Compare simple and compound interest

A fictional $2,000 deposit earns 5% per year, with no fees or payments. Compare simple interest with annual compounding. After exactly one year, which amount is larger?

Keep the $2,000 principal and 5% annual rate fixed. There are no fees or payments. Try 0, 1, 2, 3 and 6 whole years. The compound model updates annually. Read the next-year interest as well as the balances

Which explanation matches the table?

Keep assumptions separate from conclusions

“Compound is better” is incomplete. State the role, rates, term, fees and payment rules. Compare the amounts at the same endpoint, then explain what your model leaves out

1. 1State the comparison These fictional plans have fixed rates, no fees and no deposits, withdrawals or repayments during the term. Interest stays in the account; any debt is paid at the end. Hold principal and term fixed. Equal quoted rates isolate the effect of the interest model, but actual offers may have different rates
2. 2Define the symbols P is principal, t years, R the annual percentage number, k compounding periods per year, r the percentage number per period and n the period count. Amounts A include principal; interest I does not
3. 3Simple model The original principal earns the same dollar interest each year \(\displaystyle \begin{gathered}I_s=PRt/100\\A_s=P+I_s\end{gathered}\)
4. 4Compound model For a nominal annual rate, divide its percentage number among the periods. Retain calculator precision until the requested final answer \(\displaystyle \begin{gathered}r=R/k,\quad n=kt\\A_c=P(1+r/100)^n\\I_c=A_c-P\end{gathered}\)
5. 5Interpret the result With all other conditions equal, a saver prefers the larger amount and a borrower the smaller debt. Equal amounts are a tie. For the same positive rate and annual compounding, the first year ties; later whole years favour compounding for a saver

Principal

\(\displaystyle \$1{,}800\)

Annual rate

\(\displaystyle 4\%\)

Term

\(\displaystyle 3\text{ years}\)

Find the final amount under simple interest. These fictional terms have no fees or payments

Calculate interest on the original principal, then add the principal

Principal

\(\displaystyle \$1{,}800\)

Annual rate

\(\displaystyle 4\%\)

Term

\(\displaystyle 3\text{ years}\)

Find the final amount under annual compounding for this separate offer. Round only the final answer to cents; no fees or payments occur

Apply the annual multiplier three times

Simple amount

\(\displaystyle \$2{,}016.00\)

Compound amount

\(\displaystyle \$2{,}024.76\)

Find how much more the compound offer pays than the simple offer. Use these rounded final amounts

Subtract the smaller amount from the larger amount

Now treat those same fictional offers as debts, repaid only at the end with no fees. Plan A ends at $2,016.00; Plan B at $2,024.76. Which is cheaper for the borrower?

Nominal annual rate

\(\displaystyle 4.8\%\)

Periods per year

\(\displaystyle 12\)

Convert this nominal annual rate to a monthly percentage rate

Allocate the annual percentage number over the twelve months

Principal

\(\displaystyle \$2{,}500\)

Nominal annual rate

\(\displaystyle 4.8\%\)

Compounding

\(\displaystyle \text{monthly}\)

Term

\(\displaystyle 2\text{ years}\)

Find total compound interest, not the final amount. The fixed rate is nominal, no payments or fees occur and interest stays invested. Round only the final interest to cents

Count all monthly updates, calculate the full balance and subtract principal

For a $2,500 offer at 4.8% nominal interest compounded monthly over two years, a student writes these lines. Which is the first error?

Principal

\(\displaystyle \$3{,}200\)

Nominal annual rate

\(\displaystyle 6\%\)

Term

\(\displaystyle 2\text{ years}\)

Find the final debt under quarterly compounding, rounded to cents. The nominal rate is fixed, no fees or repayments occur during the term, and the debt is paid at the end

Allocate the rate over four quarters and count eight updates

Two fee-free savings offers have the same $2,000 principal and three-year term. Simple interest is 5.5% p.a.; annual compounding is 5% p.a. The final amounts are $2,330.00 and $2,315.25. Which conclusion is valid?

A statement says “compound balances must be rounded to cents after every period”. Is that the model used in this lesson?

Investigate offers with a fee

A fictional saver has $2,000 for three years. Offer A uses simple interest at 5.5% p.a.; B compounds annually at 5% p.a. Neither has payments during the term. Start without fees, then give A a $20 fee deducted at the end. Build a spreadsheet or calculator table, compare outcomes and write a recommendation. Your written reasoning needs human review

1. 1Formulate Define the variables, formulas, role and end date. Explain why the unequal quoted rates can still be compared. State the no-payments and fixed-rate assumptions
2. 2Calculate and test Show both unrounded calculations and final cents. Apply the end fee to A. Find the fee at which the two offers would be equal, using their rounded fee-free amounts
3. 3Evaluate and modify Explain whether the fee changes the recommendation. Explore terms of one to six whole years using technology. If the fee were charged at the start instead, state what would need to change before reusing the model
4. 4Communicate limits Write a short report with a labelled table or graph, calculation checks and assumptions. Real products also require access, risk, repayment and fee conditions; these invented offers are mathematical models