1. **Stating the problem:** We need to find a number such that increasing it by 14 and then doubling the result equals decreasing the number by 27 and tripling that result. 2. **Create the equation:** Let the number be $x$. Increasing the number by 14: $x + 14$ Doubling this result: $2(x + 14)$ Decreasing the number by 27: $x - 27$ Tripling this result: $3(x - 27)$ The equation is: $$2(x + 14) = 3(x - 27)$$ 3. **Solve the equation:** Expand both sides: $$2x + 28 = 3x - 81$$ 4. **Isolate $x$:** Subtract $2x$ from both sides: $$\cancel{2x} + 28 = 3x - 81 - \cancel{2x}$$ $$28 = x - 81$$ Add 81 to both sides: $$28 + 81 = x - 81 + 81$$ $$109 = x$$ 4. **Final answer:** The number is $109$.