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Give an example of, or a word for, each of the following terms: 1.1.1 Fixed income, 1.1.2 Medium-term investment, 1.1.3 A money lender who lends out money for profit, 1.1.4 The difference between the budgeted amount and the actual amount, 1.1.5 Long-term investment.

Mathematics

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1.1 Give an example of, or a word for, each of the following terms:

  • 1.1.1 Fixed income: Salary or Pension
  • 1.1.2 Medium-term investment: Fixed deposit (e.g., 1-5 years)
  • 1.1.3 A money lender who lends out money for profit: Bank or Moneylender
  • 1.1.4 The difference between the budgeted amount and the actual amount: Variance
  • 1.1.5 Long-term investment: Property or Shares

1.2 Pinkie buys a laptop priced at R13 495. She takes out a 12-month hire-purchase agreement. She pays a deposit of 20% and the interest charged on the balance is 15% per annum simple interest.

1.2.1 What will she actually pay for the laptop?

  1. Calculate the deposit amount.

    Deposit=20%×R13 495=0.20×R13 495=R2 699\text{Deposit} = 20\% \times \text{R}13\,495 = 0.20 \times \text{R}13\,495 = \text{R}2\,699

  2. Calculate the balance to be financed.

    Balance=R13 495−R2 699=R10 796\text{Balance} = \text{R}13\,495 - \text{R}2\,699 = \text{R}10\,796

  3. Calculate the simple interest on the balance for 12 months (1 year).

    The simple interest formula is I=P⋅r⋅tI = P \cdot r \cdot t.

    I=R10 796×0.15×1=R1 619.40I = \text{R}10\,796 \times 0.15 \times 1 = \text{R}1\,619.40

  4. Calculate the total amount to be repaid on the loan.

    Amount to repay=Balance+Interest=R10 796+R1 619.40=R12 415.40\text{Amount to repay} = \text{Balance} + \text{Interest} = \text{R}10\,796 + \text{R}1\,619.40 = \text{R}12\,415.40

  5. Calculate the total amount Pinkie will pay for the laptop.

    Total paid=Deposit+Amount to repay=R2 699+R12 415.40=R15 114.40\text{Total paid} = \text{Deposit} + \text{Amount to repay} = \text{R}2\,699 + \text{R}12\,415.40 = \text{R}15\,114.40

    Pinkie will actually pay for the laptop: R15 114.40\boxed{\text{R}15\,114.40}

    1.2.2 What will her monthly payment be?

  6. Use the total amount to be repaid on the loan from 1.2.1 and divide by the number of months.

    Monthly payment=R12 415.4012=R1 034.62\text{Monthly payment} = \frac{\text{R}12\,415.40}{12} = \text{R}1\,034.62

    Her monthly payment will be: R1 034.62\boxed{\text{R}1\,034.62}

    1.3 Themba won R 5 400 in a talent show. He was given two options to invest this money.

    • OPTION A: for 3 years @ 6,4% compound interest, compounded annually
    • OPTION B: for 3 years @ 7% simple interest. Which option should he choose? Give a reason for your answer and show ALL calculations.

  7. Calculate the future value for Option A (Compound Interest).

    The formula for compound interest is A=P(1+i)nA = P(1 + i)^n.

    AA=R5 400(1+0.064)3A_A = \text{R}5\,400(1 + 0.064)^3 AA=R5 400(1.064)3A_A = \text{R}5\,400(1.064)^3 AA=R5 400(1.204161)A_A = \text{R}5\,400(1.204161) AA=R6 499.47A_A = \text{R}6\,499.47

  8. Calculate the future value for Option B (Simple Interest).

    The formula for simple interest is A=P(1+r⋅t)A = P(1 + r \cdot t).

    AB=R5 400(1+0.07×3)A_B = \text{R}5\,400(1 + 0.07 \times 3) AB=R5 400(1+0.21)A_B = \text{R}5\,400(1 + 0.21) AB=R5 400(1.21)A_B = \text{R}5\,400(1.21) AB=R6 534A_B = \text{R}6\,534

  9. Compare the two options and make a choice.

    Option A yields R6 499.47.
    Option B yields R6 534.
    Themba should choose Option B because it yields a higher return.
    Themba should choose: Option B\boxed{\text{Option B}}
    Reason: Option B yields a higher return of R6 534 compared to Option A's R6 499.47.

    1.4 An amount of R1 200 accumulates to R2 600 after 3 years. Find the interest rate if the investment earned compound interest, compounded annually.

  10. Write down the compound interest formula and substitute the given values.

    The formula for compound interest is A=P(1+i)nA = P(1 + i)^n.

    R2 600=R1 200(1+i)3\text{R}2\,600 = \text{R}1\,200(1 + i)^3

  11. Isolate the term (1+i)3(1 + i)^3.

    2 6001 200=(1+i)3\frac{2\,600}{1\,200} = (1 + i)^3 2612=(1+i)3\frac{26}{12} = (1 + i)^3 136=(1+i)3\frac{13}{6} = (1 + i)^3 2.16666...=(1+i)32.16666... = (1 + i)^3

  12. Take the cube root of both sides.

    2.16666...3=1+i\sqrt[3]{2.16666...} = 1 + i 1.29386...=1+i1.29386... = 1 + i

  13. Solve for ii and convert to a percentage.

    i=1.29386...−1i = 1.29386... - 1 i=0.29386...i = 0.29386... i≈29.39%i \approx 29.39\%

    The interest rate is: 29.39%\boxed{29.39\%}

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