Compound Interest — Hard Practice Quiz

A Financial Math cheat sheet for Compound Interest — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

The future value when interest compounds each period. \(A\) is the future value, \(P\) the principal, \(i = r/n\) the periodic rate, and \(N = nt\) the total number of periods, where \(r\) is the annual rate, \(n\) the compounding periods per year, and \(t\) the years.

$$A = P(1 + i)^{N} \quad\text{or}\quad A = P\left(1 + \dfrac{r}{n}\right)^{nt}$$

The principal (present value) needed today to reach a future value \(A\) with compounding, where \(i = r/n\) is the periodic rate and \(N = nt\) the total number of periods.

$$P = \dfrac{A}{(1 + i)^{N}} \quad\text{or}\quad P = \dfrac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$$

The future value under continuous compounding, using \(e \approx 2.718\). \(P\) is the principal, \(r\) the annual rate, and \(t\) the time in years.

$$A = Pe^{rt}$$

Practice quiz

  1. An investor has two options for an initial principal $P$ invested for $10$ years. Option A offers an annual rate of $5.0\%$ compounded quarterly. Option B offers an annual rate of $4.9\%$ compounded continuously. Which option yields a higher future value, and by what approximate percentage of the initial principal $P$?

    • Option A by approximately $1.13\%$ of $P$
    • Option B by approximately $1.13\%$ of $P$
    • Option A by approximately $0.70\%$ of $P$
    • Option B by approximately $0.70\%$ of $P$

    Answer: Option A by approximately $1.13\%$ of $P$

  2. An investment of $10,000$ needs to grow to $15,000$ in $5$ years. If the interest is compounded semi-annually, what annual interest rate $r$ (to two decimal places) is required?

    • Approximately $8.28\%$
    • Approximately $8.16\%$
    • Approximately $7.95\%$
    • Approximately $8.42\%$

    Answer: Approximately $8.28\%$

  3. How many full years will it take for an investment to triple in value if it earns an annual interest rate of $6\%$ compounded monthly?

    • $18$ years
    • $19$ years
    • $20$ years
    • $17$ years

    Answer: $19$ years

  4. An individual wants to have $50,000$ in $8$ years. In Scenario 1, they invest in an account offering $4\%$ annual interest compounded semi-annually. In Scenario 2, they find a better account offering $5\%$ annual interest compounded quarterly. How much less principal would they need to invest initially in Scenario 2 compared to Scenario 1 to reach the same future value?

    • Approximately $2,821.70$
    • Approximately $2,500.00$
    • Approximately $3,150.50$
    • Approximately $2,980.20$

    Answer: Approximately $2,821.70$

  5. An initial investment of $20,000$ is made. Option X offers $6\%$ annual interest compounded monthly for $7$ years. Option Y offers $5.95\%$ annual interest compounded continuously for $7$ years. Which option yields a higher future value, and what is the difference (to the nearest dollar)?

    • Option X by approximately $75$
    • Option Y by approximately $75$
    • Option X by approximately $120$
    • Option Y by approximately $120$

    Answer: Option X by approximately $75$

  6. Consider an investment with a fixed principal $P$, annual rate $r$, and time $t$. If the number of compounding periods per year, $n$, increases indefinitely, what happens to the future value $A$?

    • $A$ increases without bound.
    • $A$ approaches a finite limit, which is $Pe^{rt}$.
    • $A$ decreases and approaches $P$.
    • $A$ remains constant after a certain point.

    Answer: $A$ approaches a finite limit, which is $Pe^{rt}$.

  7. Given the formula for future value under continuous compounding, $A = Pe^{rt}$, express the principal $P$ in terms of $A$, $r$, and $t$.

    • $P = Ae^{rt}$
    • $P = \frac{A}{e^{rt}}$
    • $P = A - e^{rt}$
    • $P = A \ln(rt)$

    Answer: $P = \frac{A}{e^{rt}}$

  8. A student needs $25,000$ for college tuition in $4$ years. They initially plan to invest in an account with $3.5\%$ annual interest compounded monthly. If, after $2$ years, the interest rate drops to $3\%$ compounded monthly, how much more would they need to invest at that point to still reach their $25,000$ goal in the remaining $2$ years?

    • Approximately $206.00$
    • Approximately $185.00$
    • Approximately $225.00$
    • Approximately $190.00$

    Answer: Approximately $206.00$

  9. An investment $P_1$ is made for time $t_1$ at an annual rate $r_1$ with continuous compounding, yielding a future value $A_1$. If a new investment $P_2 = 2P_1$ is made for time $t_2 = t_1/2$ at the same rate $r_1$, what is the ratio of the new future value $A_2$ to the original future value $A_1$?

    • $2e^{r_1 t_1/2}$
    • $2e^{-r_1 t_1/2}$
    • $e^{r_1 t_1/2}$
    • $e^{-r_1 t_1/2}$

    Answer: $2e^{-r_1 t_1/2}$

  10. What annual interest rate, compounded quarterly, would yield the same future value as an annual rate of $4.8\%$ compounded continuously, over any given time period $t$?

    • Approximately $4.83\%$
    • Approximately $4.78\%$
    • Approximately $4.90\%$
    • Approximately $4.80\%$

    Answer: Approximately $4.83\%$

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