Final answer:
Mia will earn approximately 637.10 AUD interest over 2.5 years with a deposit of 4000 AUD at a 6% interest rate compounded semi-annually. Ella initially deposited about 3553.47 AUD to end up with 4000 AUD in 2.5 years at a 4% interest rate compounded monthly.
Step-by-step explanation:
Calculating Compound Interest
To find the amount of interest that Mia will earn over the next 2.5 years on her 4000 AUD deposit, we'll use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the initial principal balance, r is the nominal annual interest rate, n is the number of times the interest is compounded per year, and t is the time the money is invested for in years. In this case, P = 4000, r = 0.06, n = 2, and t = 2.5. Plugging these values into the formula gives us: A = 4000(1 + 0.06/2)^(2*2.5), which computes to
A = 4000(1.03)^5. Calculating further we get A ≈ 4000 * 1.159274, approximately A ≈ 4637.10 So, the interest is A - P ≈ 4637.10 - 4000 = 637.10 AUD.
For Ella's scenario, we know that the final amount A after 2.5 years will be 4000 AUD, with a nominal annual interest rate r = 0.04, compounded monthly (n = 12). We rearrange the formula to solve for the principal P: P = A / (1 + r/n)^(nt). Substituting the known values gives us P = 4000 / (1 + 0.04/12)^(12*2.5), which simplifies to P = 4000 / (1.003333)^30. Calculating the denominator, we get approximately 1.12551, therefore, P ≈ 4000 / 1.12551 ≈ 3553.47 AUD. Ella deposited about 3553.47 AUD.