If a number plus its reciprocal equals $\frac{10}{3}$, find the number.

Let the number be $x$.

$$\text{Then, } x+\frac{1}{x}=\frac{10}{3}.$$

$$\Rightarrow 3x^2 + 3 = 10x$$

$$\Rightarrow 3x^2 - 10x + 3 = 0$$

Factor: Use $ab = 9, a + b = -10 \rightarrow -1$ and $-9$

$3x^2 - 9x - x + 3 = 0$
$$\Rightarrow 3x(x - 3) - 1(x - 3) = 0$$

$$\Rightarrow (3x - 1)(x - 3) = 0$$

$$x = \frac{1}{3} \text{ or } x = 3$$

Both are valid (and they are reciprocals of each other).
The number is 3 or $\frac{1}{3}$.