document.write( "Question 1199873: The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 less than the original number. Find the number.
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Algebra.Com's Answer #833860 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "If a formal algebraic solution is not required, you can use a shortcut to solve the problem.

\n" ); document.write( "When a 2-digit number has its digits reversed and the two numbers are compared, the difference is 9 times the difference between the digits.

\n" ); document.write( "In this problem, the difference between the two numbers is 45, so the difference between the two digits is 5.

\n" ); document.write( "Now we know the sum of the digits is 11 and their difference is 5; quick reasoning and mental arithmetic tells us the two digits are 8 and 3.

\n" ); document.write( "Then, since the original number is greater, it is 83.

\n" ); document.write( "ANSWER: 83

\n" ); document.write( "It is easy to prove algebraically that the difference of the two numbers is 9 times the difference of the digits.

\n" ); document.write( "Let the original number have tens digit a and units digit b; the value of the number is 10a+b.

\n" ); document.write( "The number with the digits reversed has the value 10b+a.

\n" ); document.write( "The difference between the two numbers is

\n" ); document.write( "(10a+b)-(10b+a) = 9a-9b = 9(a-b)

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