Let's denote the amount invested at [tex]\(8\% \)[/tex] as [tex]\( X \)[/tex] dollars.
Since the total investment is [tex]\( \$12,000 \)[/tex], the amount invested at [tex]\(10.5\% \)[/tex] would be [tex]\( 12,000 - X \)[/tex] dollars.
Next, let's set up the equations based on the interest rates and the total interest earned.
1. Interest from the investment at [tex]\(8\% \)[/tex] rate:
[tex]\[
\text{Interest from } X \text{ dollars} = 0.08X
\][/tex]
2. Interest from the investment at [tex]\(10.5\% \)[/tex] rate:
[tex]\[
\text{Interest from } (12,000 - X) \text{ dollars} = 0.105(12,000 - X)
\][/tex]
3. Total interest earned:
According to the problem, the total interest earned from both investments last year was [tex]\( \$1,145 \)[/tex]. So, we can set up the following equation:
[tex]\[
0.08X + 0.105(12,000 - X) = 1,145
\][/tex]
Now, we solve the equation step-by-step:
1. Distribute [tex]\(0.105\)[/tex] in the second term on the left-hand side:
[tex]\[
0.08X + 0.105 \times 12,000 - 0.105X = 1,145
\][/tex]
Simplify:
[tex]\[
0.08X + 1,260 - 0.105X = 1,145
\][/tex]
2. Combine like terms:
[tex]\[
(0.08X - 0.105X) + 1,260 = 1,145
\][/tex]
Simplify the coefficients of [tex]\( X \)[/tex]:
[tex]\[
-0.025X + 1,260 = 1,145
\][/tex]
3. Isolate the term with [tex]\( X \)[/tex] by subtracting 1,260 from both sides:
[tex]\[
-0.025X = 1,145 - 1,260
\][/tex]
[tex]\[
-0.025X = -115
\][/tex]
4. Solve for [tex]\( X \)[/tex] by dividing both sides by [tex]\( -0.025 \)[/tex]:
[tex]\[
X = \frac{-115}{-0.025}
\][/tex]
[tex]\[
X = 4,600
\][/tex]
So, the amount invested at [tex]\(8\% \)[/tex] is [tex]\( \$4,600 \)[/tex].
Finally, we determine the amount invested at [tex]\(10.5\% \)[/tex]:
[tex]\[
12,000 - 4,600 = 7,400
\][/tex]
Therefore, the amount invested at [tex]\(10.5\% \)[/tex] is [tex]\( \$7,400 \)[/tex].
In summary:
- The amount invested at [tex]\(8\% \)[/tex] is [tex]\( \$4,600 \)[/tex].
- The amount invested at [tex]\(10.5\% \)[/tex] is [tex]\( \$7,400 \)[/tex].
