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Number and Algebra - Financial mathematics

Grade 12IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Currency conversion involves using exchange rates to move between different currencies. Banks often charge a commission, which is either a flat fee or a percentage of the amount being exchanged.

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Simple Interest is calculated only on the initial principal amount PP. The interest II is constant for each time period.

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Compound Interest is calculated on the principal plus any accumulated interest. In IB AI, we often use the GDC (Graphic Display Calculator) TVM Solver where NN is the number of years, I%I\% is the annual interest rate, PVPV is the present value, PMTPMT is the payment per period, FVFV is the future value, P/YP/Y is payments per year, and C/YC/Y is compounding periods per year.

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Nominal interest rate is the stated annual rate, while the effective interest rate represents the actual interest earned/paid per year due to compounding more than once annually.

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Depreciation (reducing balance method) is when an asset loses value at a fixed percentage rate rr each year. This follows the same structure as compound interest but with a negative growth rate.

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Inflation reduces the purchasing power of money over time. To find the 'real' value of an investment, we adjust the future value by the inflation rate.

📐Formulae

I=P×r×n100I = \frac{P \times r \times n}{100}

FV=PV×(1+r100k)nkFV = PV \times \left(1 + \frac{r}{100k}\right)^{nk}

FV=PV×(1−r100)nFV = PV \times \left(1 - \frac{r}{100}\right)^{n}

Real Value=Nominal Value(1+inflation rate)n\text{Real Value} = \frac{\text{Nominal Value}}{(1 + \text{inflation rate})^n}

💡Examples

Problem 1:

A person exchanges $800 USD into EUR at an exchange rate of $1 USD = 0.92 EUR. The bank charges a 2%2\% commission on the original USD amount before conversion. Calculate the amount of EUR received.

Solution:

  1. Calculate commission in USD: 0.02×800=160.02 \times 800 = 16 USD.
  2. Subtract commission: 800−16784\begin{array}{r} 800 \\ - 16 \\ \hline 784 \end{array}
  3. Convert to EUR: 784×0.92=721.28784 \times 0.92 = 721.28 EUR.

Explanation:

Commission is deducted from the source currency first. The remaining amount is then multiplied by the exchange rate.

Problem 2:

Calculate the future value of $5000 invested for 4 years at an annual interest rate of 3.5%3.5\% compounded monthly.

Solution:

Using the compound interest formula where PV=5000PV = 5000, r=3.5r = 3.5, n=4n = 4, and k=12k = 12: FV=5000×(1+3.5100×12)4×12FV = 5000 \times \left(1 + \frac{3.5}{100 \times 12}\right)^{4 \times 12} FV=5000×(1.002916...)48≈5751.13FV = 5000 \times (1.002916...)^{48} \approx 5751.13

Explanation:

Since interest is compounded monthly, we divide the annual rate by 12 and multiply the number of years by 12 to get the total number of periods.

Problem 3:

A car is purchased for $25000 and depreciates at a rate of 15%15\% per year. Find its value after 3 years.

Solution:

Using the reducing balance formula: FV=25000×(1−15100)3FV = 25000 \times \left(1 - \frac{15}{100}\right)^{3} FV=25000×(0.85)3FV = 25000 \times (0.85)^{3} FV=25000×0.614125=15353.13FV = 25000 \times 0.614125 = 15353.13

Explanation:

Depreciation uses the decay formula (1 - r). After 3 years, the car retains approximately 61.4%61.4\% of its original value.