To get the necessary funds for a planned expansion, a small company took out three loans totaling $25,000. The company was able to borrow some money at 8%. It borrowed $2000 more than half the amount of the 8% loan at 10% and the rest at 9%. The total annual interest was $2200. How much did the company borrow at each rate?

x+y+z=25000

y = x/2 + 2000
.08x + .10y + .09z = 2200

(x,y,z) = (14000,9000,2000)

Let's start solving the problem step-by-step.

Step 1: Let's assume the amount borrowed at 8% as 'x'.

Step 2: The company borrowed $2000 more than half the amount of the 8% loan at 10%.

So, the amount borrowed at 10% = (1/2 * x) + $2000.

Step 3: The rest of the loan (25,000 - x - [(1/2 * x) + $2000]) was borrowed at 9%.

Step 4: The total annual interest was $2200.

The interest on the loan at 8% = (0.08 * x).
The interest on the loan at 10% = (0.10 * ((1/2 * x) + $2000)).
The interest on the loan at 9% = (0.09 * (25,000 - x - [(1/2 * x) + $2000])).

Step 5: Summing up all the interest, we have:

(0.08 * x) + (0.10 * ((1/2 * x) + $2000)) + (0.09 * (25,000 - x - [(1/2 * x) + $2000])) = $2200.

Now, let's solve this equation to find the values of 'x' and the other amounts.

(0.08 * x) + (0.10 * ((1/2 * x) + $2000)) + (0.09 * (25,000 - x - [(1/2 * x) + $2000])) = $2200.

Simplifying the equation further:

0.08x + 0.10((1/2)x + 2000) + 0.09(25000 - x - (1/2)x - 2000) = 2200.

0.08x + 0.10(0.5x + 2000) + 0.09(23000 - 1.5x) = 2200.

0.08x + 0.05x + 200 + 0.09(23000 - 1.5x) = 2200.

0.08x + 0.05x + 200 + 2070 - 0.135x = 2200.

0.195x + 2270 - 0.135x = 2200.

0.06x = 30.

x = 30/0.06 = 500.

Hence, the company borrowed $500 at 8%.

The amount borrowed at 10% = (1/2 * x) + $2000 = (1/2 * 500) + 2000 = 250 + 2000 = $2250.

The remaining amount borrowed at 9% = 25,000 - x - [(1/2 * x) + $2000] = 25000 - 500 - (250 + 2000) = 25000 - 500 - 2250 = $22,250.

Therefore, the company borrowed $500 at 8%, $2250 at 10%, and $22,250 at 9%.

To solve this problem, we need to set up a system of equations based on the given information.

Let's represent the amount borrowed at 8% as "x".
According to the problem, the company borrowed $2000 more than half the amount borrowed at 8% at 10%. So, the amount borrowed at 10% is (1/2)x + $2000.
The remaining amount is borrowed at 9%, which can be represented as the difference between the total borrowed and the sum of the other two loans, which is $25000 - (x + (1/2)x + $2000).

Now, we can set up the equation for the total interest earned:
0.08x + 0.10((1/2)x + $2000) + 0.09($25000 - (x + (1/2)x + $2000)) = $2200

Simplifying this equation step by step:
0.08x + 0.10(0.5x + $2000) + 0.09($25000 - x - (0.5x + $2000)) = $2200
0.08x + 0.10(0.5x + $2000) + 0.09($25000 - x - 0.5x - $2000) = $2200
0.08x + 0.10(0.5x + $2000) + 0.09($25000 - 1.5x - $2000) = $2200
0.08x + 0.10(0.5x + $2000) + 0.09($25000 - 1.5x - $2000) = $2200
0.08x + 0.10(0.5x + $2000) + 0.09($25000 - 1.5x - $2000) = $2200
0.08x + 0.05x + $1000 + 0.09($25000 - 1.5x - $2000) = $2200

Now, we can solve this equation to find the value of "x," which represents the amount borrowed at 8%.

0.08x + 0.05x + $1000 + 0.09($25000 - 1.5x - $2000) = $2200
0.13x + $1000 + 0.09($25000 - 1.5x - $2000) = $2200
0.13x + $1000 + $0.09($25000 - 1.5x - $2000) = $2200
0.13x + $1000 + $2250 - 0.135x - $180 = $2200
0.13x - 0.135x + $1000 - $180 + $2250 = $2200
-0.005x + $3070 = $2200
-0.005x = $2200 - $3070
-0.005x = -$870

Dividing both sides of the equation by -0.005 gives us:
x = -$870 / -0.005
x = $174,000

So, the company borrowed $174,000 at 8%.

Now, we can find the amount borrowed at 10% and 9%.

Amount borrowed at 10% = (1/2)x + $2000
= (1/2) * $174,000 + $2000
= $87,000 + $2000
= $89,000

Amount borrowed at 9% = Total borrowed - (Amount borrowed at 8% + Amount borrowed at 10%)
= $25000 - ($174,000 + $89,000)
= $25000 - $263,000
= -$238,000

It seems that there is no loan borrowed at 9%. It is likely a sign in the given information. Please double-check the problem statement or the calculations.