Formulate a system of equations

Question

Mr. and Mrs. Garcia have a total of $100,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 6%/year, while the bonds and the money market account pay 4%/year and 2%/year, respectively. The Garcias have stipulated that the amount invested in the money market account should be equal to the sum of 20% of the amount invested in stocks and 10% of the amount invested in bonds. How should the Garcias allocate their resources if they require an annual income of $5,000 from their investments?

Answer 1

Let S = amount invested in stocks

Let B = amount invested in bonds

Let M = amount invested in money market

 

Total of $100,000  ==>  S + B + M = 100000

 

Money market equals sum of 20% of stocks and 10% of bonds  ==>  

M = 0.2*S + 0.1*B  

 

Annual income $5,000  ==>  0.06*S + 0.04*B + 0.02*M = 5000  

 

Plug the value for M in to the other 2 equations 

 

S + B + (0.2*S + 0.1*B) = 100000

1.2*S + 1.1*B = 100000           call this equation A

 

0.06*S + 0.04*B + 0.02*(0.2*S + 0.1*B) = 5000

0.06*S + 0.04*B + 0.004*S + 0.002*B = 5000

0.064*S + 0.042*B = 5000  

 

Multiply this last equation by -1.2/0.064 = -18.75 and add it to equation A

 

-1.2*S - 0.7875*B = -93750

 1.2*S +      1.1*B = 100000

 

0.3125*B = 6250

B = 20000 

 

Plug this in to equation A 

 

1.2*S + 1.1*(20000) = 100000

1.2*S + 22000 = 100000

1.2*S = 78000

S = 65000

 

Plug S and B in to the original 1st equation 

 

65000 + 20000 + M = 100000

M = 15000

 

So they should invest $65000 in stocks, $20000 in bonds and $15000 in money market  

 

Check:  65000 + 20000 + 15000 = 100000

           .2*(65000) + .1*(20000) = 13000 + 20000 = 15000

          .06*(65000) + .04*(20000) + .02*(15000) = 3900 + 800 + 300 = 5000