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 a \$10,000 investment over 7 years. Option A offers an annual interest rate of $5.5\%$ compounded quarterly. Option B offers an annual interest rate of $5.3\%$ compounded continuously. What is the difference in the future value between Option A and Option B, rounded to the nearest cent? (Option A - Option B)

    • The difference is \$165.82
    • The difference is \$54.96
    • The difference is \$166.21
    • The difference is \$-165.82

    Answer: The difference is \$165.82

  2. A future value of \$50,000 is desired in 10 years. The investment account offers an annual interest rate of $6\%$ compounded semi-annually. However, after 4 years, a withdrawal of \$10,000 will be made from the account. What initial principal must be invested today to achieve the \$50,000 target? Round to the nearest cent.

    • \$35,577.88
    • \$19,789.70
    • \$33,220.55
    • \$27,683.79

    Answer: \$35,577.88

  3. An investment of \$15,000 grows to \$25,000 in an account that compounds interest monthly at an annual rate of $4.8\%$. How many years did it take for the investment to reach this value? Round your answer to two decimal places.

    • 10.66 years
    • 0.91 years
    • 127.96 years
    • 10.90 years

    Answer: 10.66 years

  4. 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 (approaching continuous compounding), what happens to the future value $A$?

    • The future value $A$ approaches a finite limit, $Pe^{rt}$.
    • The future value $A$ increases indefinitely.
    • The future value $A$ decreases and approaches $P$.
    • The future value $A$ remains constant, as $n$ only affects the periodic rate.

    Answer: The future value $A$ approaches a finite limit, $Pe^{rt}$.

  5. An investment offers an annual nominal rate of $7\%$ compounded daily (use $n=365$). Calculate the Effective Annual Rate (EAR) to four decimal places. Then, use this EAR to find the future value of a \$20,000 investment after 5 years, assuming interest is compounded annually at the EAR. Round the final future value to the nearest cent.

    • \$28,380.40
    • \$28,051.03
    • \$31,218.44
    • \$8,441,590,313,158,517,000,000,000,000,000,000,000,000,000,000,000,000,000,000.00

    Answer: \$28,380.40

  6. A principal of \$5,000 is invested for 3 years at an annual rate of $4\%$ compounded semi-annually. At the end of these 3 years, the entire accumulated amount is reinvested for another 5 years at an annual rate of $6\%$ compounded monthly. What is the final future value of the investment? Round to the nearest cent.

    • \$7,595.12
    • \$7,586.36
    • \$7,535.30
    • \$12,375.06

    Answer: \$7,595.12

  7. An initial investment of \$8,000 grows to \$12,000 in 6 years with continuous compounding. What is the annual interest rate $r$? Give your answer as a percentage rounded to two decimal places.

    • 6.76\%
    • 6.99\%
    • 2.93\%
    • 8.33\%

    Answer: 6.76\%

  8. A company needs to accumulate \$100,000 in 8 years for a new project. They plan to make an initial investment today in an account earning $5\%$ annual interest compounded quarterly. Additionally, they anticipate needing to deposit an extra \$15,000 into the account at the end of the 3rd year. What initial principal must the company invest today to meet their \$100,000 goal? Round to the nearest cent.

    • \$54,275.78
    • \$57,118.65
    • \$59,703.36
    • \$80,121.04

    Answer: \$54,275.78

  9. An investment is made with continuous compounding. If the annual interest rate $r$ is halved, how does the time $t$ required for the investment to double change?

    • The time $t$ doubles.
    • The time $t$ is halved.
    • The time $t$ remains the same.
    • The time $t$ increases by a factor of $\ln(2)$.

    Answer: The time $t$ doubles.

  10. An investment of \$20,000 grows to \$35,000 in 5 years. If the interest is compounded semi-annually, what is the annual interest rate $r$? Give your answer as a percentage rounded to two decimal places.

    • 11.51\%
    • 11.84\%
    • 23.69\%
    • 15.00\%

    Answer: 11.51\%

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