Solving 0x 120 + 1/2 X 0.1 X 120^2 A Step-by-Step Guide

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Hey guys! Ever stumbled upon a math problem that looks like it’s written in code? Well, today, we're cracking one of those intriguing puzzles: 0x 120 + 1/2 x 0.1 x 120^2. This isn't just a random string of numbers and symbols; it's a mathematical expression waiting to be solved. In this guide, we'll break down each component, walk through the steps to solve it, and make sure you understand the underlying concepts. Whether you're a student brushing up on your algebra or just a curious mind, you're in the right place. Let's dive in and transform this mathematical mystery into a piece of cake!

Understanding the Components: What Does It All Mean?

Before we jump into solving the equation, let's dissect it piece by piece. Our expression is 0x 120 + 1/2 x 0.1 x 120^2, and it's crucial to understand what each part signifies.

0x 120: The Hexadecimal Connection

The first part, 0x 120, might look a bit unusual if you're not familiar with hexadecimal notation. The 0x prefix is a common indicator that the number following it is in hexadecimal, a base-16 number system. In our everyday math, we use the decimal system (base-10), which uses digits 0-9. Hexadecimal, on the other hand, uses 16 symbols: 0-9 and A-F, where A represents 10, B is 11, and so on up to F, which is 15. So, how do we convert 0x 120 from hexadecimal to decimal? Here’s how:

  • Each position in a hexadecimal number represents a power of 16. From right to left, we have 16^0, 16^1, 16^2, and so on.
  • For 0x 120, the rightmost digit (0) is in the 16^0 place, the middle digit (2) is in the 16^1 place, and the leftmost digit (1) is in the 16^2 place.
  • So, we calculate it as follows: (1 * 16^2) + (2 * 16^1) + (0 * 16^0) = (1 * 256) + (2 * 16) + (0 * 1) = 256 + 32 + 0 = 288.

Therefore, 0x 120 in hexadecimal is equivalent to 288 in decimal. This conversion is crucial because we need to work with decimal numbers to solve the rest of the equation accurately.

1/2 x 0.1 x 120^2: The Arithmetic Part

The second part of our expression, 1/2 x 0.1 x 120^2, involves basic arithmetic operations: fractions, decimals, multiplication, and exponentiation. Let’s break it down:

  • 1/2: This is a simple fraction, representing one-half.
  • 0.1: This is a decimal, representing one-tenth.
  • x: This is the multiplication symbol.
  • 120^2: This means 120 squared, or 120 raised to the power of 2, which is 120 * 120.

Understanding the order of operations (PEMDAS/BODMAS) is crucial here. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This tells us the sequence in which we should perform the operations to get the correct result.

In this part of the expression, we first need to calculate 120^2, then perform the multiplications from left to right. This order ensures we follow the correct mathematical procedure and arrive at the accurate answer. By understanding each component and the operations involved, we set the stage for solving the entire expression step by step.

Step-by-Step Solution: Cracking the Code

Now that we've dissected the components of the expression 0x 120 + 1/2 x 0.1 x 120^2, it's time to put the pieces together and solve it. We'll go through each step meticulously, ensuring that you understand the process and can apply it to similar problems.

Step 1: Convert Hexadecimal to Decimal

As we discussed earlier, 0x 120 is a hexadecimal number. To work with it in our equation, we need to convert it to decimal. We already did this in the previous section, but let's recap:

  • 0x 120 = (1 * 16^2) + (2 * 16^1) + (0 * 16^0)
  • = (1 * 256) + (2 * 16) + (0 * 1)
  • = 256 + 32 + 0
  • = 288

So, 0x 120 is equal to 288 in decimal. This conversion is our starting point for solving the entire expression.

Step 2: Calculate the Exponent

Next, we tackle the exponent in the second part of the expression: 120^2. This means 120 raised to the power of 2, or 120 multiplied by itself.

  • 120^2 = 120 * 120
  • = 14400

So, 120^2 equals 14400. This value will be used in the subsequent multiplication steps.

Step 3: Perform the Multiplications

Now we have the expression 1/2 x 0.1 x 14400. Following the order of operations (PEMDAS/BODMAS), we perform the multiplications from left to right.

  • First, let's multiply 1/2 by 0.1:
    • (1/2) x 0.1 = 0.5 x 0.1
    • = 0.05
  • Next, we multiply the result (0.05) by 14400:
    • 0.05 x 14400 = 720

So, the result of 1/2 x 0.1 x 120^2 is 720. We've now simplified the second part of the expression to a single number.

Step 4: Add the Results

Finally, we add the results from Step 1 and Step 3:

  • 288 + 720 = 1008

Therefore, the solution to the expression 0x 120 + 1/2 x 0.1 x 120^2 is 1008. We've successfully cracked the code by breaking down the problem into manageable steps and following the correct order of operations.

Key Concepts Revisited: PEMDAS/BODMAS and Hexadecimal Conversion

To truly master problems like 0x 120 + 1/2 x 0.1 x 120^2, it’s essential to understand the underlying concepts. Let’s revisit two key principles: the order of operations (PEMDAS/BODMAS) and hexadecimal conversion.

PEMDAS/BODMAS: The Order of Operations

The acronym PEMDAS (or BODMAS in some regions) is a mnemonic device that helps us remember the correct order in which to perform mathematical operations. It stands for:

  • Parentheses (or Brackets)
  • Exponents (or Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This order is crucial because performing operations in the wrong sequence can lead to incorrect results. In our problem, we first handled the exponent (120^2) before multiplication and addition, following PEMDAS. If we had added before multiplying, we would have arrived at a completely different answer.

Understanding and applying PEMDAS ensures that you maintain mathematical consistency and accuracy. It’s like having a roadmap for solving equations – it guides you through the steps in the correct order.

Hexadecimal Conversion: Decoding 0x

Hexadecimal, or base-16, is a number system that uses 16 symbols: 0-9 and A-F. The prefix 0x is a common notation to indicate that a number is in hexadecimal. Converting hexadecimal numbers to decimal (base-10) is essential when the equation involves decimal operations.

To convert a hexadecimal number to decimal, you multiply each digit by 16 raised to the power of its position (starting from 0 on the right) and then add the results. For example, with 0x 120:

  • The rightmost digit (0) is in the 16^0 place.
  • The middle digit (2) is in the 16^1 place.
  • The leftmost digit (1) is in the 16^2 place.

Thus, we calculate it as (1 * 16^2) + (2 * 16^1) + (0 * 16^0), which equals 288. This conversion allows us to integrate the hexadecimal number into the decimal-based arithmetic of the equation.

Mastering hexadecimal conversion not only helps in solving mathematical problems but also has practical applications in computer science, where hexadecimal is commonly used to represent memory addresses and color codes. By understanding these key concepts, you’re better equipped to tackle a wide range of mathematical and computational challenges.

Real-World Applications: Why This Matters

You might be wondering,