 A university invests $1,400,000 

at simple interest, part at 5%, half that amount at 

3.5%, and the rest at 5.5%. What is the most that the 

university can invest at 3.5% and still be guaranteed at 

least $68,000 in interest per year?

To solve this problem, we'll use the formula for simple interest:

\[
\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
\]

Since we're calculating the interest for one year, the formula simplifies to:

\[
\text{Interest} = \text{Principal} \times \text{Rate}
\]

Let:
- $ x $ be the amount invested at 5%.
- $ y $ be the amount invested at 3.5%.
- $ z $ be the amount invested at 5.5%.

According to the problem:
- The total amount invested is $1,400,000:

\[
x + y + z = 1{,}400{,}000
\]

- Half of the amount invested at 5% is invested at 3.5%:

\[
y = \frac{x}{2}
\]

- The university wants at least $68,000 in interest per year:

\[
0.05x + 0.035y + 0.055z \geq 68{,}000
\]

Now, let's solve the problem step by step.

1. **Substitute $ y = \frac{x}{2} $ into the equation** $ x + y + z = 1{,}400{,}000 $:

\[
x + \frac{x}{2} + z = 1{,}400{,}000
\]

2. **Combine like terms**:

\[
\frac{3x}{2} + z = 1{,}400{,}000
\]

\[
z = 1{,}400{,}000 - \frac{3x}{2}
\]

3. **Substitute $ y = \frac{x}{2} $ and $ z = 1{,}400{,}000 - \frac{3x}{2} $ into the interest equation**:

\[
0.05x + 0.035\left(\frac{x}{2}\right) + 0.055\left(1{,}400{,}000 - \frac{3x}{2}\right) \geq 68{,}000
\]

4. **Simplify the equation**:

\[
0.05x + 0.0175x + 0.055(1{,}400{,}000) - 0.055\left(\frac{3x}{2}\right) \geq 68{,}000
\]

\[
0.0675x + 77{,}000 - 0.0825x \geq 68{,}000
\]

5. **Combine like terms**:

\[
-0.015x + 77{,}000 \geq 68{,}000
\]

6. **Solve for $ x $**:

\[
-0.015x \geq 68{,}000 - 77{,}000
\]

\[
-0.015x \geq -9{,}000
\]

\[
x \leq \frac{9{,}000}{0.015}
\]

\[
x \leq 600{,}000
\]

Since $ y = \frac{x}{2} $, the maximum amount that can be invested at 3.5% is:

\[
y = \frac{600{,}000}{2} = 300{,}000
\]

So, the most the university can invest at 3.5% and still guarantee at least $68,000 in interest per year is **$300,000**.

Would you like further details or have any other questions?

Here are some related questions you might find interesting:

1. What if the total interest required was $70,000? How would that affect the maximum investment at 3.5%?
2. How would the solution change if the interest rates were different, for example, 4% and 6% instead of 3.5% and 5.5%?
3. What would be the effect of increasing the total investment to $1,500,000?
4. Can the university invest the entire amount at one interest rate to meet the $68,000 requirement?
5. How would compound interest affect the calculation if the investment was for more than one year?

**Tip:** When dealing with problems involving multiple interest rates, it's often helpful to express one or more variables in terms of another to reduce the complexity of the problem.

Learn how to calculate the maximum amount that can be invested at 3.5% to guarantee at least $68,000 in interest per year. This problem involves simple interest calculations with different interest rates and a total investment of $1,400,000.

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