Difference between revisions of "2010 AMC 8 Problems/Problem 22"
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The result must hold for any three-digit number with hundreds digit being <math>2</math> more than the units digit. <math>301</math> is such a number. Evaluating, we get <math>301-103=198</math>. Thus, the units digit in the final result is <math>\boxed{\textbf{(E)}\ 8}</math> | The result must hold for any three-digit number with hundreds digit being <math>2</math> more than the units digit. <math>301</math> is such a number. Evaluating, we get <math>301-103=198</math>. Thus, the units digit in the final result is <math>\boxed{\textbf{(E)}\ 8}</math> | ||
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+ | ==Solution 3== | ||
+ | Set the units digit of the original number as <math> x </math>. Thus, its hundreds digit is <math> x+2 </math>. After the digits are reversed, the hundreds digit of the original number will be the units digit of the new number. Since <math> x-(x+2) = -2 </math>, we can do regrouping and "borrow" <math> 1 </math> from the tens digit and bring it to the units digit as a <math> 10 </math>. Therefore, the units digit will end up as <math> -2 + 10 = \boxed{\textbf{(E)}\ 8}</math> | ||
+ | |||
+ | ~[[User:Bloggish|Bloggish]] | ||
==Video by MathTalks== | ==Video by MathTalks== |
Revision as of 05:19, 27 January 2023
Problem
The hundreds digit of a three-digit number is more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
Solution 1
Let the hundreds, tens, and units digits of the original three-digit number be , , and , respectively. We are given that . The original three-digit number is equal to . The hundreds, tens, and units digits of the reversed three-digit number are , , and , respectively. This number is equal to . Subtracting this expression from the expression for the original number, we get . Thus, the units digit in the final result is
Solution 2
The result must hold for any three-digit number with hundreds digit being more than the units digit. is such a number. Evaluating, we get . Thus, the units digit in the final result is
Solution 3
Set the units digit of the original number as . Thus, its hundreds digit is . After the digits are reversed, the hundreds digit of the original number will be the units digit of the new number. Since , we can do regrouping and "borrow" from the tens digit and bring it to the units digit as a . Therefore, the units digit will end up as
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See Also
2010 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.