Largest possible value for a math test

Question

Last week, Adam and Bronson sat for a Mathematics test, of which the full mark was 100. Adam scored x marks and Bronson scored y marks, where x and y are whole numbers. Given that 8x = 5y, find the largest possible value of Adam's score.

I get the answer x = 60, y = 96 via trial and error but whats the correct way to solve this question?

Answer 1

$8x=5y$, so $x$ is a multiple of $\ldots$

$x=\frac{5y}{8}$ and $y\le\ldots\,$, therefore $x\le\ldots$

See if you can fill in the gaps and finish the problem.

Answer 2

Whole numbers means that $5$ divides x and 8 divides y (as gcd(8,5)=1). So $x=5t, y=8s \implies t=s$. What are the biggest for $t, s$ so that $x, y$ remains $\leq100$?

Answer 3

For every $8$ points Bronson got, Adam only got $5$. To maximize Adam's score, we need to maximize Bronson's score. Bronson can score a maximum of $8\lfloor\frac{100}{8}\rfloor=8\cdot12=96$, giving Adam a score of $5\cdot12=60$.

Answer 4

Try thinking about equivalent fractions. If $8x=5y$, then $\frac{x}{y} = \frac{5}{8} = \frac{5n}{8n}$.

What would be the largest value of $n$ such that $8n\lt 100$? What would be the values of $5n$ and $8n$ in that case?