Can someone help me solve this word problem using a diophantine equation?
A man has $4.55 in change composed entirely of dimes and quarters. What are the maximum and minimum number of coins that he can have? Is it possible for the number of dimes to equal the number of quarters.
So far I set up what was given so I have:
x= # of dimes
y= # of quarters
455=total
10=cost of dime
25=cost of quarter
equation: 10x+25y=455
so I found the gcd(25,10)=5 and since gcd(25,10)|455 there is a solution.
where do I go from here?
Best Answer
For the first question.
You can write the equation as $$10x+25y=455$$ or equivalently as $$2x+5y=91$$ So you can see that the maximal admissible value for $y$ is one such that $5y=85$ which gives $y=17$. Then $x=3$. Accordingly, the maximum value for $x$ is one such that $2x=86$ which gives $x=43$ and $y=1$. In the first case you have the minimum number of coins $(20)$ (since it corresponds to the maximum possible value of quarters - the higher valued coin) and in the second the maximum $(44)$ (since it corresponds to the maximum possible value of dimes - the lower valued coin).
For the second question. Just set $x=y$ and see if you can solve the equation (which you can for $x=y=13$).