Final answer:
To find the sum borrowed from a loan with semi-annual installments and semi-annual compound interest, you can use the present value of an annuity formula. With an installment of R 1458 and an interest rate of 16% per annum compounded semi-annually, the sum borrowed comes out to approximately R 26282.
Step-by-step explanation:
The sum borrowed for a loan that is repaid in two equal semi-annual instalments with a semi-annual compound interest rate of 16% per annum can be determined using the present value of an annuity formula. Since each instalment is R 1458, the calculation is based on the fact that the present value of each instalment is equal to the sum borrowed.
Let's denote the sum borrowed as 'P'. The present value (PV) of annuity formula is:
PV = R [ (1 - (1 + i)^(-n)) / i ]
Where 'R' is the instalment amount, 'i' is the semi-annual interest rate, and 'n' is the number of semi-annual periods. We have R = 1458, i = 0.08 (which is 16% per annum compounded semi-annually, or 8% per period), and n = 2 (as there are two instalments in total).
Thus,
PV = 1458 [(1 - (1 + 0.08)^(-2)) / 0.08]
Plugging in the numbers gives,
PV = 1458 [(1 - (1 + 0.08)^(-2)) / 0.08]PV = 1458 [(1 - 1/1.1664) / 0.08]PV = 1458 [0.1441 / 0.08]PV = 1458 * 18.0125PV = 26281.72
Therefore, the sum borrowed 'P' is roughly R 26282.