1. **State the problem:** We have a bouquet with red, purple, and orange balloons. The number of each color is given by: - Red: $2p$ - Orange: $p - 2$ - Purple: $p$ The total number of balloons is 18. We need to find the value of $p$, the number of purple balloons. 2. **Write the equation for the total number of balloons:** $$2p + (p - 2) + p = 18$$ 3. **Simplify the equation:** $$2p + p - 2 + p = 18$$ $$4p - 2 = 18$$ 4. **Isolate the variable term:** $$4p - 2 + 2 = 18 + 2$$ $$4p = 20$$ 5. **Divide both sides by 4 to solve for $p$:** $$\frac{4p}{\cancel{4}} = \frac{20}{\cancel{4}}$$ $$p = 5$$ 6. **Interpret the result:** There are 5 purple balloons in the bouquet. **Final answer:** $\boxed{5}$