Question 120354: I am really struggling with this problem. Can anyone please help?
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 amounts she invested.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Call one of the investments x dollars and the other investment y dollars.
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The way Melinda has her money invested she makes 6% on investment x and 5% on investment y.
This is equivalent to 0.06*x and 0.05*y. These two returns add up to $234.50. So we can
write this in equation form as:
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0.06*x + 0.05*y = 234.50
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If the investments were interchanged, then x would return 5% and y would return 6%. So this
time the returns would be written 0.05*x and 0.06*y. If you add these two together, then
the return would be $5.10 more than she got before. This means the return would now be
$234.50 + $5.10 = $239.60. So, in equation form, this time the equation is:
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0.05*x + 0.06*y = 239.60
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So we have two equations:
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0.06*x + 0.05*y = 234.50 and
0.05*x + 0.06*y = 239.60
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Let's use variable elimination to solve these two equations. Multiply the top equation (all
terms on both sides) by 500 to get:
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30*x + 25*y = 117250
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Next multiply the bottom equation by 600 (all terms on both sides) and the equation
becomes:
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30*x + 36*y = 143760
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So we have converted the two equations to:
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30*x + 25*y = 117250 and
30*x + 36*y = 143760
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Notice now that if we subtract the two equations in vertical columns the term 30*x will
disappear from the resulting equation because we made it such that both equations had
a common x-term. (We did that by our choice of the multipliers for each equation.)
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Subtracting the two equations vertically, we get:
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-11*y = -26510
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Solve for y by dividing both sides of this equation by -11 and you get:
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y = -26510/-11 = 2410
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So one of the sums of money (the one she has invested at 5%) is $2410.00
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We can now return to either of the original equations we developed and solve for x by
substituting $2410 for y. Let's return to the equation:
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0.06*x + 0.05*y = 234.50
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and substituting $2410 for y we get:
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0.06*x + (0.05)(2410) = 234.50
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Multiply out the second term on the left side and the equation becomes:
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0.06*x + 120.50 = 234.50
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Get rid of the 120.50 on the left side by subtracting 120.50 from both sides. When you do
that the equation reduces to:
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0.06*x = 114
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Solve for x by dividing both sides by 0.06 and you get:
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x = 114/0.06 = 1900
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This tells you that she has $1900.00 invested at 6%.
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So the answer to this problem is that she has $1900.00 invested at 6% and $2410.00 invested
at 5%.
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Hope this helps you to understand the problem.
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