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 xx be the amount invested at 4.5%, yy be the amount invested at 5%, and zz be the amount invested at 8%.

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

xx dollars invested at 4.5% earns 0.045x0.045x dollars in interest, yy dollars invested at 5% earns 0.05y0.05y dollars in interest, and zz dollars invested at 8% earns 0.08z0.08z dollars in interest. The total amount of interest earned on the three investments is then 0.045x+0.05y+0.08z0.045x+0.05y+0.08z dollars. Since the total interest earned is $1275\$1275, then 0.045x+0.05y+0.08z=12750.045x+0.05y+0.08z=1275.

If the amount invested at 5% is $4,000\$4,000 less than the amount invested at 4.5%, then y=x4,000y=x-4,000.

If the amount invested at 8% is $10,000\$10,000 less than the amount invested at 4.5%, then z=x10,000z=x-10,000.

The system of equations is then

{x+y+z=25,0000.045x+0.05y+0.08z=1275y=x4,000z=x10,000\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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