3/4 Of A Number Is 27 What's The Number

3/4 of a Number is 27: What's the Number? A Comprehensive Guide to Solving This Math Problem and Similar Equations

This seemingly simple math problem, "3/4 of a number is 27, what's the number?", serves as an excellent gateway to understanding fundamental algebraic concepts. While the solution might seem straightforward, delving deeper reveals valuable insights into problem-solving techniques and their applications in various fields. This article provides a detailed explanation of how to solve this specific problem, explores different methods for solving similar equations, and offers practical applications to solidify your understanding.

Understanding the Problem

The core of the problem lies in translating the word problem into a mathematical equation. "3/4 of a number" can be represented algebraically as (3/4)x, where 'x' represents the unknown number. The phrase "is 27" translates to an equals sign (=). Therefore, the complete equation becomes:

(3/4)x = 27

This simple equation forms the foundation for our solution. Let's explore several methods to solve for 'x'.

Method 1: Solving Using Multiplication

This is the most direct and intuitive method. To isolate 'x', we need to eliminate the fraction (3/4) from the left side of the equation. We can achieve this by multiplying both sides of the equation by the reciprocal of (3/4), which is (4/3):

(4/3) * (3/4)x = 27 * (4/3)

The fractions on the left side cancel each other out, leaving:

x = 27 * (4/3)

Simplifying the right side:

x = (27 * 4) / 3

x = 108 / 3

x = 36

Therefore, the number is 36.

Method 2: Solving Using Division

An alternative approach involves first dividing both sides of the original equation by (3/4). While this might seem less intuitive than multiplication by the reciprocal, it highlights the relationship between fractions and division. Remember that dividing by a fraction is equivalent to multiplying by its reciprocal.

(3/4)x = 27

Dividing both sides by (3/4):

x = 27 / (3/4)

This is equivalent to:

x = 27 * (4/3)

Following the same steps as in Method 1, we arrive at the same solution:

x = 36

Method 3: Solving Using Decimals

We can also solve this equation by converting the fraction (3/4) into its decimal equivalent, which is 0.75. The equation then becomes:

0.75x = 27

To isolate 'x', we divide both sides by 0.75:

x = 27 / 0.75

Using a calculator or performing long division, we get:

x = 36

This method offers a different perspective and demonstrates the interchangeability between fractions and decimals in solving algebraic equations.

Verifying the Solution

It's crucial to verify our solution by substituting the value of 'x' (which we found to be 36) back into the original equation:

(3/4) * 36 = 27

Simplifying the left side:

(3 * 36) / 4 = 27

108 / 4 = 27

27 = 27

The equation holds true, confirming that our solution, x = 36, is correct.

Expanding on the Concept: Solving Similar Equations

The principles applied to solving "(3/4)x = 27" can be generalized to solve a wide range of similar equations. Consider the following examples:

• (2/5)x = 10: Following the same methods, multiply both sides by (5/2) to find x = 25.
• (7/8)x = 49: Multiply both sides by (8/7) to find x = 56.
• 0.6x = 18: Divide both sides by 0.6 to find x = 30.
• (a/b)x = c: The general solution is x = (b/a)c

These examples illustrate that the core principle remains consistent: isolate the variable 'x' by performing the inverse operation on both sides of the equation.

Real-World Applications

This type of equation is surprisingly versatile and finds practical applications in various fields:

• Percentage Calculations: Many percentage problems can be framed as fraction equations. For example, "30% of a number is 15" can be written as 0.3x = 15, easily solvable using the methods described above.
• Scaling and Proportion: In engineering, architecture, and cooking, scaling recipes or blueprints involves solving similar proportional equations.
• Financial Calculations: Interest calculations, discounts, and profit margins often involve solving fractional equations to determine unknown values.
• Data Analysis: When dealing with fractions or percentages in datasets, solving equations similar to this helps in extracting meaningful insights.

Advanced Concepts and Further Learning

For those looking to delve deeper, understanding the following concepts will enhance your problem-solving skills:

• Linear Equations: This problem falls under the category of linear equations, a fundamental concept in algebra. Learning more about linear equations will open doors to solving more complex problems.
• Simultaneous Equations: Solving systems of equations involving multiple variables and equations builds upon the foundation established here.
• Inequalities: Expanding your knowledge to include inequalities will enable you to solve problems involving ranges of values rather than single solutions.

Conclusion

Solving the equation "(3/4)x = 27" might seem trivial at first, but it lays the groundwork for a comprehensive understanding of algebra and its practical applications. Mastering the techniques outlined in this article, and further exploring related concepts, will significantly enhance your mathematical abilities and problem-solving skills across various disciplines. The ability to translate word problems into mathematical equations and solve them efficiently is a valuable skill applicable throughout life, both personally and professionally. Remember to practice regularly and explore different methods to solidify your understanding and build confidence.