This problem involves solving a system of equations based on simple interest.

We are given the following:

• Phyllis invested $48,000 in total.
• A part of the money earns 5% simple interest, and the other part earns 6%.
• After one year, the total interest earned is $2,650.

Let:

• $x$ be the amount invested at 5%.
• $48000 - x$ be the amount invested at 6%.

We know that:

• The interest from the 5% investment is $0.05x$.
• The interest from the 6% investment is $0.06(48000 - x)$.

The total interest is given as $2,650, so we can set up the equation:

$0.05x + 0.06(48000 - x) = 2650$

Now, let's solve this equation step by step.

Step 1: Distribute the 0.06

$0.05x + 0.06(48000) - 0.06x = 2650$ $0.05x + 2880 - 0.06x = 2650$

Step 2: Combine like terms

$(0.05x - 0.06x) + 2880 = 2650$ $-0.01x + 2880 = 2650$

Step 3: Move the 2880 to the right side

$-0.01x = 2650 - 2880$ $-0.01x = -230$

Step 4: Solve for $x$

$x = \frac{-230}{-0.01} = 23000$

So, Phyllis invested $23,000 at 5%.

Step 5: Find the amount invested at 6%

$48000 - 23000 = 25000$

Thus, Phyllis invested:

• $23,000 at 5%, and
• $25,000 at 6%.

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore further:

1. What is simple interest, and how does it differ from compound interest?
2. How would the problem change if the interest were compounded annually instead of simple interest?
3. Can you solve the problem using a graph of the two interest rates?
4. What would happen if the total amount invested was different, say $60,000?
5. How do different interest rates affect investment decisions?

Tip: To check if your solution is correct, always verify the total interest by plugging the amounts back into the interest formulas.