Verified Answer :

  • Convert the binary numbers \(101_2\), \(110_2\), and \(1000_2\) into their decimal values \(5\), \(6\), and \(8\).
  • Solve the equation \(\frac{x}{x+5} = \frac{6}{8}\) by cross-multiplying to find the decimal value \(x=15\).
  • Convert the decimal \(15\) back to binary to obtain the final result.
  • The binary representation of the solution is \(\boxed{1111}\).

Explanation

  1. Converting Binary to Decimal

The problem involves binary numbers, where the base is 2. The given equation is \(\frac{x}{x + 101_2} = \frac{110_2}{1000_2}\). To solve for \(x\), we first convert these binary numbers into their base-10 (decimal) equivalents.

  • \(101_2 = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5_{10}\).
  • \(110_2 = 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6_{10}\).
  • \(1000_2 = 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 8 + 0 + 0 + 0 = 8_{10}\).

Now, our equation in decimal form is \(\frac{x}{x + 5} = \frac{6}{8}\).

  1. Solving the Algebraic Equation

With the equation \(\frac{x}{x + 5} = \frac{6}{8}\), we can simplify the right side: \(\frac{6}{8} = \frac{3}{4}\).
Now the equation is \(\frac{x}{x + 5} = \frac{3}{4}\).
To solve for \(x\), we cross-multiply:
\(4x = 3(x + 5)\)
\(4x = 3x + 15\)
Subtract \(3x\) from both sides:
\(4x - 3x = 15\)
\(x = 15\)
So, the decimal value of \(x\) is \(15_{10}\).

  1. Converting Result back to Binary

We now convert the decimal value \(15\) back into binary representation.

  • \(15 \div 2 = 7\) with a remainder of \(1\)
  • \(7 \div 2 = 3\) with a remainder of \(1\)
  • \(3 \div 2 = 1\) with a remainder of \(1\)
  • \(1 \div 2 = 0\) with a remainder of \(1\)

Reading the remainders from bottom to top, we get \(1111_2\). Thus, \(15_{10} = 1111_2\). The solution is \(\boxed{1111}\).

Examples

Binary systems are the foundation of computer science and digital communication. Imagine you are working with digital signals and need to balance load distribution across network nodes. Understanding how to handle equations in binary ensures you can correctly configure devices and optimize data flow in binary-logic environments. Being comfortable with these conversions allows for direct manipulation of digital data at the hardware level.

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