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Question 26750: My problem is the following, I can work out the answer by trial and error, but I figure there must be an algebraic way of solving the question.
John Invests $23000 at the bank.
The bank places some of the money at 8% and the rest at 5%.
After one year he has gained $1000. How much did the bank place at 8%
Answer by bmauger(101)   (Show Source):
You can put this solution on YOUR website! Just so you know (and maybe you made this problem up) but your problem doesn't work. Even if the whole amount is invested at 5%, John would still end up making more than $1000.  That said, let's try to solve it anyway...
Presuming there's no compounding of interest (and since you didn't mention any I'll presume there isn't) This can be solved by a system of equations. Let's call the amount invested at 8% "x" and the amount invested at 5% "y". You know that the amount the bank invested must add up to $23000, thus our first equation is:
The other equation involves the rates of investment, which when multiplied by the principle amount, gives the total value from each investment. In other words, the amount gained from investment x = x*.08 and from y = y*.05. The problem tells us that when we add these investments, they'll total $1000. To "write that in math" we would write:
Now that we have our two equations, we can solve this system by substitution or elimination. Using elimination we can rewrite the first equation for x as:
Putting that in for the second equation (replacing x) gives:
 Distribute to get:
 Rearrange to get:
 Combine y's and get:
 Divide both sides by -.03 and get:
Putting back into this eqation:
using y = 28000 gives:
 Subtract and get:
Seeing this number should raise some flags, since they invested $28000 at 5% and somehow invested NEGATIVE $5000 at 8%. Your bank is ripping you off, and giving you an average rate of LESS than the 5% they promised. To be exact, they're investing your money at an average rate of:
But this should give you help as to the methods used to solve it. Try it again where you earn $1390 after a year instead of $1000. You should get $8000 invested at 8% and $15000 at 5%.
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