# 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 ANSWERED 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%.