In a two-digit number the tons digit is twite the unit digit. If the digits are interchanged the difference between the original number and new number is twenty-seven. Determine the original and new number.

Let's denote the two-digit number as \( 10t + u \), where \( t \) is the tens digit and \( u \) is the units digit. According to the problem, we have the following conditions: 1. The tens digit \( t \) is twice the units digit \( u \): \[ t = 2u \] 2. If the digits are interchanged, the new number becomes \( 10u + t \). The difference between the original number and the new number is 27: \[ (10t + u) - (10u + t) = 27 \] Now, let's simplify the second equation: \[ 10t + u - 10u - t = 27 \] \[ 9t - 9u = 27 \] Dividing the entire equation by 9 gives: \[ t - u = 3 \] Now we have a system of equations: 1. \( t = 2u \) 2. \( t - u = 3 \) We can substitute the first equation into the second equation. Let's solve this system step by step. Substituting \( t = 2u \) into \( t - u = 3 \): \[ 2u - u = 3 \] \[ u = 3 \] Now, substituting \( u = 3 \) back into the first equation to find \( t \): \[ t = 2u = 2 \times 3 = 6 \] Now we have both digits: - Tens digit \( t = 6 \) - Units digit \( u = 3 \) Thus, the original number is: \[ 10t + u = 10 \times 6 + 3 = 60 + 3 = 63 \] The new number, after interchanging the digits, is: \[ 10u + t = 10 \times 3 + 6 = 30 + 6 = 36 \] Finally, we can summarize the results: - The original number is \( 63 \). - The new number is \( 36 \). Let's verify the difference: \[ 63 - 36 = 27 \] This confirms that our solution is correct. **Final Answer:** - Original number: \( 63 \) - New number: \( 36 \)