1. Investment Allocation

I am really having trouble breaking this problem down into a formula. It just doesn't feel like there is enough information, but I know that there is. My instructor provides no examples and the book has none on this one specifically. I am hoping someone can help me solve it, or atleast get it started. Any help much appreciated.

A company invests a total of \$30,000 of surplus funds at two annual rates of interest: 5% and 6.75%. It wishes an annual yield of no less than 6.5%. What is the least amount of money that the company must invest at the 6.75% rate.

2. Let x denote the amount invested at 6.75%. That means that (30,000 - x) will be invested at 5%.

The allocation of funds between the two investments which would produce an aggregate return of exactly 6.5% would be described by the equation

0.065 = 0.0675x + 0.05(30,000 - x). Solving for x gives the amount invested at 6.75% (with the remainder invested at 5%) that would yield exactly 6.5%.

Technically, the question would be modeled as an inequality, since the company needs the portfolio to yield at least 6.5%...

0.065 <= 0.0675x + 0.05(30,000 - x) which would give you an answer of the form that x must be >= to some amount. But in this case, after solving the equality, it's common sense that this would represent the minimum that must be invested at 6.75%, because any shifting of funds from the 6.75% over to the 5% would decrease the overall yield below the required target.

3. Thank you SO much!

4. Very glad to help, Rizz....and I'm sure you've caught my typo by now; viz

30,000(0.065) = 0.0675x + 0.05(30,000 - x)

...is a bit more like it.

Also, be sure to check your result to verify that it's correct.

Side note: If it happens that "two equations and two unknowns" is more your cup of tea, you could also have attacked it that way. Letting x and y denote the amounts invested at 6.75% and 5%, respectively, your problem tells you two things:

1. The total amount invested is 30K; hence x + y = 30,000
2. The required portfolio yield is 6.5%; thus 0.0675x + .05y = 1,950

But if you solve it this way, you'll quickly see that it really amounts to the same thing as the first method; i.e., you'll see a lot of similarities in your resulting expressions.

5. Yes, I did catch the type-o but the overall concept was exactly what I needed. I have a long string of homework problems and I just needed to figure out how it was broken down. Thanks for the other method as well they are equally useful in different scenarios.