If $101_{\text {two }}+12 y=23_{\text {five}}$, find the value of $y$.
Answer (2)
The value of y in the equation 10 1 two + 12 y = 2 3 five is 3 2 , after converting both numbers to base 10 and solving for y .
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Convert 10 1 two to base 10: 10 1 two = 5 .
Convert 2 3 five to base 10: 2 3 five = 13 .
Substitute the base 10 values into the equation: 5 + 12 y = 13 .
Solve the equation for y : y = 12 8 = 3 2 .
3 2
Explanation
Understanding the problem We are given the equation 10 1 two + 12 y = 2 3 five and we need to find the value of y . First, we need to convert the numbers in base 2 and base 5 to base 10 so we can solve for y .
Converting from base 2 to base 10 Converting 10 1 two to base 10: 10 1 two = 1 × 2 2 + 0 × 2 1 + 1 × 2 0 = 4 + 0 + 1 = 5 .
Converting from base 5 to base 10 Converting 2 3 five to base 10: 2 3 five = 2 × 5 1 + 3 × 5 0 = 10 + 3 = 13 .
Substituting the values Now we substitute these values into the equation: 5 + 12 y = 13 .
Isolating the term with y Subtracting 5 from both sides, we get: 12 y = 13 − 5 12 y = 8 .
Solving for y Dividing both sides by 12, we get: y = 12 8 = 3 2 .
Final Answer Therefore, the value of y is 3 2 .
Examples
Base conversions are used in computer science to represent numbers in different formats. For example, converting binary numbers (base 2) to decimal numbers (base 10) is essential for displaying data in a human-readable format. Similarly, converting between different bases is used in cryptography and data compression algorithms. Understanding base conversions helps in designing efficient and secure systems.
