1. (a) If [tex]$123_{2 x-2}=313_x$[/tex], find the value of [tex]$x$[/tex].
(b) Express [tex]$2.51 \overline{6}$[/tex] as a mixed fraction.
Answer (1)
Solve for x by converting 12 3 2 x − 2 and 31 3 x to base 10, setting them equal, and solving the resulting quadratic equation: x 2 − 5 x = 0 , which gives x = 5 .
Express 2.51 6 as a mixed fraction by setting y = 2.51 6 , then calculating 100 y = 251. 6 and 1000 y = 2516. 6 .
Subtract 100 y from 1000 y to get 900 y = 2265 , so y = 900 2265 = 60 151 .
Convert the improper fraction 60 151 to a mixed fraction: 2 60 31 .
Explanation
Problem Analysis We are given two problems: (a) Find the value of x if 12 3 2 x − 2 = 31 3 x .
(b) Express 2.51 6 as a mixed fraction.
Converting to Base 10 (a) We need to convert both numbers to base 10. The number 12 3 2 x − 2 in base 10 is 1 ⋅ ( 2 x − 2 ) 2 + 2 ⋅ ( 2 x − 2 ) 1 + 3 ⋅ ( 2 x − 2 ) 0 . The number 31 3 x in base 10 is 3 ⋅ x 2 + 1 ⋅ x 1 + 3 ⋅ x 0 . Setting these equal to each other, we get:
( 2 x − 2 ) 2 + 2 ( 2 x − 2 ) + 3 = 3 x 2 + x + 3
Solving for x Expanding and simplifying the equation:
4 x 2 − 8 x + 4 + 4 x − 4 + 3 = 3 x 2 + x + 3
4 x 2 − 4 x + 3 = 3 x 2 + x + 3 x 2 − 5 x = 0 x ( x − 5 ) = 0
So, x = 0 or x = 5 . Since the base must be greater than the largest digit, we have 3"> 2 x − 2 > 3 and 3"> x > 3 . Thus, x = 5 is the only valid solution.
Expressing as a Mixed Fraction (b) Let y = 2.51 6 . To express this as a mixed fraction, we can write:
100 y = 251. 6 1000 y = 2516. 6
Subtracting the two equations:
1000 y − 100 y = 2516. 6 − 251. 6 900 y = 2265 y = 900 2265 = 180 453 = 60 151
Now, we express 60 151 as a mixed fraction: 60 151 = 2 60 31 .
Final Answer Therefore, the value of x is 5 and 2.51 6 as a mixed fraction is 2 60 31 .
Examples
Understanding different number bases is crucial in computer science, where binary (base 2), octal (base 8), and hexadecimal (base 16) systems are frequently used. Converting between these bases and base 10 is a fundamental skill for programmers. Similarly, expressing repeating decimals as fractions is important in financial calculations, where accuracy is paramount. For example, calculating the exact interest on a loan or the precise amount of a recurring payment often involves converting repeating decimals to fractions to avoid rounding errors. These skills ensure precision and reliability in both technical and financial applications.
