Question

George is considering two different investment options. The first option offers 7.4% per year simple interest on the

initial deposit. The second option offers a 6.5% interest rate but is compounded quarterly. He may not withdraw any of
the money for three years after the initial deposit. Once the minimum 3 years is reached, he can choose to withdraw his
money or continue to collect interest. Suppose that George opens one of each type of account and deposits $10,000
into each.
Part A: Determine the value of the simple interest investment at the end of three years. Use the formula
A = P + Prt, where A represents the value of the investment, P represents the original amount, r represents the
rate, and t represents the time in years. Show your work.
Part B: Determine the value of the compound-interest investment at the end of three years. Use the formula
A = P(1+) , where A represents the value of the investment, P represents the original amount, r represents
the rate compounded n times per year, and t represents the time in years.
Show your work.
Part : Which investment is better over the first three years?
Explain your answer by using your work from Parts A and B as support.
Part D: How would you advise George to invest his money if he is unsure how long he will keep the money in the
account? Justify your reasoning using a graph or table.

Answer 1

Part A: The value of the simple interest investment at the end of three years is $12,220

Part B: The value of the compounded quarterly interest investment at the end of three years is $12,134.08

Part C: The simple interest investment is better over the first three years

Part D: I advise George to invest his money in the compounded interest investment if he will keep the money for a long time

Explanation:

Part A:

A = P + P r t, where

• A represents the value of the investment
• P represents the original amount
• r represents the rate in decimal
• t represents the time in years

∵ George deposits $10,000

∴ P = 10,000

∵ First option offers 7.4% per year simple interest

∴ r = 7.4% = 7.4 ÷ 100 = 0.074

∵ He may not withdraw any of the money for three years after

the initial deposit

∴ t = 3

- Substitute all of these values in the formula above

∴ A = 10,000 + 10,000(0.074)(3)

∴ A = 10,000 + 2,220

∴ A = 12,220

The value of the simple interest investment at the end of three years is $12,220

Part B:

`A=P(1+(r)/(n))^(nt)`, where

• A represents the value of the investment
• P represents the original amount
• r represents the rate in decimal
• n is a number of periods of a year
• t represents the time in years

∵ George deposits $10,000

∴ P = 10,000

∵ The second option offers a 6.5% interest rate compounded quarterly

∴ r = 6.5% = 6.5 ÷ 100 = 0.065

∴ n = 4 ⇒ quarterly

∵ He may not withdraw any of the money for three years after

the initial deposit

∴ t = 3

- Substitute all of these values in the formula above

∴
`A=10,000(1+(0.065)/(4))^((4)(3))`

∴
`A=10,000(1.01625)^(12)`

∴ A = 12,134.08

The value of the compounded quarterly interest investment at the end of three years is $12,134.08

Part C:

∵ 12,220 > 12,134.08

∴ The simplest interest investment is better than the compounded

interest investment at the end of three years

The simple interest investment is better over the first three years

Part D:

I advise George to invest his money in the compounded interest investment if he will keep the money for a long time

Look to the attached graph below

• The red line represents the simple interest investment
• The blue curve represents the compounded interest investment
• (Each 1 unit in the vertical axis represents $1000)
• After 0 years and before 4.179 years the red line is over the blue curve, that means the simple interest is better because it gives more money than the compounded interest
• After that the blue curve is over the red line that means the compounded quarterly is better because it gives more money than the simple interest