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

3/4 of a Number is 27: What's the Number? A Deep Dive into Solving Algebraic Equations

This seemingly simple question, "3/4 of a number is 27, what's the number?", opens a door to a fascinating world of algebra and problem-solving. While the answer might seem immediately apparent to some, understanding the underlying principles and methods involved is crucial for tackling more complex mathematical challenges. This article will not only provide the solution but will also explore various approaches to solving this type of problem, highlighting the importance of algebraic thinking and its applications in everyday life.

Understanding the Problem: Translating Words into Equations

The key to solving this problem lies in translating the word problem into a mathematical equation. The phrase "3/4 of a number" can be represented algebraically. Let's use 'x' to represent the unknown number. Therefore, "3/4 of a number" translates to (3/4)x or 0.75x.

The problem states that this fraction of the number is 27. The word "is" signifies equality. Thus, our complete equation becomes:

(3/4)x = 27

This simple equation forms the foundation for solving the problem. Now let's explore different methods to find the value of 'x'.

Method 1: Solving Using Multiplication

The most straightforward approach involves isolating 'x' by performing the inverse operation. Since 'x' is being multiplied by 3/4, we can undo this multiplication by multiplying both sides of the equation by the reciprocal of 3/4, which is 4/3.

(3/4)x = 27

Multiply both sides by 4/3:

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

The fractions on the left-hand side cancel each other out:

x = 27 * (4/3)

Simplifying the right-hand side:

x = (27 * 4) / 3

x = 108 / 3

x = 36

Therefore, the number is 36.

Method 2: Solving Using Decimal Representation

Alternatively, we can convert the fraction 3/4 into its decimal equivalent, 0.75. The equation then becomes:

0.75x = 27

To solve for x, we divide both sides of the equation by 0.75:

x = 27 / 0.75

Performing the division:

x = 36

Again, we arrive at the same solution: the number is 36.

Method 3: Understanding Proportions

This problem can also be approached using the concept of proportions. We can set up a proportion:

3/4 = 27/x

This states that the ratio of 3 to 4 is equal to the ratio of 27 to the unknown number (x). To solve for x, we can cross-multiply:

3x = 4 * 27

3x = 108

Dividing both sides by 3:

x = 108 / 3

x = 36

Verification: Checking Our Answer

It's crucial to verify our solution. Let's substitute x = 36 back into the original equation:

(3/4) * 36 = 27

Simplifying:

(3 * 36) / 4 = 27

108 / 4 = 27

27 = 27

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

Expanding the Concept: Applying Algebraic Problem-Solving

The seemingly simple problem of "3/4 of a number is 27" provides a fundamental illustration of algebraic problem-solving. This approach isn't limited to simple fraction problems. The same principles can be applied to a wide range of real-world scenarios, including:

• Calculating discounts and sales tax: Determining the original price of an item after a discount or calculating the final price after adding sales tax.
• Financial calculations: Figuring out interest earned on savings, loan repayments, or investment growth.
• Measurement conversions: Converting between different units of measurement (e.g., kilometers to miles, liters to gallons).
• Scientific applications: Solving equations in physics, chemistry, and other scientific fields.
• Engineering problems: Calculating dimensions, forces, and other parameters in engineering design.

Beyond the Basics: More Complex Fraction Problems

While this problem focuses on a straightforward fraction, the core principles extend to more complex scenarios involving multiple fractions, decimals, and even variables. For instance, consider a problem like:

(2/3)x + (1/4)x = 17

This equation requires combining like terms and then applying similar algebraic manipulation to isolate 'x' and find its value. The techniques discussed above provide a solid foundation for tackling such more complex problems.

Conclusion: Mastering Algebra for Real-World Success

The question, "3/4 of a number is 27, what's the number?" may appear simple at first glance, but its solution reveals the power and versatility of algebra. By understanding the process of translating word problems into equations and employing appropriate algebraic techniques, we can solve a wide variety of mathematical problems with confidence. This fundamental skill extends far beyond the classroom, proving invaluable in various professional and personal aspects of life. The ability to translate real-world situations into mathematical models and solve for unknowns is a critical skillset for success in many fields. Mastering these basic algebraic principles opens doors to understanding and solving far more complex problems in the future. Therefore, taking the time to thoroughly understand this seemingly simple problem lays a crucial foundation for more advanced mathematical endeavors.