To determine how to invest $8,800 such that the interest earned from a low-risk investment at 9% is equal to the interest earned from a high-risk investment at 13%, we will follow these steps:
1. Define the Variables:
Let's denote:
- \( x \) as the amount invested at the low-risk rate of 9%.
- \( y \) as the amount invested at the high-risk rate of 13%.
2. Relationship Between Variables:
Since the total investment is $8,800, we can write:
[tex]\[
x + y = 8800
\][/tex]
3. Interest Earned from Each Investment:
The interest earned from the low-risk investment is \( x \times 0.09 \).
The interest earned from the high-risk investment is \( y \times 0.13 \).
4. Equal Interest Condition:
According to the problem, the interest from both investments should be equal. Thus, we can set up the equation:
[tex]\[
x \times 0.09 = y \times 0.13
\][/tex]
5. Substitute \( y \):
From the total investment equation \( x + y = 8800 \), we can express \( y \) in terms of \( x \):
[tex]\[
y = 8800 - x
\][/tex]
Substituting \( y \) in the interest equation gives:
[tex]\[
x \times 0.09 = (8800 - x) \times 0.13
\][/tex]
6. Solve for \( x \):
Let's solve the equation for \( x \):
[tex]\[
0.09x = 0.13 \times (8800 - x)
\][/tex]
Expanding and simplifying,
[tex]\[
0.09x = 1144 - 0.13x
\][/tex]
[tex]\[
0.09x + 0.13x = 1144
\][/tex]
[tex]\[
0.22x = 1144
\][/tex]
[tex]\[
x = \frac{1144}{0.22}
\][/tex]
[tex]\[
x = 5200
\][/tex]
7. Find \( y \):
Using \( y = 8800 - x \),
[tex]\[
y = 8800 - 5200
\][/tex]
[tex]\[
y = 3600
\][/tex]
Thus, to ensure that the interest earned from each investment is the same:
- $5,200 should be invested at the low-risk rate of 9%.
- $3,600 should be invested at the high-risk rate of 13%.
