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Difference between revisions of "1993 AJHSME Problems/Problem 22"

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The number of two-digit numbers with a two in the tens place is <math>10</math> and the number with a two in the ones place is <math>9</math>. Thus the digit two is used <math>10+9=19</math> times for the two digit numbers.
 
The number of two-digit numbers with a two in the tens place is <math>10</math> and the number with a two in the ones place is <math>9</math>. Thus the digit two is used <math>10+9=19</math> times for the two digit numbers.
  
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Now, Pat Peano only has <math>22-1 (because "22" was counted twice) -19=2</math> remaining twos. The last numbers with a two that he can write are <math>102</math> and <math>112</math>. He can continue numbering the last couple pages without a two until <math>120</math>, with the last number he writes being <math>\boxed{\text{(D)}\ 119}</math>.
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Now, Pat Peano only has <math>22-1-19=2</math> remaining twos. You must subtract 1 because 22 is counted twice. The last numbers with a two that he can write are <math>102</math> and <math>112</math>. He can continue numbering the last couple pages without a two until <math>120</math>, with the last number he writes being <math>\boxed{\text{(D)}\ 119}</math>.
  
 
==See Also==
 
==See Also==
 
{{AJHSME box|year=1993|num-b=21|num-a=23}}
 
{{AJHSME box|year=1993|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:49, 9 December 2020

Problem

Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?

$\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199$

Solution

There is $1$ two in the one-digit numbers.

The number of two-digit numbers with a two in the tens place is $10$ and the number with a two in the ones place is $9$. Thus the digit two is used $10+9=19$ times for the two digit numbers.

Now, Pat Peano only has $22-1-19=2$ remaining twos. You must subtract 1 because 22 is counted twice. The last numbers with a two that he can write are $102$ and $112$. He can continue numbering the last couple pages without a two until $120$, with the last number he writes being $\boxed{\text{(D)}\ 119}$.

See Also

1993 AJHSME (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

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