To find the rate of interest, we can use the formula for Simple Interest ($SI$):

$$SI = \frac{P \times R \times T}{100}$$

Where:

$P$ = Principal (the money borrowed)

$R$ = Rate of interest per annum

$T$ = Time period in years

Step 1: Determine the Simple Interest

Let the principal amount borrowed be $P$.

After $3$ years ($T = 3$), the total amount ($A$) returned to the lender is $\frac{4}{3}$ of the borrowed money:

$$A = \frac{4}{3}P$$

The Simple Interest is the difference between the total amount returned and the principal:

$$SI = A - P$$

$$SI = \frac{4}{3}P - P = \frac{1}{3}P$$

Step 2: Substitute the values into the formula

Now, substitute $SI = \frac{1}{3}P$ and $T = 3$ into the simple interest formula:

$$\frac{1}{3}P = \frac{P \times R \times 3}{100}$$

Step 3: Solve for $R$

Since $P$ is on both sides, we can divide both sides by $P$:

$$\frac{1}{3} = \frac{3R}{100}$$

Cross-multiply to solve for $R$:

$$3 \times 3R = 100$$

$$9R = 100$$

$$R = \frac{100}{9}$$

$$R = 11\frac{1}{9}\% \text{ or approximately } 11.11\%$$

The rate of interest was $11\frac{1}{9}\%$ (or $11.11\%$ per annum).