Solutions

Solutions are only provided for even-numbered questions.


  1. There is no solution provided for this question.
  2. Chequing accounts, savings accounts, and GICs all have guaranteed (but modest) interest rates; this also means that there is no risk associated with these accounts.

    Chequing accounts have the lowest interest rate since they offer the most convenience in terms of accessing your money with cheques, e-transfers, etc.

    Savings accounts have the next lowest rate since they still offer quick access to your money, but not with the same flexibility as chequing accounts.

    GICs have the highest interest rate of the three types of accounts with guaranteed rates since you cannot access your money for a specified period of time.

    Mutual fund returns are not guaranteed, so they have a higher risk level and they have a wide range of possible returns. In the long run, average annual returns will normally be higher than the returns from guaranteed accounts, but there may be years in which the returns are less than in a guaranteed account, or even negative.

    Therefore, we complete the graph as shown:

    A blank axis where Risk is along the horizontal and Possible Returns is along the vertical. Risk cannot be negative. The following have no risk: GICs, Savings Accounts, Chequing Accounts, where your possible returns are positive. Mutual funds are higher risk but include a range of possible returns including negative values.

    Different mutual funds have different risk levels. In general, the higher the risk level, the more variation there is in annual returns but there will also normally be higher average long-term annual returns.

    This graph shows only the rate of return, not the purchasing power. If inflation is considered, the actual purchasing power of money in chequing or savings accounts decreases in most years. 

  3. There is no solution provided for this question.
  4. Notice that the accumulated amount in Joshua's account starts lower than the accumulated amount in Zaira's account (\(1000\) compared to \(1200\)), but Joshua's accumulated amount grows more quickly since his interest rate is greater than Zaira's (\(4.5\% \) compared to \(3\%\)).

    We can use the formula \(A=P(1+rt)\) to write linear relations for each account.

    • Zaira: \(A=1200(1+0.03t)=1200+36t\)
    • Joshua: \(A=1000(1+0.045t)=1000+45t\)

    Now we have a system of equations and we find when the two accounts will have the same accumulated amount:

    \(\begin{align*} 1000 + 45t &= 1200+36t\\ 1000 +9t &= 1200\\ 9t &= 200\\ t &\approx 22.2 \end{align*}\)

    Since interest is paid to each account at the end of each year, the accumulated amount in Joshua's account will first be greater than in Zaira's account in the \(23\)rd year.

  5. There is no solution provided for this question.

  6. Let's compare each situation using the formula \(I=Prt\).

    Doubling the Principal

    • The new principal is \(2P\).
    • The new interest is \(I=(2P)rt=2Prt\)

    Doubling the Interest Rate

    • The new interest rate is \(2r\).
    • The new interest is \(I=P(2r)t=2Prt\)

    Doubling the Time

    • The new time is \(2t\).
    • The new interest is \(I=Pr(2t)=2Prt\)

    So we see that the result is the same for each situation since each scenario yields \(I=2Prt\). Doubling any one of the principal, interest rate, or time will double the amount of interest earned.

  7. There is no solution provided for this question.

  8. Option 1

    We are given:

    • \(P=4500\)
    • \(r=4.5\%=0.045\)
    • \(t=6\)

    Applying \(A=P(1+rt)\): 

    \(\begin{align*} A&=P(1+rt)\\ &=4500(1+0.045(6))\\ &=5715 \end{align*}\)

    So at the end of six years, Percy would have \($5715\) if he chooses Option 1.

    Option 2

    We need to do three separate calculations, one for each \(2\)-year GIC.

    For the first two years, we have:

    • \(P=4500\)
    • \(r=4.2\%=0.042\)
    • \(t=2\)

    Applying \(A=P(1+rt)\): 

    \(\begin{align*} A&=P(1+rt)\\ &=4500(1+0.042(2))\\ &=4878 \end{align*}\)

    So at the end of the first two years, Percy would have \($4878\).

    He reinvests this amount for another two years, so we have:

    • \(P=4878\)
    • \(r=4.2\%=0.042\)
    • \(t=2\)

    Applying \(A=P(1+rt)\): 

    \(\begin{align*} A&=P(1+rt)\\ &=4878(1+0.042(2))\\ &=5287.75 \end{align*}\)

    So at the end of the first four years, Percy would have \($5287.75\).

    He reinvests this amount for another two years so we have:

    • \(P=5287.75\)
    • \(r=4.2\%=0.042\)
    • \(t=2\)

    Applying \(A=P(1+rt)\): 

    \(\begin{align*} A&=P(1+rt)\\ &=5287.75(1+0.042(2))\\ &= 5731.92 \end{align*}\)

    So at the end of the six years, Percy would have \($5731.92\) if he chooses Option 2.

    This is slightly more than he would have with Option 1, so Option 2 is the better choice.