5.2 Simple Interest and Compound Interest
Key Takeaways
- Simple interest accrues strictly on the original principal balance over time according to $I = Prt$, yielding an accumulated future value of $A = P(1 + rt)$, where time $t$ must always be measured in years ($t = \text{months}/12$ or $t = \text{days}/365$).
- Periodic compound interest calculates interest on both the initial principal and all previously accumulated interest according to $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $n$ represents the compounding frequency per year.
- Common compounding frequencies $n$ are: annual ($n=1$), semiannual ($n=2$), quarterly ($n=4$), monthly ($n=12$), and daily ($n=365$); increasing $n$ increases the accumulated future value, but with diminishing marginal returns.
- Simple interest exhibits linear capital growth ($y = mt + b$), whereas compound interest exhibits exponential capital growth ($y = a b^t$), creating vast divergence over multi-decade investment horizons.
- To evaluate compound interest on the TI-30XS MultiView without intermediate rounding errors, input the formula as a single unbroken expression using parentheses and the exponent key [^], or store the periodic interest rate $\frac{r}{n}$ into memory using [sto->].
5.2 Simple Interest and Compound Interest
Interest is the fee paid for the use of borrowed capital or the financial return earned on invested funds. The CLEP College Mathematics examination places heavy emphasis on distinguishing between simple interest (where interest accrues linearly on the principal alone) and compound interest (where interest is reinvested to generate exponential capital accumulation).
1. Simple Interest Mechanics & Time Conversion
Under Simple Interest, the interest earned ($I$) is directly proportional to three variables: the principal investment ($P$), the annual nominal interest rate ($r$, expressed as a decimal), and the duration of the loan or deposit ($t$, expressed strictly in years).
The Governing Formulas
Where:
- $I$ = Total simple interest accrued ($)
- $P$ = Principal amount (initial investment or loan balance)
- $r$ = Annual interest rate in decimal form ($r = \frac{\text{APR}}{100}$)
- $t$ = Time duration expressed in years
- $A$ = Accumulated total future value (Principal + Interest)
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| SIMPLE INTEREST ALGEBRAIC ROTATION |
| |
| To solve for any individual variable given the other three: |
| |
| 1. Interest Earned: I = P * r * t |
| 2. Principal Needed: P = I / (r * t) or P = A / (1 + r * t) |
| 3. Annual Rate: r = I / (P * t) |
| 4. Time in Years: t = I / (P * r) |
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Converting Non-Year Time Units into Fractional Years
A frequent testing trap involves providing time in months, days, or weeks. Time $t$ must always be converted to fractional years:
| Provided Time Unit | Exact Conversion to $t$ (Years) | Example for $t$ |
|---|---|---|
| Months ($m$) | $t = \frac{m}{12}$ | 18 months $\implies t = \frac{18}{12} = 1.5\text{ years}$ |
| Days ($d$) — Exact Interest | $t = \frac{d}{365}$ | 146 days $\implies t = \frac{146}{365} = 0.40\text{ years}$ |
| Days ($d$) — Banker's Rule / Ordinary | $t = \frac{d}{360}$ | 90 days $\implies t = \frac{90}{360} = 0.25\text{ years}$ |
| Weeks ($w$) | $t = \frac{w}{52}$ | 26 weeks $\implies t = \frac{26}{52} = 0.50\text{ years}$ |
Worked Example: Solving for Principal Under Simple Interest
Problem: A corporate treasury note promises to pay a total accumulated maturity value of $A = 8500$ dollars at the end of 9 months. If the note earns an annual simple interest rate of 6.25%, what initial principal $P$ must be deposited today?
- Convert time to fractional years: $t = \frac{9}{12} = 0.75\text{ years}$.
- Rearrange the future value formula for principal $P$:
2. Periodic Compound Interest Mechanics
Unlike simple interest, compound interest credits interest payments to the principal balance at regular conversion intervals. In subsequent periods, interest is earned on both the initial principal and all previously accrued interest.
The Compound Interest Formula
Where:
- $P$ = Principal amount (present value)
- $r$ = Annual nominal interest rate (in decimal form)
- $n$ = Number of compounding conversion periods per year
- $t$ = Total duration in years
- $nt$ = Total number of compounding periods ($N$)
- $\frac{r}{n}$ = Periodic interest rate per conversion period ($i$)
- Total Interest Earned: $I = A - P$
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| STANDARD COMPOUNDING FREQUENCIES (n) |
| |
| Frequency Terminology Periods per Year (n) Periodic Rate (i = r/n)|
| -------------------------- -------------------- ---------------------- |
| Annually n = 1 i = r / 1 |
| Semiannually n = 2 i = r / 2 |
| Quarterly n = 4 i = r / 4 |
| Monthly n = 12 i = r / 12 |
| Weekly n = 52 i = r / 52 |
| Daily n = 365 i = r / 365 |
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Worked Example: Quarterly Compounding
Problem: An investor places $P = 8000$ dollars into an account earning an annual nominal rate of 6% compounded quarterly ($n = 4$) for 5 years ($t = 5$). What is the future value $A$ and the total compound interest earned?
