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.
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Let the unit digit be \( x \). Then, the tons (tens) digit, being twice the unit digit, is \( 2x \). The original two-digit number can be expressed as \( 10(2x) + x = 20x + x = 21x \). When the digits are interchanged, the new number becomes \( 10x + 2x = 12x \). According to the problem, the difference between the original number and the new number is 27: \[ 21x - 12x = 27 \] \[ 9x = 27 \] \[ x = 3 \] Now we can find the digits: - Unit digit: \( x = 3 \) - Tens digit: \( 2x = 6 \) So, the original number is \( 63 \) and the new number after interchanging the digits is \( 36 \). To summarize, the original number is \( 63 \) and the new number is \( 36 \).
