Businesses deposit large sums of money into bank accounts. Imagine an account with $10 million dollars in it.
a. How much would the account earn in one year of simple interest at a rate of 2.12%? Round to the nearest cent. 
$212,000.00
 
b. How much would the account earn in one year at 2.12% if the interest was compounded daily? Round to the nearest cent. 
 
c. How much more interest is earned by interest compounded daily compared to simple interest? 
 


Let's break this down:

### Part A: Simple Interest

The simple interest formula is:

\[
I = P \times r \times t
\]

Where:
- $ P $ is the principal (initial amount of money) = $10,000,000
- $ r $ is the annual interest rate = 2.12% = 0.0212
- $ t $ is the time in years = 1 year

Using the formula:

\[
I = 10,000,000 \times 0.0212 \times 1 = 212,000
\]

So the account earns **$212,000** in one year of simple interest, which is given.

### Part B: Compound Interest (Daily)

The formula for compound interest is:

\[
A = P \times \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the amount after interest
- $ P $ is the principal = $10,000,000
- $ r $ is the annual interest rate = 0.0212
- $ n $ is the number of times interest is compounded per year = 365 (daily compounding)
- $ t $ is the time in years = 1

Now, let's calculate:

\[
A = 10,000,000 \times \left(1 + \frac{0.0212}{365}\right)^{365 \times 1}
\]

I'll calculate this.The amount of interest earned with daily compounding is **$214,256.88**.

### Part C: Difference Between Compound Interest and Simple Interest

To find how much more interest is earned with daily compounding compared to simple interest, we subtract the simple interest from the compound interest:

\[
\text{Difference} = 214,256.88 - 212,000 = 2,256.88
\]

So, **$2,256.88** more is earned by compounding daily compared to simple interest.

---

Would you like more details or have any questions? 

Here are 5 related questions:
1. How does the interest amount change with monthly compounding?
2. What would the interest be after 5 years with simple interest?
3. How does a higher interest rate, say 5%, affect the results for both simple and compound interest?
4. How does continuous compounding compare to daily compounding in this case?
5. What would the interest difference be if the compounding was semi-annual instead of daily?

**Tip**: Compounded interest grows faster over time compared to simple interest due to the "interest on interest" effect.

Businesses deposit large sums of money into bank accounts. Imagine an account with $10 million dollars in it.
a. How much would the account earn in one year of simple interest at a rate of 2.12%? Round to the nearest cent.
$212,000.00

b. How much would the account earn in one year at 2.12% if the interest was compounded daily? Round to the nearest cent.

c. How much more interest is earned by interest compounded daily compared to simple interest?

Calculate the simple interest and compound interest earned on $10 million at a rate of 2.12%. Learn how simple and daily compounded interest differ, including a detailed breakdown of the formulas and an interest difference of $2,256.88. Suitable for high school students.

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