IB Mathematics AA SL Compound interest Study Notes
IB Mathematics AA SL Compound interest Study Notes Offer a clear explanation of Compound interest , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Compound interest
Compound Interest
Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
The formula for calculating the compound amount \( A \) after \( n \) years is:
\( A = P(1 + r)^n \)
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = annual interest rate (decimal).
- \( n \) = the number of years the money is invested or borrowed.
The compound interest \( CI \) can be calculated using:
\( CI = A – P \)
For example, if you invest \( P = \$1000 \) at an annual interest rate of \( r = 5\% \) for \( n = 10 \) years, the accumulated amount will be:
\( A = 1000(1 + 0.05)^{10} \approx \$1628.89 \)
Thus, the compound interest earned would be:
\( CI = 1628.89 – 1000 \approx \$628.89 \)
Properties of Compound Interest
Compound interest has several key properties that differentiate it from simple interest:
1. Interest on Interest: Unlike simple interest, which is calculated only on the principal amount, compound interest is calculated on both the principal and the accumulated interest.
2. Exponential Growth: The value of an investment or loan grows exponentially over time due to the effect of compounding.
3. Frequency of Compounding: The frequency of compounding (annually, semi-annually, quarterly, monthly, or daily) significantly affects the total amount of interest earned or paid. More frequent compounding results in higher total interest.
4. Time Value of Money: The longer the money is invested or borrowed, the more substantial the effect of compound interest becomes.
5. Rule of 72: A quick way to estimate how long it will take for an investment to double is to divide 72 by the annual interest rate. For example, at an interest rate of 6%, it will take approximately \( \frac{72}{6} = 12 \) years for the investment to double.
IB Mathematics AA SL Compound interest Exam Style Worked Out Questions
Question
Ravi invests $5,000 in a savings account that offers an annual compound interest rate of 4%. What will be the balance in his account after 3 years if the interest is compounded annually
▶️Answer/Explanation
To find the balance, use the compound interest formula:
\( A = P \cdot (1 + \frac{r}{n})^{n \cdot t} \)
where:
- \( P = 5000 \) (principal)
- \( r = 0.04 \) (interest rate)
- \( n = 1 \) (compounded annually)
- \( t = 3 \) (time in years)
Substituting these values:
\( A = 5000 \cdot (1 + \frac{0.04}{1})^{1 \cdot 3} = 5000 \cdot (1.04)^3 \approx 5624.32 \)
The balance after 3 years is approximately $5,624.32.
Question
Priya invested $2,500 in an account with a compound interest rate of 6% compounded semi-annually. How much will she have in her account after 5 years?
▶️Answer/Explanation
Apply the compound interest formula:
\( A = P \cdot (1 + \frac{r}{n})^{n \cdot t} \)
where:
- \( P = 2500 \) (principal)
- \( r = 0.06 \) (interest rate)
- \( n = 2 \) (compounded semi-annually)
- \( t = 5 \) (time in years)
Substituting these values:
\( A = 2500 \cdot (1 + \frac{0.06}{2})^{2 \cdot 5} = 2500 \cdot (1.03)^{10} \approx 3367.28 \)
The balance after 5 years is approximately $3,367.28.