QCE General Mathematics - Unit 4 - Loans, investments and annuities 1
Compound Interest Loans and Investments | QCE General Mathematics
Learn recurrence relations, compound interest formulae, effective annual interest and practical finance calculations for QCE General Mathematics.
Updated 2026-05-18 - 5 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Use a recurrence relation to model a compound interest loan or investment: $A_{n+1}=rA_n$, where $A_{n+1}$ is the total amount at the beginning of the $(n+1)$th period, $A_n$ is the total amount at the beginning of the $n$th period, and $r=1+i$ where $i$ is interest rate per compounding period.
- Use the compound interest formula to model a compound interest loan or investment: $A=P(1+i)^n$, where $A$ is total amount, $P$ is principal, $i$ is interest rate per compounding period and $n$ is number of compounding periods.
- Calculate the effective annual rate of interest, $i_{\text{effective}}$, and use the results to compare interest on loans or investments when interest is paid or charged for different compounding periods, including daily, monthly, quarterly and six-monthly: $i_{\text{effective}}=(1+i)^k-1$.
- Solve practical problems involving compound interest loans or investments, including determining the total amount of the loan or investment, total interest, principal, interest rate per year and per compounding period, and the effect of the interest rate and number of compounding periods on the total amount.
Compound interest means interest is added to the balance, then future interest is calculated on the new balance. This can model investments growing or loans becoming larger before repayments are made.
Original Sylligence diagram for general compound interest flow.
Recurrence model
The recurrence relation is:
$ A_{n+1}=rA_n $
where $r=1+i$ and $i$ is the interest rate per compounding period.
If the balance is $2500$ and monthly interest is $0.6\%$, then $i=0.006$ and $r=1.006$.
Compound interest formula
The direct formula is:
$ A=P(1+i)^n $
where $P$ is principal, $i$ is the interest rate per period and $n$ is the number of periods.
Effective annual rate
The effective annual interest rate compares different compounding frequencies:
$ i_{\text{effective}}=(1+i)^k-1 $
where $i$ is the rate per compounding period and $k$ is the number of compounding periods in a year.
Worked example
Common traps
Recurrence table versus direct formula
The recurrence relation shows the period-by-period process:
| Period | Balance | |---:|---:| | $0$ | $P$ | | $1$ | $P(1+i)$ | | $2$ | $P(1+i)^2$ | | $3$ | $P(1+i)^3$ |
The direct formula $A=P(1+i)^n$ is the shortcut for the same repeated multiplication.
Matching periods
| Compounding | Periodic rate from annual nominal rate | Periods in $t$ years | |---|---|---| | annually | divide by $1$ | $t$ | | quarterly | divide by $4$ | $4t$ | | monthly | divide by $12$ | $12t$ | | daily | divide by $365$ unless told otherwise | $365t$ |
The rate and number of periods must use the same time unit.
Comparing effective annual rates
Two investments can have the same nominal annual rate but different effective annual rates because one compounds more often.
Depth: compounding periods
Compound interest applies interest to both the original principal and previously earned interest. The key setup is matching the rate and number of periods.
| Compounding | Periodic rate $i$ | Periods per year | |---|---|---:| | annually | annual rate | 1 | | quarterly | annual rate divided by 4 | 4 | | monthly | annual rate divided by 12 | 12 | | fortnightly | annual rate divided by 26 | 26 | | weekly | annual rate divided by 52 | 52 |
The formula is:
$ A=P(1+i)^n $
where $i$ is the rate per compounding period and $n$ is the number of compounding periods.
Effective annual rate
The effective annual rate compares different compounding frequencies by converting them into the equivalent one-year growth rate:
$ i_{\text{eff}}=\left(1+\frac{r}{m}\right)^m-1 $
where $r$ is the nominal annual rate and $m$ is the number of compounding periods per year.
If two accounts have the same nominal annual rate, the one compounded more frequently has a slightly higher effective annual rate.
Loans versus investments
The same compound-interest formula can model investments or debts. The interpretation changes:
- investment: $A$ is the future value of savings
- loan without repayments: $A$ is the amount owed
- depreciation: use a multiplier less than $1$
Calculator discipline
Keep rates as decimals in calculations. Do not round the periodic rate too early unless the question instructs you to. For money, final answers are usually rounded to the nearest cent, but intermediate values should keep more precision.
Depth: solving for time or rate
Some questions reverse the formula. If $P$, $A$ and $i$ are known, technology or logarithms can be used to solve for $n$:
$ A=P(1+i)^n $
For General Mathematics, technology is usually the most practical method. When interpreting $n$, remember that it counts compounding periods, not necessarily years.
Rounding direction depends on context. Reaching a savings target usually rounds time up. Finding the last full period before a balance exceeds a limit may round down.
For comparison questions, calculate both final balances using the same time unit and rounding convention. A small difference in periodic rate can become noticeable over many compounding periods.