a) R5 s00 at \( 9,1 \% \) compounded annee Por 5 years? b) R100 000 at \( 11,2 \% \quad p \cdot a \cdot C \) I for years 3.fit unat intaest rate must \( \begin{array}{llll}K 20 & 00 & \text { be invested to grow to } \\ \text { K27 000 in is years. If CI usec }\end{array} \) Ci. Calculaze the origimal amount invested, if you have R150 000 in years it the interest vas \( 5,25 \% \) pa CI

Let's break down each part of the problem step by step. ### Part a We need to calculate the future value of an investment of R5, compounded annually at a rate of \( 9.1\% \) for 5 years. The formula for compound interest is given by: \[ A = P(1 + r)^n \] where: - \( A \) = the amount of money accumulated after n years, including interest. - \( P \) = the principal amount (the initial amount of money). - \( r \) = the annual interest rate (decimal). - \( n \) = the number of years the money is invested or borrowed. **Given:** - \( P = 5 \) - \( r = 9.1\% = 0.091 \) - \( n = 5 \) Now, we can substitute these values into the formula. ### Part b We need to calculate the future value of an investment of R100,000 at \( 11.2\% \) compounded annually for 3 years. Using the same formula: \[ A = P(1 + r)^n \] **Given:** - \( P = 100000 \) - \( r = 11.2\% = 0.112 \) - \( n = 3 \) ### Part c We need to find the interest rate \( r \) that must be invested to grow R20,000 to R27,000 in 5 years using compound interest. We can rearrange the compound interest formula to solve for \( r \): \[ A = P(1 + r)^n \implies 1 + r = \left(\frac{A}{P}\right)^{\frac{1}{n}} \implies r = \left(\frac{A}{P}\right)^{\frac{1}{n}} - 1 \] **Given:** - \( A = 27000 \) - \( P = 20000 \) - \( n = 5 \) ### Part d We need to calculate the original amount invested if R150,000 is accumulated in 5 years at an interest rate of \( 5.25\% \) compounded annually. Using the rearranged formula: \[ P = \frac{A}{(1 + r)^n} \] **Given:** - \( A = 150000 \) - \( r = 5.25\% = 0.0525 \) - \( n = 5 \) Now, let's perform the calculations for each part. Calculate the value by following steps: - step0: Calculate: \(100000\left(1+0.112\right)^{3}\) - step1: Add the numbers: \(100000\times 1.112^{3}\) - step2: Convert the expressions: \(100000\left(\frac{139}{125}\right)^{3}\) - step3: Simplify: \(100000\times \frac{139^{3}}{125^{3}}\) - step4: Rewrite the expression: \(3125\times 32\times \frac{139^{3}}{125^{3}}\) - step5: Rewrite the expression: \(5^{5}\times 32\times \frac{139^{3}}{5^{9}}\) - step6: Reduce the numbers: \(32\times \frac{139^{3}}{5^{4}}\) - step7: Multiply: \(\frac{32\times 139^{3}}{5^{4}}\) - step8: Simplify: \(\frac{32\times 139^{3}}{625}\) Calculate or simplify the expression \( 150000/(1 + 0.0525)^5 \). Calculate the value by following steps: - step0: Calculate: \(\frac{150000}{\left(1+0.0525\right)^{5}}\) - step1: Add the numbers: \(\frac{150000}{1.0525^{5}}\) - step2: Convert the expressions: \(\frac{150000}{\left(\frac{421}{400}\right)^{5}}\) - step3: Simplify the expression: \(\frac{150000}{\frac{421^{5}}{400^{5}}}\) - step4: Rewrite the expression: \(\frac{150000\times 400^{5}}{421^{5}}\) Calculate or simplify the expression \( (27000/20000)^(1/5) - 1 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{27000}{20000}\right)^{\frac{1}{5}}-1\) - step1: Reduce the fraction: \(\left(\frac{27}{20}\right)^{\frac{1}{5}}-1\) - step2: Rewrite the expression: \(\frac{27^{\frac{1}{5}}}{20^{\frac{1}{5}}}-1\) - step3: Calculate: \(\frac{27^{\frac{1}{5}}\times 20^{\frac{4}{5}}}{20}-1\) - step4: Reduce fractions to a common denominator: \(\frac{27^{\frac{1}{5}}\times 20^{\frac{4}{5}}}{20}-\frac{20}{20}\) - step5: Transform the expression: \(\frac{27^{\frac{1}{5}}\times 20^{\frac{4}{5}}-20}{20}\) - step6: Simplify: \(\frac{2\sqrt[5]{5000}\times \sqrt[5]{27}-20}{20}\) - step7: Reduce the fraction: \(\frac{\sqrt[5]{135000}-10}{10}\) Calculate or simplify the expression \( 5*(1 + 0.091)^5 \). Calculate the value by following steps: - step0: Calculate: \(5\left(1+0.091\right)^{5}\) - step1: Add the numbers: \(5\times 1.091^{5}\) - step2: Convert the expressions: \(5\left(\frac{1091}{1000}\right)^{5}\) - step3: Simplify: \(5\times \frac{1091^{5}}{1000^{5}}\) - step4: Rewrite the expression: \(5\times \frac{1091^{5}}{125^{5}\times 8^{5}}\) - step5: Rewrite the expression: \(5\times \frac{1091^{5}}{5^{15}\times 8^{5}}\) - step6: Reduce the numbers: \(1\times \frac{1091^{5}}{5^{14}\times 8^{5}}\) - step7: Multiply the fractions: \(\frac{1091^{5}}{32768\times 5^{14}}\) ### Part a The future value of R5 compounded annually at \( 9.1\% \) for 5 years is calculated as follows: \[ A = 5(1 + 0.091)^5 \approx 7.73 \] So, the amount after 5 years will be approximately **R7.73**. ### Part b The future value of R100,000 compounded annually at \( 11.2\% \) for 3 years is: \[ A = 100000(1 + 0.112)^3 \approx 137503.69 \] Thus, the amount after 3 years will be approximately **R137,503.69**. ### Part c To find the interest rate \( r \) that grows R20,000 to R27,000 in 5 years: \[ r = \left(\frac{27000}{20000}\right)^{\frac{1}{5}} - 1 \approx 0.06186 \] This means the interest rate must be approximately **6.19%**. ### Part d To calculate the original amount invested if R150,000 is accumulated in 5 years at an interest rate of \( 5.25\% \): \[ P = \frac{150000}{(1 + 0.0525)^5} \approx 116139.71 \] Thus, the original amount invested was approximately **R116,139.71**. ### Summary of Results: - **Part a:** R7.73 - **Part b:** R137,503.69 - **Part c:** 6.19% - **Part d:** R116,139.71