1. **Stating the problem:** We are given the equation $[x] - \{x\} = \frac{x}{3}$, where $[x]$ is the greatest integer function (floor function) and $\{x\}$ is the fractional part of $x$. We need to find the value of $\frac{10\lambda}{3}$, where $\lambda$ is the sum of all values of $x$ satisfying the equation. 2. **Recall definitions and formula:** For any real number $x$, we can write: $$x = [x] + \{x\}$$ where $[x]$ is an integer and $0 \leq \{x\} < 1$. 3. **Rewrite the given equation:** $$[x] - \{x\} = \frac{x}{3}$$ Substitute $x = [x] + \{x\}$: $$[x] - \{x\} = \frac{[x] + \{x\}}{3}$$ 4. **Multiply both sides by 3:** $$3[x] - 3\{x\} = [x] + \{x\}$$ 5. **Bring all terms to one side:** $$3[x] - 3\{x\} - [x] - \{x\} = 0$$ $$2[x] - 4\{x\} = 0$$ 6. **Express fractional part in terms of floor:** $$2[x] = 4\{x\}$$ $$\{x\} = \frac{2[x]}{4} = \frac{[x]}{2}$$ 7. **Recall that $0 \leq \{x\} < 1$, so:** $$0 \leq \frac{[x]}{2} < 1 \implies 0 \leq [x] < 2$$ Since $[x]$ is an integer, possible values are $[x] = 0$ or $[x] = 1$. 8. **Find $x$ for each case:** - For $[x] = 0$: $$\{x\} = \frac{0}{2} = 0$$ $$x = [x] + \{x\} = 0 + 0 = 0$$ - For $[x] = 1$: $$\{x\} = \frac{1}{2} = 0.5$$ $$x = 1 + 0.5 = 1.5$$ 9. **Sum of all $x$ values:** $$\lambda = 0 + 1.5 = 1.5$$ 10. **Calculate $\frac{10\lambda}{3}$:** $$\frac{10 \times 1.5}{3} = \frac{15}{3} = 5$$ **Final answer:** $$\boxed{5}$$