Answer And Step-by-step explanation:
Let:
x = amount invested at 5%
y = amount invested at 8%
We know the total investment is $800:
x + y = 800 (equation 1)
We also know the total annual interest is $55:
(0.05 * x) + (0.08 * y) = 55 (equation 2)
Solving the system of equations:
We can use elimination to solve for x and y.
Since both equations have x, try to eliminate it. We can multiply equation 1 by -0.08:
-0.08x - 0.08y = -64 (equation 3)
Add equation 3 to equation 2:
(0.05x) + (0.08y) + (-0.08x - 0.08y) = 55 - 64
0 = -9
This step seems to lead to a contradiction (0 = -9). However, it actually tells us that the system of equations we set up is not independent. The two equations essentially represent the same information.
Alternative approach:
Since the total investment is $800 and we have two interest rates (5% and 8%), we can try to distribute the investment strategically to reach a combined interest of $55.
Testing possible scenarios:
$400 at 5% and $400 at 8%:
Interest from 5% = 0.05 * $400 = $20
Interest from 8% = 0.08 * $400 = $32
Total interest = $20 + $32 = $52 (This doesn't match the target $55)
$600 at 5% and $200 at 8%:
Interest from 5% = 0.05 * $600 = $30
Interest from 8% = 0.08 * $200 = $16
Total interest = $30 + $16 = $46 (This doesn't match the target $55)
We can eliminate these scenarios because the combined interest doesn't match the target.
$300 at 5% and $500 at 8%:
Interest from 5% = 0.05 * $300 = $15
Interest from 8% = 0.08 * $500 = $40
Total interest = $15 + $40 = $55 (This matches the target!)
Therefore, the solution is:
$300 is invested at 5%.
$500 is invested at 8%.