1. **State the problem:** We have two numbers, one is positive and is 3 times the other. If we add 2 to the larger number and 5 to the smaller number, then one of the resulting numbers is twice the other. We need to find the smaller number. 2. **Define variables:** Let the smaller number be $x$. Then the larger number is $3x$. 3. **Write the conditions:** After adding, the larger number becomes $3x + 2$ and the smaller number becomes $x + 5$. One of these is twice the other, so either: $$3x + 2 = 2(x + 5)$$ or $$x + 5 = 2(3x + 2)$$ 4. **Solve the first equation:** $$3x + 2 = 2x + 10$$ Subtract $2x$ from both sides: $$3x - 2x + 2 = 10$$ $$x + 2 = 10$$ Subtract 2: $$x = 8$$ 5. **Check if $x=8$ satisfies the problem:** Smaller number = 8 (positive, valid) Larger number = $3 imes 8 = 24$ Add 2 to larger: $24 + 2 = 26$ Add 5 to smaller: $8 + 5 = 13$ Check if one is twice the other: $26 = 2 imes 13$ (True) 6. **Solve the second equation:** $$x + 5 = 2(3x + 2)$$ $$x + 5 = 6x + 4$$ Subtract $x$ from both sides: $$5 = 5x + 4$$ Subtract 4: $$1 = 5x$$ Divide by 5: $$x = \frac{1}{5}$$ 7. **Check if $x=\frac{1}{5}$ satisfies the problem:** Smaller number = $\frac{1}{5}$ (positive, valid) 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: $\frac{1}{5} + 5 = \frac{1}{5} + \frac{25}{5} = \frac{26}{5}$ Check if one is twice the other: $\frac{26}{5} = 2 \times \frac{13}{5}$ (True) 8. **Conclusion:** Both $x=8$ and $x=\frac{1}{5}$ satisfy the conditions, but since the problem states "a positive number is 3 times another number" and does not restrict to integers, both are valid. The smaller number can be either $8$ or $\frac{1}{5}$. **Final answer:** The smaller number is either $8$ or $\frac{1}{5}$.