MHB How to Solve Investment Word Problems Involving Interest Rates and Profits?

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To solve the investment problem involving a trust fund, the equation derived shows that an additional $32,000 should be invested at 8.5% to achieve an overall return of 8% on the total investment. For the second scenario, where a businessman invests $12,000 in two ventures with different profit and loss rates, the equations x + y = 12,000 and 0.08x - 0.04y = 120 can be used to find the amounts invested in each venture. By substituting and eliminating variables, the solution can be derived. The method involves solving for one variable and substituting it back to find the other. This approach effectively addresses both investment scenarios involving interest rates and profits.
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1. A trust fund has invested \$8000 at 6% annual interest.
How much additional money should be invested at
8.5% to obtain a return of 8% on the total amount
invested?

my solution,

let
$n=$amount invested @ 8.5%

$0.06(8000)+0.085n=0.08(8000+n)$

$480+0.085n=640+0.08n$

$0.085n-0.08n=640-480$

$n=32,000$

\$32,000 is the addtional money should be invested at 8.5%. is this correct?

2. A businessman invested a total of \$12,000 in two ven-tures. In one he made a profit of 8% and in the other he
lost 4%. If his net profit for the year was $120, how
much did he invest in each venture?

in this problem i don't know how will i represent the amount invested in the other venture. please help. thanks!
 
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1.) Correct.

2.) Let $x$ represent the first investment amount and $y$ represent the other. So we know:

$$x+y=12000$$

And we also know:

$$0.08x-0.04y=120$$

which I would rewrite as:

$$2x-y=3000$$

Now, add the two equations to eliminate $y$, then solve for $x$, then use this value of $x$ in either of the two equations to determine $y$.
 
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