1. We are given that Rhianna wants to invest ₱50,000 to accumulate to ₱65,000 in 4 years with interest compounded semiannually. 2. The formula for compound interest is $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ where: * $A$ = the amount after time $t$ * $P$ = principal amount * $r$ = annual interest rate (decimal) * $n$ = number of compounding periods per year * $t$ = time in years 3. Substitute the values: * $A = 65000$ * $P = 50000$ * $n = 2$ (because interest is compounded semiannually) * $t = 4$ Using the formula: $$65000 = 50000 \left(1 + \frac{r}{2}\right)^{2 \times 4} = 50000 \left(1 + \frac{r}{2}\right)^8$$ 4. Divide both sides by 50000: $$\frac{65000}{50000} = \left(1 + \frac{r}{2}\right)^8$$ $$1.3 = \left(1 + \frac{r}{2}\right)^8$$ 5. Take the 8th root on both sides: $$\left(1 + \frac{r}{2}\right) = 1.3^{\frac{1}{8}}$$ 6. Calculate the 8th root: $$1.3^{\frac{1}{8}} \approx 1.03308$$ 7. Solve for $r$: $$1 + \frac{r}{2} = 1.03308$$ $$\frac{r}{2} = 1.03308 - 1 = 0.03308$$ $$r = 0.03308 \times 2 = 0.06616$$ 8. Convert to percentage: $$r = 0.06616 \times 100 = 6.616\%$$ 9. Rounding to two decimal places, the rate is approximately 6.62%. 10. Among the given options, the closest rate is 6.57% which is slightly less than our calculation, but the nearest correct choice by rounding is 6.67%. Hence, the best answer is: Rhianna should invest her money at 6.67% compounded semiannually.