Digit Exchange

A three-digit number increases by nine if you exchange the second and third digits. The same three-digit number increases by 90 if you exchange the first and second digits. By how much will the number increase if you exchange the first and third digits of the same number?
Source: mathcontest.olemiss.edu 11/14/2011

SOLUTION
Exchange second and third digits
Suppose the three-digit number is a49. If we exchange the second and third digits, the resulting number is a94. The increase a94-a49=45 is much greater than 9.
So we try a smaller number for example a45 and exchange the second and third digits to obtain a54. The increase equals a54-a45=9 which is the correct amount. But, this is true also of a23, a34, etc.

Exchange first and second digits
From what we learned above the first digit should be 1 less than the second digit if we want the increase to equal 90. For example, if we start with the three-digit number 123 and exchange the first and second digits to obtain 213, the increase equals 213-123=90 as expected. But, this is true of the many more numbers as shown below.

123 213 213 – 123 = 90
234 324 324 – 234 = 90
345 435 435 – 345 = 90
456 546 546 – 456 = 90
567 657 657 – 567 = 90
678 768 768 – 678 = 90
789 879 879 – 789 = 90

Exchange first and third digits
If we exchange the first and third digits, the increase equals 198.

123 321 321 – 123 = 198
234 432 432 – 234 = 198
345 543 543 – 345 = 198
456 654 654 – 456 = 198
567 765 765 – 567 = 198
678 876 876 – 678 = 198
789 987 987 – 789 = 198

Answer: 198.

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About mvtrinh

Retired high school math teacher.
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