\text{The HCF and the LCM of two numbers are 15 and 300, respectively. If one of the numbers is }\frac{5}{4}\text{ times the other, what is the larger number?}

To find the larger number, we can use the fundamental relationship between the Highest Common Factor (HCF), the Least Common Multiple (LCM), and the two numbers.

Let the two numbers be $x$ and $y$.

We are given that:
HCF = 15
LCM = 300

The relationship between two numbers and their HCF and LCM is given by the formula:
$x\times y=\text{HCF}\times \text{LCM}$

Substituting the given values into the formula, we get:
$x\times y=15\times 300$
$x\times y=4500$

We are also given that one of the numbers is $\frac{5}{4}$ times the other. Let:
$y=\frac{5}{4}x$

Now, substitute this expression for $y$ into the product equation:
$x\times \left(\frac{5}{4}x\right)=4500$
$\frac{5}{4}{x}^2=4500$

To solve for $x^2$, multiply both sides by $\frac{4}{5}$:
$x^2=4500\times \frac{4}{5}$
$x^2=900\times 4$
$x^2=3600$

Taking the positive square root on both sides:
$x=\sqrt{3600}$
$x=60$

Now, find the other number $y$:
$y=\frac{5}{4}\times 60$
$y=5\times 15$
$y=75$

Comparing the two numbers, 60 and 75, the larger number is 75.

Therefore, the correct option is 75.