Question

A trust fund has $200,000 to invest. Three alternative investments have been identified, earning 10 percent, 7 percent, and 8 percent, respectively. A

Equations and inequalities
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asked 2020-11-10
A trust fund has $200,000 to invest. Three alternative investments have been identified, earning 10 percent, 7 percent, and 8 percent, respectively. A goal has been set to earn an annual income of $16,000 on the total investment. One condition set by the trust is that the combined investment in alternatives 2 and 3 should be triple the amount invested in alternative 1. Determine the amount of money which should be invested in each option to satisfy the requirements of the trust fund.

Answers (1)

2020-11-11
Let x be the amount invested at 10%, yy be the amount invested at 7%, and z be the amount invested at 8%.
The total investment is $200000:
\(\displaystyle{x}+{y}+{z}={200000}{\left({1}\right)}\)
The annual income is $16,000:
\(\displaystyle{0.10}{x}+{0.07}{y}+{0.08}{z}={16000}{\left({2}\right)}\)
The combined investment in alternatives 2 and 3 should be triple the amount invested in alternative 1 so:
\(\displaystyle{y}+{z}={3}{x}\)
or
\(\displaystyle−{3}{x}+{y}+{z}={0}−{3}{x}+{y}+{z}={0}{t}{a}{g}{\left\lbrace{3}\right\rbrace}\$\$\)
Subtract each side of (1) and (3) then solve for xx:
\(\displaystyle{4}{x}={200000}\)
\(\displaystyle{x}={50000}\)
Substitute x=50000 to (2) and simplify to obtain (4):
\(\displaystyle{0.10}{\left({50000}\right)}+{0.07}{y}+{0.08}{z}={16000}\)
\(\displaystyle{5000}+{0.07}{y}+{0.08}{z}={16000}\)
\(\displaystyle{0.07}{y}+{0.08}{z}={11000}{\left({4}\right)}\)
Substitute x=50000 to (1) and simplify to obtain (5):
\(\displaystyle{50000}+{y}+{z}={200000}\)
\(\displaystyle{y}+{z}={150000}{\left({5}\right)}\)
Eliminate zz. Multiply (5) by 0.08 to obtain (6):
\(\displaystyle{0.08}{y}+{0.08}{z}={12000}{\left({6}\right)}\)
Subtract each side of (4) and (5) then solve for x:
\(\displaystyle−{0.01}{y}=−{1000}\)
\(\displaystyle{y}={100000}\)
Solve for zz using (5):
\(\displaystyle{100000}+{z}={150000}\)
\(\displaystyle{z}={50000}\)
So, $50000 was invested at 10%, $100000 was invested at 7%, and $50000 was invested at 8%.
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