Question

I have this practice question from my ACT prep guide, it’s from pre calc I believe. (Note: This is a lengthy problem)Below you will find the questions to this problem ( includes 5 questions )1. What is the balance of Albert’s $2000 after 10 years? 2. What is the balance of Marie’s $2000 after 10 years? 3. What is the balance of Han’s $2000 after 10 years?4. What is the balance of Max’s $2000 after 10 years? And lastly, 5. Who is $10,000 richer at the end of the competition?

Answer 1

.We have to calculate the future value of the portfolio of each of the participants.

If we have many investments, the future value is equal to the sum of the future values of each of the investments.

1) Albert:

His investments are:

• $1000 earned 1.2% annual interest compounded monthly.

,

• $500 lost 2% over the course of 10 years.

,

• $500 grew compounded continuosly at rate of 0.8% annually.

The first investment has a future value that can be calculated as:

`\begin{gathered} FV=PV(1+(r)/(m))^(n\cdot m) \\ FV=1000\cdot(1+(0.012)/(12))^(10\cdot12) \\ FV=1000\cdot(1+0.001)^(120) \\ FV=1000\cdot1.001^(120) \\ FV\approx1000\cdot1.12743 \\ FV\approx1127.43 \end{gathered}`

The second investment is a lost: he lost 2% in the 10-year period. Then, the investment will be 98% of the original investment. We can then calculate the future value as:

`\begin{gathered} FV=PV\cdot(1-0.02) \\ FV=500\cdot0.98 \\ FV=490 \end{gathered}`

His last investment is compounded continously, so it can be calculated as:

`\begin{gathered} FV=PV\cdot e^(rt) \\ FV=500\cdot e^(0.008\cdot10) \\ FV=500\cdot e^(0.08) \\ FV\approx500\cdot1.083287 \\ FV\approx541.64 \end{gathered}`

Then, the future value of its portfolio can be calculated as the sum of the future value of each of the investments:

`\begin{gathered} FV_p=FV_1+FV_2+FV_3 \\ FV_p=1127.43+490+541.64 \\ FV_p=2159.10 \end{gathered}`

The portfolio of Albert at the end of the 10 years will be $2159.10.

2) Marie:

Her investments are:

• $1500 earned 1.4% annual interest compounded quarterly (m = 4).

,

• $500 gained 4% over the course of 10 years. In this case, the total variation is just 4%.

We then can calculate the future value of each of the investments as:

`\begin{gathered} FV_1=PV\cdot(1+(r)/(m))^(m\cdot n) \\ FV_1=1500\cdot(1+(0.014)/(4))^(4\cdot10) \\ FV_1=1500\cdot(1.0035)^(40) \\ FV_1\approx1500\cdot1.15 \\ FV_1\approx1725 \end{gathered}`
`\begin{gathered} FV_2=PV\cdot(1+\alpha) \\ FV_2=500\cdot(1+0.04) \\ FV_2=500\cdot1.04 \\ FV_2=520 \end{gathered}`

Then, we can calculate the portfolio value as:

`\begin{gathered} FV_p=FV_1+FV_2 \\ FV_p=1725+520 \\ FV_p=2245 \end{gathered}`

The portfolio of Marie will have a value of $2245.

3) Han:

In this case, the portfolio is one investment of $2000 that is compounded continuously at a rate of 0.9% annually.

Then, the value of the portfolio can be calculated as:

`\begin{gathered} FV_p=PV\cdot e^(rt) \\ FV_p=2000\cdot e^(0.009\cdot10) \\ FV_p=2000\cdot e^(0.09) \\ FV_p\approx2000\cdot1.09417 \\ FV_p\approx2188.35 \end{gathered}`

The portfolio of Han will have a value of $2188.35.

4) Max:

In this case, we have 2 investments:

• $1000 that decrease in value exponentially at a rate of 0.5% annually.

,

• $1000 earned 1.8% annual interest compounded biannually (m = 2).

The first investment has negative interest rate (r = -0.005), so we can calculate the final value as:

`\begin{gathered} FV_1=PV\cdot e^(rt) \\ FV_1=1000\cdot e^(-0.005\cdot10) \\ FV_1=1000\cdot e^(-0.05) \\ FV_1\approx1000\cdot0.951229 \\ FV_1\approx951.23 \end{gathered}`

The second investment can be calculated as: