Simple Interest — Hard Practice Quiz

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

Formulas & key concepts

Simple interest earned, calculated only on the original principal. Here \(I\) is the total interest, \(P\) the principal (present value), \(r\) the annual nominal rate written as a decimal, and \(t\) the time in years.

$$I = P \times r \times t$$

The ending balance (future value) under simple interest. \(A\) is the future value, \(P\) the principal, \(r\) the annual rate, and \(t\) the time in years.

$$A = P(1 + rt)$$

The principal that grows to a given future value \(A\) under simple interest, where \(r\) is the annual rate and \(t\) the time in years.

$$P = \dfrac{A}{1 + rt}$$

Practice quiz

  1. An investor places a principal $P$ into an account earning simple interest at an annual rate $r$ for $t$ years. The total interest earned is $I$, and the future value is $A$. If the investor doubles the initial principal to $2P$ while keeping the rate $r$ and time $t$ constant, how will the new total interest earned $I'$ and the new future value $A'$ compare to the original $I$ and $A$?

    • $I' = 2I$ and $A' = 2A$
    • $I' = 2I$ and $A' = A + P(1+rt)$
    • $I' = 2I$ and $A' = P(1+2rt)$
    • $I' = I + Prt$ and $A' = 2A$

    Answer: $I' = 2I$ and $A' = 2A$

  2. An initial investment of $P_1$ grows to a future value of $A_1$ in $t_1$ years under simple interest. If a different principal $P_2$ is invested at the same annual simple interest rate for $t_2$ years, what will be the total interest $I_2$ earned?

    • $I_2 = P_2 \frac{A_1 - P_1}{P_1 t_1} t_2$
    • $I_2 = P_2 \frac{A_1}{P_1 t_1} t_2$
    • $I_2 = (A_1 - P_1) \frac{P_2 t_2}{P_1 t_1}$
    • $I_2 = (A_1 - P_1) \frac{P_2}{P_1} + P_2 \frac{t_2}{t_1}$

    Answer: $I_2 = P_2 \frac{A_1 - P_1}{P_1 t_1} t_2$

  3. Investor X invests principal $P$ for time $t$ at a simple interest rate $r_X$. Investor Y invests principal $2P$ for time $t/2$ at a simple interest rate $r_Y$. If their future values are equal, which of the following relationships between $r_X$ and $r_Y$ is true?

    • $r_X = r_Y + \frac{1}{t}$
    • $r_X = 2r_Y$
    • $r_X = r_Y - \frac{1}{t}$
    • $r_X = \frac{1}{2} r_Y + \frac{1}{t}$

    Answer: $r_X = r_Y + \frac{1}{t}$

  4. An investment grows to a future value of $A$ in $t$ years, earning a total simple interest of $I$. Which expression correctly represents the annual simple interest rate $r$?

    • $r = \frac{I}{(A-I)t}$
    • $r = \frac{A-I}{It}$
    • $r = \frac{A}{It} - \frac{1}{t}$
    • $r = \frac{A}{Pt}$

    Answer: $r = \frac{I}{(A-I)t}$

  5. A principal $P$ is invested at a simple interest rate $r$. How many years $t$ will it take for the total interest earned to be equal to the initial principal $P$?

    • $t = \frac{1}{r}$
    • $t = r$
    • $t = \frac{P}{r}$
    • $t = \frac{2}{r}$

    Answer: $t = \frac{1}{r}$

  6. An investor wants to accumulate a future value of $A$ in $t$ years. They have $P_0$ available to invest, which is only half of the required principal $P$ to reach $A$ at rate $r$ in time $t$. If they invest $P_0$ at the same rate $r$, how much additional time $t_{add}$ (beyond the original $t$) would be required to reach the target future value $A$?

    • $t_{add} = t + \frac{1}{r}$
    • $t_{add} = t$
    • $t_{add} = \frac{1}{r}$
    • $t_{add} = 2t - \frac{1}{r}$

    Answer: $t_{add} = t + \frac{1}{r}$

  7. An investor divides a total principal $P_{total}$ into two parts: $P_1$ and $P_2$, such that $P_1 = 2P_2$. $P_1$ is invested at a simple interest rate $r_1$ for $t$ years, and $P_2$ is invested at a simple interest rate $r_2$ for the same $t$ years. If the total interest earned from both investments is $I_{total}$, what is the effective overall simple interest rate $r_{eff}$ for the entire $P_{total}$ over time $t$?

    • $r_{eff} = \frac{2r_1 + r_2}{3}$
    • $r_{eff} = \frac{r_1 + r_2}{2}$
    • $r_{eff} = \frac{r_1 + 2r_2}{3}$
    • $r_{eff} = \frac{r_1 r_2}{r_1 + r_2}$

    Answer: $r_{eff} = \frac{2r_1 + r_2}{3}$

  8. An investment earns a simple interest $I$ over $t$ years at an annual rate $r$. If the principal is increased by $50\%$ and the time period is halved, what is the new simple interest $I'$ earned in terms of the original interest $I$?

    • $I' = 0.75 I$
    • $I' = 1.5 I$
    • $I' = 0.5 I$
    • $I' = 1.25 I$

    Answer: $I' = 0.75 I$

  9. An investor wants to achieve a future value $A$ by investing a principal $P$ for $t$ years at a simple interest rate $r$. If they decide to invest for $2t$ years instead, and the principal available is $P/2$, what new simple interest rate $r'$ would be required to achieve the same future value $A$?

    • $r' = r + \frac{1}{2t}$
    • $r' = r - \frac{1}{2t}$
    • $r' = 2r + \frac{1}{t}$
    • $r' = \frac{r}{2} + \frac{1}{t}$

    Answer: $r' = r + \frac{1}{2t}$

  10. An investment of $P$ dollars is made for $t$ years. If the simple interest rate is $r_1$, the future value is $A_1$. If the simple interest rate is $r_2$, the future value is $A_2$. If $A_1 = 2A_2$, and $r_1 = 3r_2$, what is the relationship between $t$ and $r_2$?

    • $t = \frac{1}{r_2}$
    • $t = \frac{1}{2r_2}$
    • $t = \frac{2}{r_2}$
    • $t = \frac{3}{r_2}$

    Answer: $t = \frac{1}{r_2}$

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