We are given that we need to find a number $x$ such that when it is multiplied by the numerator of the fraction $\frac{2}{3}$ and added to the denominator, the resulting fraction equals $\frac{4}{3}$.

Let's set up the equation based on the given information:

$\frac{x \cdot 2}{x + 3} = \frac{4}{3}$

Step-by-step solution:

1. Multiply both sides by $3(x + 3)$ to eliminate the denominators:

$3(x \cdot 2) = 4(x + 3)$

2. Simplify the equation:

$6x = 4x + 12$

3. Subtract $4x$ from both sides:

$2x = 12$

4. Divide by 2:

$x = 6$

So, the number is $6$.

Do you want further details or have any questions? Here are some related questions to practice:

1. What number multiplied by the numerator and added to the denominator of $\frac{3}{5}$ makes the resulting fraction $\frac{7}{5}$?
2. Solve for $x$: $\frac{x \cdot 4}{x + 2} = \frac{8}{5}$.
3. How do you set up a similar equation for $\frac{1}{2}$ and the resulting fraction being $\frac{5}{6}$?
4. What is the process to cross-multiply fractions in equations?
5. Can you derive the equation for a general fraction $\frac{a}{b}$ and another fraction $\frac{c}{d}$?

Tip: Always check your solution by plugging the value of $x$ back into the original equation.