Convert \( 5.111 \ldots \) into a ratio of two integers. Express your answer as a fraction in lowest terms.

Let \( x = 5.111 \ldots \). This number can be expressed as: \[ x = 5.111111\ldots \] To eliminate the repeating decimal, we can multiply \( x \) by 10: \[ 10x = 51.111111\ldots \] Now, we have two equations: 1. \( x = 5.111111\ldots \) 2. \( 10x = 51.111111\ldots \) Next, we subtract the first equation from the second: \[ 10x - x = 51.111111\ldots - 5.111111\ldots \] This simplifies to: \[ 9x = 51.111111\ldots - 5.111111\ldots \] Calculating the right side: \[ 51.111111\ldots - 5.111111\ldots = 51 - 5 = 46 \] Thus, we have: \[ 9x = 46 \] Now, we solve for \( x \): \[ x = \frac{46}{9} \] Next, we need to check if this fraction is in lowest terms. The numerator is 46 and the denominator is 9. The prime factorization of 46 is \( 2 \times 23 \), and the prime factorization of 9 is \( 3^2 \). Since there are no common factors between 46 and 9, the fraction is already in lowest terms. Therefore, the final answer is: \[ \boxed{\frac{46}{9}} \]