problems?

A3: Absolutely! Calculators simplify calculations, especially with larger numbers or decimals. However, understanding the underlying principles is crucial, even when using a calculator.

Q4: Are there other ways to represent "15 of 90"?

A4: Yes, you could also use a decimal (0.1667), a ratio (1:6), or describe it verbally as "one-sixth" or "16.67 percent."

Conclusion: Mastering Fundamental Mathematical Concepts

Understanding the relationship between 15 and 90 through the lens of fractions, percentages, and ratios reinforces the fundamental mathematical concepts crucial for various aspects of daily life and professional endeavors. This seemingly simple question opens up a world of practical applications, demonstrating the power of basic math in solving everyday problems and navigating complex situations. By mastering these concepts, you equip yourself with valuable tools for critical thinking and problem-solving. The more you practice, the more intuitive and effortless these calculations will become, enhancing your numerical literacy and confidence in tackling mathematical challenges. Remember, the key lies not just in finding the answer but in comprehending the underlying mathematical logic.