Question

# An inheritance of $25,000 is divided among three investments yielding a total of$1275 in interest per year. The interest rates for the three investments are 4.5%, 5%, and 8%. The amounts invested at 5% and 8% are $4000 and$10,000 less than the amount invested at 4.5%, respectively. Find the amount invested at each rate.

Solution

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Let $x$ be the amount invested at 4.5%, $y$ be the amount invested at 5%, and $z$ be the amount invested at 8%.

The total investment is $\25,000$ so $x+y+z=25,000$.

$x$ dollars invested at 4.5% earns $0.045x$ dollars in interest, $y$ dollars invested at 5% earns $0.05y$ dollars in interest, and $z$ dollars invested at 8% earns $0.08z$ dollars in interest. The total amount of interest earned on the three investments is then $0.045x+0.05y+0.08z$ dollars. Since the total interest earned is $\1275$, then $0.045x+0.05y+0.08z=1275$.

If the amount invested at 5% is $\4,000$ less than the amount invested at 4.5%, then $y=x-4,000$.

If the amount invested at 8% is $\10,000$ less than the amount invested at 4.5%, then $z=x-10,000$.

The system of equations is then

\left\{\begin{aligned}x+y+z&=25,000\\0.045x+0.05y+0.08z&=1275\\y&=x-4,000\\z&=x-10,000\end{aligned}\right.

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