Difference between revisions of "2016 AMC 10A Problems/Problem 6"
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+ | == Problem == | ||
Ximena lists the whole numbers <math>1</math> through <math>30</math> once. Emilio copies Ximena's numbers, replacing each occurrence of the digit <math>2</math> by the digit <math>1</math>. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's? | Ximena lists the whole numbers <math>1</math> through <math>30</math> once. Emilio copies Ximena's numbers, replacing each occurrence of the digit <math>2</math> by the digit <math>1</math>. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's? | ||
<math>\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110</math> | <math>\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110</math> | ||
+ | |||
+ | == Solution == | ||
+ | |||
+ | For every tens digit 2, we subtract 10, and for every units digit 2, we subtract 1. Because 2 appears 10 times as a tens digit and 2 appears 3 times as a units digit, the answer is <math>10\cdot 10+1\cdot 3=\boxed{\textbf{(D) }103.}</math> | ||
+ | |||
+ | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
+ | https://youtu.be/qrgPsA9AQLA. (short and simple!) | ||
+ | |||
+ | Education, the Study of Education. | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/XXX4_oBHuGk | ||
+ | https://youtu.be/8i_NYG3aRxE | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2016|ab=A|num-b=5|num-a=7}} | ||
+ | {{MAA Notice}} |
Latest revision as of 10:07, 16 June 2025
Contents
Problem
Ximena lists the whole numbers through
once. Emilio copies Ximena's numbers, replacing each occurrence of the digit
by the digit
. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
Solution
For every tens digit 2, we subtract 10, and for every units digit 2, we subtract 1. Because 2 appears 10 times as a tens digit and 2 appears 3 times as a units digit, the answer is
Video Solution (HOW TO THINK CRITICALLY!!!)
https://youtu.be/qrgPsA9AQLA. (short and simple!)
Education, the Study of Education.
Video Solution
https://youtu.be/XXX4_oBHuGk https://youtu.be/8i_NYG3aRxE
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 AMC 10 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.