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

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## Problem

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?

## Solution

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: $r-n=4$ $r=n+4$

Which we substitute into the third equation $s+2a+n+4+h=18$. $s+2a+n+h=14$.

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 $\boxed{14}$.

## Credits

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