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Question

# An inheritance of $20,000 is divided among three investments yielding a total of$1780 in interest per year. The interest rates for the three investments are 7%, 9%, and 11%. The amounts invested at 9% and 11% are $3000 and$1000 less than the amount invested at 7%, respectively. Find the amount invested at each rate.

Solution

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Let $x$ be the amount invested at 7%, $y$ be the amount invested at 9%, and $z$ be the amount invested at 11%.

Since the total investment is $\20,000$, then $x+y+z=20,000$.

$x$ dollars invested at 7% will yield $0.07x$ dollars in interest, $y$ dollars invested at 9% will yield $0.09y$ dollars in interest, and $z$ dollars invested at 11% will yield $0.11z$ dollars in interest. The total amount of interest is then $0.07x+0.09y+0.11z$ dollars. Since the investments earn a total of $\1780$ in interest, then $0.07x+0.09y+0.11z=1780$.

If the amount invested at 9% is $\3000$ less than the amount invested at 7%, then $y=x-3000$.

If the amount invested at 11\$is$\1000$ less than the amount invested at 7%, then $z=x-1000$.

The system of equations is then

\left\{\begin{aligned}x+y+z&=20,000\\0.07x+0.09y+0.11z&=1780\\y&=x-3000\\z&=x-1000\end{aligned}\right.

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