Pregunta
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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.

Ask by Colon Harmon. in South Africa
Mar 12,2025

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The original number is 63 and the new number is 36.

Solución

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: \( 36 \)

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Beyond the Answer

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 \( 36 \).

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