1. **State the problem:** Theo starts with 18 and saves 4.5 each week. James starts with 0 and saves 4 each week. We want to find the number of weeks $w$ when their savings are in the ratio 15:8. 2. **Set up the expressions for their savings after $w$ weeks:** Theo's savings: $18 + 4.5w$ James's savings: $0 + 4w = 4w$ 3. **Write the ratio equation:** $$\frac{18 + 4.5w}{4w} = \frac{15}{8}$$ 4. **Cross multiply to solve for $w$:** $$8(18 + 4.5w) = 15(4w)$$ $$144 + 36w = 60w$$ 5. **Isolate $w$:** $$144 = 60w - 36w$$ $$144 = 24w$$ $$w = \frac{144}{24} = 6$$ 6. **Interpretation:** After 6 weeks, Theo and James will have savings in the ratio 15:8. **Final answer:** $\boxed{6}$ weeks.