1. **State the problem:** We have two positive numbers where one is 3 times the other. If we add 2 to the larger number and 5 to the smaller number, then one of these new numbers becomes twice the other. 2. **Define variables:** Let the smaller number be $x$. Then the larger number is $3x$. 3. **Write the condition after adding:** Adding 2 to the larger number: $3x + 2$ Adding 5 to the smaller number: $x + 5$ 4. **Set up the equation based on the condition:** One of these numbers is twice the other. So either: $$3x + 2 = 2(x + 5)$$ or $$x + 5 = 2(3x + 2)$$ 5. **Solve the first equation:** $$3x + 2 = 2x + 10$$ Subtract $2x$ from both sides: $$3x - 2x + 2 = 10$$ $$x + 2 = 10$$ Subtract 2: $$x = 8$$ Check if $x=8$ is valid: Smaller number = 8, larger number = 24 Add 2 to larger: $24 + 2 = 26$ Add 5 to smaller: $8 + 5 = 13$ Is one twice the other? $26 = 2 imes 13$ yes. 6. **Solve the second equation:** $$x + 5 = 2(3x + 2)$$ $$x + 5 = 6x + 4$$ Bring all terms to one side: $$x + 5 - 6x - 4 = 0$$ $$-5x + 1 = 0$$ $$-5x = -1$$ $$x = \frac{1}{5}$$ Check if $x=\frac{1}{5}$ is valid: Smaller number = $\frac{1}{5}$, larger number = $3 \times \frac{1}{5} = \frac{3}{5}$ Add 2 to larger: $\frac{3}{5} + 2 = \frac{3}{5} + \frac{10}{5} = \frac{13}{5}$ Add 5 to smaller: $\frac{1}{5} + 5 = \frac{1}{5} + \frac{25}{5} = \frac{26}{5}$ Is one twice the other? $\frac{26}{5} = 2 \times \frac{13}{5}$ yes. 7. **Final answers:** The two possible pairs of numbers are: - Smaller = 8, Larger = 24 - Smaller = $\frac{1}{5}$, Larger = $\frac{3}{5}$ Both satisfy the problem conditions.