Question below: This is a practice question for a bigger test, I need an explanation so I can understand. Thanks

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 $10000 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.

Part B: Determine the value of the compound-interest investment at the end of three years. Use the formula A=P(1+r/n)^nt, 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.

Part C: Which investment is better over the first three years, why?

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.

I have most of my answers, but for some of them, I'm not sure if Im correct. I really need an explanation.

Answer 1

Hi Rada,

part 1) A = p ( 1 + rt)

A = 10000( 1 + 0.074 *3 ) = 12220

part 2) A = p ( 1 + r/n) ^nt

A = 100000( 1 + 0.065/4) ^ 3*4 = 12134.07

part 3) you see after 3 years which investment is better simple interest is better

part 4) you need to make a table

T year	Part1	Part2
4	12960	12864.6
5	13700	13700
6	14440	14591
7	15180	15539.9

Now, you can make a decision if he invested yearly and compound interest after 6 years is better than simple interest.

I hope it is useful,

Minoo

Answer 2

Hi Rada,

Thanks for the question, below is an explanation of each part!

A: This question is about simple interest and gives you the equation to solve, A = P + Prt. Our original amount, P, is $10,000. Our interest rate, r, is 7.4% or in mathematical terms 7.4/100 or 0.074. Finally, the time elapsing, t, is 3 years. Plugging these in we come to:

The Value of the Investment = A = $10,000 + $10,000 × 0.074 × 3 = $12,220.

In other words, George made $2,220 over 3 years or $740 per year.

B: This question is about compound interest and gives you the equation to solve, A = P(1 + r/n)^nt. Our original amount, P, is still $10,000. Our interest rate, r, is now 6.5% or in mathematical terms 6.5/100 or 0.065. The time elapsing, t, is still 3 years. For compound interest, we also need to know how many times the interest is compounded each year, n. The question root states that the interest "is compounded quarterly". As the word quarterly suggest, each fiscal year has four financial quarters so the number of time the interested will be compounded yearly, n, is 4. Plugging these in we come to:

The Value of the Investment = A = $10,000 × (1 + 0.065 / 4) ^{(4 × 3)}

Simplification 1: A = $10,000 x (1 + 0.01625) ⁽¹²⁾

Simplification 2: A = $10,000 x (1.01625)¹²

Simplification 3: A = $10,000 x 1.2134 (This is a rounded version of the actual value of 1.01625¹²)

Final Answer: A = $12,130 (This is rounded to the whole $ amount)

In other words, George made roughly $2,130 over 3 years. Although this averages to $710 per year, it is important to understand that George actually makes more money on his investment each year than he did the previous year. In this example, he actually made less than $710 during his first year of investing and more than $710 his third year of investing.

C: The goal of an investment is to make the most money, so over a 3 year period George is better off choosing the simply 7.4% interest that we calculated as $2,220 in part A vs. the 6.5% compound interest we calculated as $2,130 in part B.

D. This is the fun part of the question. As I mentioned in part B, with compound interest, George will make more money each year than he did before. Meanwhile, with simple interest George will make the same amount of money each year, $720.

Let's try a longer time period of 5 years for both the simple and compound interest investments. To do this we simply replace the value of t, our time period, with 5 instead of 3.

Simple Interest over 5 Years: A = $10,000 + $10,000 × 0.074 × 5 = $13,700

Over 5 years of simple interest, George makes $3,700 which still averages to $740 per year.

Compound Interest over 5 Years: A = $10,000 × (1 + 0.065 / 4) ^{(4 × 5)}

Simplification 1: A = $10,000 x (1 + 0.01625) ⁽²⁰⁾

Simplification 2: A = $10,000 x (1.01625)²⁰

Simplification 3: A = $10,000 x 1.3804 (This is a rounded version of the actual value of 1.01625¹²)

Final Answer: A = $13,800 (This is rounded to the whole $ amount)

Over 5 years of compounding interest, George makes $3,800 or $760 per year. As we said above, it is still true that George made less than $710 his first year. But, since we know he makes more money each year than he did the previous year, we can be sure that he made over $760 his 5th year of investing.

Initial conclusion: While after 3 years of investing $10,000 in each of these strategies, George made more money via simple interest, at the end 5 years his compounded interest investment has yielded more money. This question asks for you to show your results in a table or graph so below is a table showing the return on each type of investment for each of the first six years. (I chose 6 to prove that the compound interest investment will do better long-term)

Investing Strategy 1 yr (t = 1) 2 yrs (t = 2) 3 yrs (t = 3) 4 yrs (t = 4) 5 yrs (t=5) 6 yrs (t = 6)

Simple Interest $10,740, $11,480, $12,220, $12,960, $13,700, $14,440

Compound Interest $10,670, $11,380, $12,130, $12,940, $13,800, $14,720

Final Conclusion: As we can see, for the first 4 full years of the investment, George is better off using simple interest. Somewhere in his 4th year of investing however, the rate he will make through compound interest surpasses that of simple interest. (To be specific, this happens after just one quarter of the 4th year).

Answer: As a friend of George, I would recommend that George invests using compound interest if he thinks there is even a small chance of keeping his investment for 5+ years. While he will make $20 - $100 less with this strategy if he takes out his investment within the first 4 years, by the 6th year he will make $280 more through compound interest and this difference will only grow with time. I would also encourage George to commit to keeping his money in the compound interest account as long as possible. If George is positive his investment will last 4 years or less, then I would recommend the simple interest option.

I hope this helps!

Answer 3

Why don't you go ahead and provide your answers? I'm happy to help you starting with a double-check.