1. **State the problem:** Hashim leased equipment worth 45000 for 8 years. The cost of borrowing is 6.10% compounded semi-annually. We need to find the size of the lease payment made at the beginning of each half-year. 2. **Identify the formula:** Since payments are made at the beginning of each period, this is an annuity due problem. The formula for the present value of an annuity due is: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ where: - $PV$ is the present value (45000), - $P$ is the payment per period, - $r$ is the interest rate per period, - $n$ is the total number of payments. 3. **Calculate parameters:** - Annual interest rate = 6.10% = 0.061 - Compounded semi-annually means 2 periods per year, so: - $r = \frac{0.061}{2} = 0.0305$ - $n = 8 \times 2 = 16$ 4. **Substitute values into the formula:** $$45000 = P \times \frac{1 - (1 + 0.0305)^{-16}}{0.0305} \times (1 + 0.0305)$$ 5. **Calculate the annuity factor:** Calculate $(1 + 0.0305)^{-16}$: $$ (1.0305)^{-16} = \frac{1}{(1.0305)^{16}} \approx \frac{1}{1.6047} \approx 0.6233 $$ Calculate the fraction: $$ \frac{1 - 0.6233}{0.0305} = \frac{0.3767}{0.0305} \approx 12.3525 $$ Multiply by $(1 + 0.0305) = 1.0305$: $$ 12.3525 \times 1.0305 \approx 12.726 $$ 6. **Solve for $P$:** $$ P = \frac{45000}{12.726} \approx 3534.88 $$ **Final answer:** The lease payment required at the beginning of each half-year is approximately **3534.88**.