- Determine the periodic interest rate: $i = \frac{r}{n} = \frac{0.06}{4} = 0.015$.
- Determine the total number of compounding periods: $N = nt = 4 \times 5 = 20$.
- Compute accumulated future value:
- Compute total compound interest earned:
Comparison with Simple Interest: At 6% simple interest, $I_{\text{simple}} = 8000 \times 0.06 \times 5 = 2400.00$ dollars. Compounding quarterly yields an additional 374.84 dollars in interest due to reinvestment.
3. Linear vs. Exponential Growth: Long-Horizon Trajectory
The fundamental mathematical difference between simple and compound interest is the function family governing their growth curves over time:
- Simple Interest is a Linear Function: $A(t) = (Pr)t + P$, with constant slope $m = Pr$.
- Compound Interest is an Exponential Function: $A(t) = P(1 + i)^{nt} = P(b)^t$, where the base $b = (1 + r/n)^n > 1$.
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| GROWTH OF $10,000 AT 8.00% APR OVER TIME |
| |
| Years (t) Simple Interest Compounded Annually Compounded Monthly |
| --------- --------------- ------------------- ------------------ |
| 1 Year $10,800.00 $10,800.00 $10,830.00 |
| 5 Years $14,000.00 $14,693.28 $14,898.46 |
| 10 Years $18,000.00 $21,589.25 $22,196.40 |
| 20 Years $26,000.00 $46,609.57 $49,268.03 |
| 30 Years $34,000.00 $100,626.57 $109,357.30 |
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Balance ($)
^
$120k| * (Monthly Compound)
$100k| *
$80k| *
$60k| *
$40k| *
$20k| * * - - - (Simple Linear)
$0k+--------------*------------*------------*------------*------------>
0 10 20 30 40 Years (t)
Over short periods ($t \le 1$), the difference is minimal. However, as $t$ grows, exponential compounding accelerates rapidly because each new period's interest generates its own future interest in perpetuity.
4. Step-by-Step TI-30XS MultiView Calculator Operations
To ensure exact answers and avoid rounding errors on the CLEP examination, evaluate compound interest expressions directly on the TI-30XS MultiView without recording intermediate decimal steps.
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| TI-30XS KEYSTROKE PROTOCOL FOR COMPOUND INTEREST |
| |
| Mathematical Goal: Evaluate A = 5000 * (1 + 0.075 / 12)^(12 * 6) |
| |
| Method 1: Direct Single-Expression Input (MathPrint Mode) |
| Key Sequence: |
| 5 0 0 0 [ * ] [ ( ] 1 [ + ] 0 . 0 7 5 [ / ] 1 2 [ ) ] [ ^ ] [ ( ] 1 2 |
| [ * ] 6 [ ) ] [enter] |
| Display Output: 7830.5872... ===> Round to $7,830.59 |
| |
| Method 2: Register Storage Method ([sto->]) |
| Step 1: Compute Periodic Rate: 0 . 0 7 5 [ / ] 1 2 [sto->] [x] [enter] |
| Step 2: Evaluate Expression: 5 0 0 0 [ * ] [ ( ] 1 [ + ] [x] [ ) ] |
| [ ^ ] 7 2 [enter] |
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Common Test Traps to Avoid
- Unparenthesized Exponents on Older Syntax: Typing
5000 * (1.005) ^ 12 * 4without parentheses around the exponent will cause the calculator to raise the base to the $12^{\text{th}}$ power first and then multiply the entire answer by 4. Always type^ (12 * 4)or compute the total exponent 48 before keying. - Manual Rounding of Periodic Rate: If $r = 7%$ and $n = 12$, typing $0.07 / 12 \approx 0.0058$ discards significant repeating digits ($0.0058333\dots$). Over 360 periods, this truncation can introduce discrepancies of hundreds of dollars.
- Mixing Rate Units with Time Units: If interest rate is quoted annually, time $t$ must be in years. Do not input $t = 18$ when a problem states 18 months.
A small business owner secures a short-term working capital loan of $12,000 at an annual simple interest rate of 7.5%. The loan terms require full repayment of principal and interest after 18 months. How much total interest is due, and what is the total accumulated repayment amount?
An investor deposits $5,000 into a certificate of deposit (CD) earning an annual nominal interest rate of 6% compounded monthly (n = 12). What will be the total accumulated balance in the account at the end of 4 years?
A saver has $10,000 to invest for 5 years at an annual interest rate of 8%. How much more interest will be earned if the funds are placed in an account compounding quarterly (n = 4) compared to an account paying simple interest at the same rate?