Difference between revisions of "2021 WSMO Team Round Problems/Problem 3"
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==Problem== | ==Problem== | ||
Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find <math>n.</math> | Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find <math>n.</math> | ||
| + | |||
| + | ''Proposed by pinkpig'' | ||
==Solution== | ==Solution== | ||
| + | |||
| + | Let the cost of grass be <math>g</math> dollars per pound and hay be <math>h</math> dollars per pound. Converting units, we have 32 oz = 2 lb and 96 oz = 6 lb, so: | ||
| + | <cmath>\begin{align*} | ||
| + | n &= 50g+2h = 6g+24h \\ | ||
| + | n &= 18g+16h+4. \\ | ||
| + | \end{align*}</cmath> | ||
| + | From the first equation, | ||
| + | <cmath>50g+2h = 6g+24h \Rightarrow 44g = 22h \Rightarrow h = 2g.</cmath> | ||
| + | Substituting, we have | ||
| + | <cmath>\begin{align*} | ||
| + | n &= 50g+2(2g) = 54g \\ | ||
| + | n &= 18g+16(2g)+4 = 50g+4. \\ | ||
| + | \end{align*}</cmath> | ||
| + | So, | ||
| + | <cmath>54g = 50g+4 \Rightarrow g = 1 \Rightarrow n = \boxed{54}.</cmath> | ||
| + | ~pinkpig | ||
Latest revision as of 12:51, 9 September 2025
Problem
Farmer Sam has 
dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find 
Proposed by pinkpig
Solution
Let the cost of grass be 
dollars per pound and hay be 
dollars per pound. Converting units, we have 32 oz = 2 lb and 96 oz = 6 lb, so:
From the first equation,
![\[50g+2h = 6g+24h \Rightarrow 44g = 22h \Rightarrow h = 2g.\]](http://latex.artofproblemsolving.com/c/b/3/cb391059ce0681db7487fb86fec39b427eb8b448.png)
Substituting, we have
So,
![\[54g = 50g+4 \Rightarrow g = 1 \Rightarrow n = \boxed{54}.\]](http://latex.artofproblemsolving.com/c/5/b/c5b14070165d19e3eb54f3ac048c4449166c0a43.png)
~pinkpig
