1. **Problem Statement:** Ella plans an overseas trip lasting 3 years and needs €600 at the start of every month for expenses. The annual interest rate is 4%. We need to find how much money she must have saved before the trip to fund these monthly withdrawals. 2. **Formula and Explanation:** This is a problem of finding the present value of an annuity with monthly withdrawals at the start of each month (an annuity due). The formula for the present value of an annuity due is: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ where: - $P$ = payment per period (€600) - $r$ = monthly interest rate (annual rate divided by 12) - $n$ = total number of payments (months) 3. **Calculate monthly interest rate:** $$r = \frac{4\%}{12} = \frac{0.04}{12} = 0.0033333$$ 4. **Calculate total number of payments:** $$n = 3 \text{ years} \times 12 \text{ months/year} = 36$$ 5. **Calculate present value:** $$PV = 600 \times \frac{1 - (1 + 0.0033333)^{-36}}{0.0033333} \times (1 + 0.0033333)$$ Calculate the term inside the fraction: $$1 + 0.0033333 = 1.0033333$$ Calculate the power term: $$1.0033333^{-36} = \frac{1}{1.0033333^{36}}$$ Calculate $1.0033333^{36}$: $$1.0033333^{36} \approx e^{36 \times \ln(1.0033333)} \approx e^{36 \times 0.003327} \approx e^{0.11977} \approx 1.1275$$ So: $$1.0033333^{-36} \approx \frac{1}{1.1275} = 0.8867$$ Now calculate the numerator: $$1 - 0.8867 = 0.1133$$ Calculate the fraction: $$\frac{0.1133}{0.0033333} = 33.99$$ Multiply by $(1 + r)$: $$33.99 \times 1.0033333 = 34.10$$ Finally, multiply by payment $P$: $$PV = 600 \times 34.10 = 20460$$ 6. **Answer:** Ella needs approximately **20460** euros saved before her trip to fund her expenses.