UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1


Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs $\textdollar{21}$ to monogram BRYAN, $\textdollar{25}$ to monogram BARRY and $\textdollar{18}$ to monogram SARAH, how much does it cost to monogram SHANA?


We set up the following equations:

$b+r+y+a+n= 21$

$b+a+2r+y= 25$

$s+2a+r+h= 18$

We are asked to find the price of "Shana", or $s+h+2a+n$. We notice that this expression has no $b$ or $y$, so we subtract the first equation from the second to eliminate those variables:



Which we substitute into the third equation



This matches the expression we need to solve for, so we are done. $\boxed{14}$.

Slick Solution

Notice that BRYAN is simple BARRY rearranged, and SARAH is simple SHANA rearranged. Let the cost of rearranging SHANA be $x$. We have the following equation: \[21+18=25+x\] Solving for $x$, our answer is $\boxed{14}$.


  • Credit to Football017 for writing the solution
  • Credit to yofro for writing the second solution

See also

2014 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10
All UNM-PNM Problems and Solutions