$x$ invested at $9\%$ simple interest per annum for $7$ years yields the same interest as that on $y$ invested at $5.25\%$ simple interest per annum for $16$ years. Find $x:y$.

Formula for Simple Interest is:
$$\text{SI}=\frac{P\times R\times T}{100}$$

For the first investment:

$${\text{SI}}_1=\frac{x\times 9\times 7}{100}=\frac{63x}{100}$$

For the second investment:

$${\text{SI}}_2=\frac{y\times 5.25\times 16}{100}$$

Since $5.25\times 16=84$:

$${\text{SI}}_2=\frac{84y}{100}$$

Given that the interests are equal (${\text{SI}}_1={\text{SI}}_2$):

$$\frac{63x}{100}=\frac{84y}{100}$$

$$63x=84y$$

$$\frac{x}{y}=\frac{84}{63}$$

Dividing both numerator and denominator by $21$:

$$\frac{x}{y}=\frac{4}{3}$$

Therefore, $x:y=4:3$.