SOLUTION: 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.

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.

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52786) (Show Source): You can put this solution on YOUR website!
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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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Let "ab" be the decimal presentation of the number, so "b" is the "units" digit and "a" is the "tens" digit. Then the value of the original number is (10a+b), while the value of the reversed digit number is (10b+a). From the problem, a + b = 11, (1) 10a + b = 10b + a + 45. (2) Simplify equation (2) step by step 10a + b - 10b - a = 45 9a - 9b = 45 9(a-b) = 45 a - b = 45. Thus we have this system of equations a + b = 11, (3) a - b = 5. (4) To solve the system, add equations (3) and (4) 2a = 11 + 5 = 16, a = 16/2 = 8. Then from (3) b = 11 -a = 11 - 8 = 3. ANSWER. The number is 83. CHECK. The sum of the digits is 8 + 3 = 11; the difference of the numbers is 83 - 38 = 45. ! correct !

Solved.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

If a formal algebraic solution is not required, you can use a shortcut to solve the problem.

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.

In this problem, the difference between the two numbers is 45, so the difference between the two digits is 5.

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.

Then, since the original number is greater, it is 83.

ANSWER: 83

It is easy to prove algebraically that the difference of the two numbers is 9 times the difference of the digits.

Let the original number have tens digit a and units digit b; the value of the number is 10a+b.

The number with the digits reversed has the value 10b+a.

The difference between the two numbers is

(10a+b)-(10b+a) = 9a-9b = 9(a-b)

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