### Instruction: Find the largest integral value of $x$ which solves: $\frac{1}{3}<{\frac{x}{5}}<{\frac{5}{8}}$.### Response: To solve this inequality for $x$, I need to isolate $x$ on one side of the inequality sign.
I can do this by multiplying both sides of the inequality by $5$, which is a positive number and does not change the direction of the inequality.
This gives me $5\cdot \frac{1}{3}<5\cdot \frac{x}{5}<5\cdot \frac{5}{8}$, which simplifies to $\frac{5}{3}<x<\frac{25}{8}$.
Now I need to find the largest integer value of $x$ that satisfies this inequality.
I can see that $\frac{25}{8}$ is slightly more than $3$, since $\frac{24}{8}=3$.
Therefore, the largest integer value of $x$ that is less than $\frac{25}{8}$ is $3$.
I can check that $3$ is indeed a solution by plugging it back into the original inequality: $\frac{1}{3}<\frac{3}{5}<\frac{5}{8}$, which is true.
Therefore, the largest integral value of $x$ that solves the inequality is $3$.