Desiree invested some money at 9% and $100 less than three times the amount at 7%. Her total annual interest was $83. How much did she invest at each rate?

9 % = 9 / 100 = 0.09

7 % = 7 / 100 =0.07

x = the amount of money

$100 less then three times the amount = 3 x - 100

x * 0.09 + ( 3 x - 100 ) * 0.07 = 83

0.09 x + 3 x * 0.07 - 100 * 0.07 = 83

0.09 x + 0.21 x - 7 = 83

0.3 x - 7 = 83

0.3 x = 83 + 7

0.3 x = 90

x = 90 / 0.3

x = 300

3 x - 100 = 3 * 300 - 100 = 900 - 100 = 800

$300 with interest 9% and $800 with interest 7%

$300 * 0.09 + $800 * 0.07 = $27 + $56 = $83

Well, well, well, it looks like Desiree was quite the investment guru! Let's solve this riddle, shall we?

Let's assume that Desiree invested x amount of money at 9% and y amount of money at 7%. According to the problem, we know that y = 3x - $100.

Now, let's calculate the interest from each investment. The interest from the 9% investment is 0.09x, and the interest from the 7% investment is 0.07(y).

Adding those together, we have 0.09x + 0.07(y) = $83.

Now, let's substitute y with its value from the first equation:

0.09x + 0.07(3x - $100) = $83.

Now it's time for some calculation magic!

0.09x + 0.21x - $7 = $83.

Combining like terms, we have 0.30x - $7 = $83.

Adding $7 to both sides, we get 0.30x = $90.

Now, let's divide both sides by 0.30 to find x:

x = $90 / 0.30.

Grab your calculator, and you'll find that x = $300.

So, Desiree invested $300 at 9% and y = 3x - $100, which means y = 3($300) - $100.

Again, get your trusty calculator, and voila! y = $800.

So, Desiree invested $300 at 9% and $800 at 7%. That's how the cookie crumbles!

Let's assume Desiree invested x amount at 9% interest rate.

According to the given information, she invested (3x - 100) amount at 7% interest rate.

The interest earned on the amount invested at 9% is:
0.09x

The interest earned on the amount invested at 7% is:
0.07(3x - 100)

According to the given information, the sum of the two interests earned is $83. Hence, we can write the equation as follows:

0.09x + 0.07(3x - 100) = 83

Now, let's solve this equation to find the value of x.

To solve this problem, let's use algebraic equations.

Let's represent the amount Desiree invested at 9% as x dollars.
According to the problem, she invested $100 less than three times that amount at 7%. So, the amount she invested at 7% can be represented as 3x - $100.

Now, let's set up an equation for the total interest earned:
0.09x + 0.07(3x - $100) = $83

To solve this equation, we'll distribute 0.07 to 3x and -100:
0.09x + 0.21x - $7 = $83

Combining like terms:
0.3x - $7 = $83

Now, let's isolate the variable x by adding $7 to both sides of the equation:
0.3x = $90

Finally, divide both sides of the equation by 0.3 to find the value of x:
x = $90 / 0.3
x = $300

So, Desiree invested $300 at a 9% interest rate and $100 less than three times that amount ($900 - $100 = $800) at a 7% interest rate.