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The problem involves calculating how much money a business invested in two accounts at different interest rates, given the total interest earned after one year.

Let's define the variables:
- Let $ x $ be the amount invested at the interest rate of $ 4\frac{1}{2}\% $ (or 4.5%).
- The remaining amount, $ 10,000 - x $, is invested at $ 5\% $.
- The total interest earned after one year is $480.

The simple interest formula is:

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

Where:
- $ I $ is the interest,
- $ P $ is the principal (amount invested),
- $ r $ is the rate (expressed as a decimal),
- $ t $ is the time in years.

Using this formula for each account:

1. For the account invested at 4.5%:
\[
I_1 = x \times 0.045 \times 1 = 0.045x
\]

2. For the account invested at 5%:
\[
I_2 = (10,000 - x) \times 0.05 \times 1 = 0.05(10,000 - x)
\]

The total interest earned from both accounts is $480, so:

\[
0.045x + 0.05(10,000 - x) = 480
\]

Let's solve this equation for $ x $.

\[
0.045x + 0.05(10,000 - x) = 480
\]

Simplifying:

\[
0.045x + 500 - 0.05x = 480
\]

\[
-0.005x + 500 = 480
\]

\[
-0.005x = 480 - 500
\]

\[
-0.005x = -20
\]

\[
x = \frac{-20}{-0.005} = 4000
\]

So, the business invested **$4,000** at 4.5%.

The remaining amount is:

\[
10,000 - 4,000 = 6,000
\]

Thus, the business invested **$6,000** at 5%.

### Final Answer:
- $4,000 was invested at 4.5%.
- $6,000 was invested at 5%.

Would you like more details, or do you have any further questions?

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Here are some related questions for further exploration:

1. What is the total interest after 2 years with the same investments?
2. How would the investments change if the interest rates were swapped?
3. What if the total investment was increased to $15,000 but the interest rate remained the same?
4. Can the same method be used for compounded interest? If so, how would the equations change?
5. How does the overall return change if the interest rates decrease by 1%?

**Tip:** When solving problems involving two accounts or two unknowns, setting up a system of equations based on known values helps find the solution efficiently.

A business invests a total of $10,000 at 4.5% and 5% simple interest. During one year, the two accounts earn $480. How much (in dollars) did the business invest in each account?

Learn how to calculate how much was invested in two accounts at different interest rates using the simple interest formula. A business invests $10,000 at 4.5% and 5%, earning $480 in one year. Suitable for students in Grades 9-11.

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