please help?

Question

Melinda Bowen receives an annual income of $234.50 from investing one amount of money at 6% and another amount at 5%. If the investments were interchanged, her income would increase by $5.10. Find the amount invested at each rate of return.

A.
Melinda has $1900 invested at 6% and $2410 invested at 5%.

B.
Melinda has $1980 invested at 6% and $2350 invested at 5%.

C.
Melinda has $2040 invested at 6% and $2320 invested at 5%.

D.
Melinda has $2120 invested at 6% and $2240 invested at 5%

Answer 1

First, let's set this up as a set of equations.
First sentance: P1 * 6% + P2 * 5% = $234.50
Second sentance: P1 * 6% + P2 * 5% + $5.10 = P1 * 5% + P2 * 6%

So, let's first simplify the second equation and use it to isolate the value of P1 and P2 relative to each other.
.06P1 + .05P2 + 5.1 = .05P1 + .06P2
.01P1 + 5.1 = .01 P2
P1 + 510 = P2

Now, let's substitute the value of P2 into the first equation.
.06P1 + .05P2 = $234.50
.06P1 + .05(P1 + 510) = $234.50
.06P1 + .05P1 + 25.5 = $234.50
.11P1 = 209
P1 = 1900

Substitute the known value of P1 back into our equation for P2:
P1 + 510 = P2
1900 + 510 = P2 = 2410

The answer is A.

Check using the second equation:
P1 * 5% + P2 * 6% = $234.50 + $5.10
1900 * .05 + 2410 * .06 = $239.60
95 + 144.6 = 239.6
239.6 = 239.6 (check!)