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Difference between revisions of "2014 AMC 12B Problems/Problem 8"

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It is intuitively obvious, even to the most casual observer that the problem statement can be rewritten as:
 
It is intuitively obvious, even to the most casual observer that the problem statement can be rewritten as:
  
−
<math>\overset{4}{10}A + \overset{4}{10}B + \overset{3}{10}B + \overset{3}{10}C + \overset{2}{10}B + \overset{2}{10}A + 10C + 10D + B + A = \overset{4}{10}D + \overset{3}{10}B + \overset{2}{10}D + 10D + D</math>. This equation can be simplified into:
+
<math>10^4A + \10^4B + 10^3B + 10^3C + 10^2B + 10^2A + 10C + 10D + B + A = 10^4D + 10^3B + 10^2D + 10D + D</math>. This equation can be simplified into:
  
−
<math>\overset{4}{10}A + \overset{4}{10}B + \overset{3}{10}C + \overset{2}{10}B + \overset{2}{10}A + 10C + B + A = \overset{4}{10}D + \overset{2}{10}D + D</math>.
+
10^4A + 10^4B + 10^3C + 10^2B + 10^2A + 10C + B + A = 10^4D + 10^2D + D<math>.
  
−
Now from here, it should hopefully make sense that <math>A + B = D</math> by looking at the one's digit of both equations. Factoring out <math>A + B</math> gives:
+
Now from here, it should hopefully make sense that </math>A + B = D<math> by looking at the one's digit of both equations. Factoring out </math>A + B<math> gives:
  
−
<math>\overset{4}{10}(A+B) + \overset{3}{10}C \overset{2}{10}(A+B) + 10C + (B + A) = \overset{4}{10}D + \overset{2}{10}D + D</math>.  
+
</math>10^4(A+B) + 10^3C 10^2(A+B) + 10C + (B + A) = 10^4D + 10^2D + D<math>.  
  
 
Which equals:
 
Which equals:
−
<math>\overset{4}{10}(D) + \overset{3}{10}C \overset{2}{10}(D) + 10C + D = \overset{4}{10}D + \overset{2}{10}D + D</math>.  
+
</math>10^4(D) + 10^3C 10^2(D) + 10C + D = 10^4D + 10^2D + D<math>.  
  
 
This simplifies into:
 
This simplifies into:
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<math>\overset{3}{10}C + 10C = 0</math>.
+
</math>10^3C + 10C = 0<math>.
  
−
Therefore <math>c = 0</math>.
+
Therefore </math>c = 0<math>.
  
−
This means that <math>A + B = D</math> and <math>D < 10</math> or else there would be parts carried over in the equation. The positive integers that satisfy this equation are a minimum <math>(2, 1)</math> and a maximum of <math>4, 5</math>. This means that <math>D = 3</math> <math>, 4</math> <math>, 5</math> <math>, 6</math> <math>, 7</math> <math>, 8</math> <math>, 9</math>. Giving  
+
This means that </math>A + B = D<math> and </math>D < 10<math> or else there would be parts carried over in the equation. The positive integers that satisfy this equation are a minimum </math>(2, 1)<math> and a maximum of </math>(4, 5)<math>. This means that </math>D = 3<math> </math>, 4<math> </math>, 5<math> </math>, 6<math> </math>, 7<math> </math>, 8<math> </math>, 9<math>. Giving  
−
<math>\boxed{\textbf{(C)}\ 7}</math>
+
</math>\boxed{\textbf{(C)}\ 7}$
 
~PeterDoesPhysics
 
~PeterDoesPhysics
  

Revision as of 23:42, 10 August 2024

Problem

In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?

\[\begin{tabular}{cccccc}&A&B&B&C&B\\ +&B&C&A&D&A\\ \hline &D&B&D&D&D\end{tabular}\]

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

From the first column, we see $A+B < 10$ because it yields a single digit answer. From the fourth column, we see that $C+D$ equals $D$ and therefore $C = 0$. We know that $A+B = D$. Therefore, the number of values $D$ can take is equal to the number of possible sums less than $10$ that can be formed by adding two distinct natural numbers. Letting $A=1$, and letting $B=2,3,4,5,6,7,8$, we have \[D = 3,4,5,6,7,8,9 \implies \boxed{\textbf{(C)}\ 7}\]

Solution (Equation Algorithm)

It is intuitively obvious, even to the most casual observer that the problem statement can be rewritten as:

$10^4A + \10^4B + 10^3B + 10^3C + 10^2B + 10^2A + 10C + 10D + B + A = 10^4D + 10^3B + 10^2D + 10D + D$ (Error compiling LaTeX. Unknown error_msg). This equation can be simplified into:

10^4A + 10^4B + 10^3C + 10^2B + 10^2A + 10C + B + A = 10^4D + 10^2D + D$.

Now from here, it should hopefully make sense that$ (Error compiling LaTeX. Unknown error_msg)A + B = D$by looking at the one's digit of both equations. Factoring out$A + B$gives:$10^4(A+B) + 10^3C 10^2(A+B) + 10C + (B + A) = 10^4D + 10^2D + D$.

Which equals:$ (Error compiling LaTeX. Unknown error_msg)10^4(D) + 10^3C 10^2(D) + 10C + D = 10^4D + 10^2D + D$.

This simplifies into:$ (Error compiling LaTeX. Unknown error_msg)10^3C + 10C = 0$.

Therefore$ (Error compiling LaTeX. Unknown error_msg)c = 0$.

This means that$ (Error compiling LaTeX. Unknown error_msg)A + B = D$and$D < 10$or else there would be parts carried over in the equation. The positive integers that satisfy this equation are a minimum$(2, 1)$and a maximum of$(4, 5)$. This means that$D = 3$$ (Error compiling LaTeX. Unknown error_msg), 4$$ (Error compiling LaTeX. Unknown error_msg), 5$$ (Error compiling LaTeX. Unknown error_msg), 6$$ (Error compiling LaTeX. Unknown error_msg), 7$$ (Error compiling LaTeX. Unknown error_msg), 8$$ (Error compiling LaTeX. Unknown error_msg), 9$. Giving$\boxed{\textbf{(C)}\ 7}$ ~PeterDoesPhysics

See also

2014 AMC 12B (Problems • Answer Key • Resources)
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 AMC 12 Problems and Solutions

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