Difference between revisions of "1987 AJHSME Problems/Problem 8"

(New page: ==Problem== If <math>\text{A}</math> and <math>\text{B}</math> are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is <cmath>\...)
 
m (Question 8)
 
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9 & 8 & 7 & 6 \\
 
9 & 8 & 7 & 6 \\
 
& A & 3 & 2 \\
 
& A & 3 & 2 \\
& & B & 1 \\ \hline  
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& B & 1 \\ \hline  
 
\end{tabular}</cmath>
 
\end{tabular}</cmath>
  
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The minimum possible value of this sum is when <math>A=B=1</math>, which is <cmath>9876+132+11=10019</cmath>
 
The minimum possible value of this sum is when <math>A=B=1</math>, which is <cmath>9876+132+11=10019</cmath>
  
The largest possible value of the sum is when <math>A=B=9</math>, making the sum <cmath>9876+999+91=10966</cmath>
+
The largest possible value of the sum is when <math>A=B=9</math>, making the sum <cmath>9876+932+91=10899</cmath>
  
Since all the possible sums are between <math>10019</math> and <math>10966</math>, they must have <math>5</math> digits.
+
Since all the possible sums are between <math>10019</math> and <math>10899</math>, they must have <math>5</math> digits.
  
 
<math>\boxed{\text{B}}</math>
 
<math>\boxed{\text{B}}</math>
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==See Also==
 
==See Also==
  
[[1987 AJHSME Problems]]
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{{AJHSME box|year=1987|num-b=7|num-a=9}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 01:47, 20 September 2018

Problem

If $\text{A}$ and $\text{B}$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is

\[\begin{tabular}[t]{cccc} 9 & 8 & 7 & 6 \\ & A & 3 & 2 \\ & B & 1 \\ \hline  \end{tabular}\]


$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{depends on the values of A and B}$

Solution

The minimum possible value of this sum is when $A=B=1$, which is \[9876+132+11=10019\]

The largest possible value of the sum is when $A=B=9$, making the sum \[9876+932+91=10899\]

Since all the possible sums are between $10019$ and $10899$, they must have $5$ digits.

$\boxed{\text{B}}$

See Also

1987 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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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