1. **State the problem:** We have two gym membership plans, Plan X and Plan Y, with different signup fees and monthly charges. We want to find the total cost of each plan after 6 months, given that after 6 months both plans cost the same. 2. **Define variables and formulas:** - Let $C_X$ be the total cost of Plan X after 6 months. - Let $C_Y$ be the total cost of Plan Y after 6 months. The total cost formula for each plan is: $$C = \text{signup fee} + (\text{monthly fee} \times \text{number of months})$$ 3. **Write expressions for each plan:** - Plan X: $$C_X = 30 + 10 \times 6$$ - Plan Y: $$C_Y = 10 + 15 \times 6$$ 4. **Calculate each total cost:** - Plan X: $$C_X = 30 + 60 = 90$$ - Plan Y: $$C_Y = 10 + 90 = 100$$ 5. **Check the condition:** The problem states that after 6 months both plans cost the same, but from the calculations, Plan X costs 90 and Plan Y costs 100. This suggests a misunderstanding; the problem likely means to find the month when costs are equal. 6. **Find the month $m$ when costs are equal:** Set $$30 + 10m = 10 + 15m$$ 7. **Solve for $m$:** $$30 - 10 = 15m - 10m$$ $$20 = 5m$$ $$m = 4$$ 8. **Calculate total cost at $m=4$ months:** - Plan X: $$30 + 10 \times 4 = 30 + 40 = 70$$ - Plan Y: $$10 + 15 \times 4 = 10 + 60 = 70$$ **Final answer:** After 6 months, Plan X costs 90 and Plan Y costs 100. However, the plans cost the same after 4 months, and the total cost then is 70 for each plan.