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A woman invests $800 in two savings accounts earning simple annual interest. Some of the money is invested at a simple annual interest rate of 5%, and the rest of the money is invested at a simple annual interest rate of 8%. If the combined annual interest accrued is $55, how much money is invested at each interest rate?
$500 at 5% and $300 at 8%
$300 at 5% and $500 at 8%
$400 at 5% and $400 at 8%
$600 at 5% and $200 at 8%



Answer :

RanzFromIndonesian

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%.

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