A comprehensive investment plan is made for a person with a total investment of $26,300. The money is broken into three investments. Part of the money is invested into a retirement account 5.1\% interest per year, part is invested in a saving account earning 2.8\% interest per year, and part is invested into a certificate of deposit earning 3.0\% interest per year. The goal is to earn $1133.60 in interest at the end of one year. One last piece of information is that the broker recommends that you invest twice as much into savings as you do into the certificate of deposit. How much money went into each account?Lets start by defining our variables:x: Amount invested in retirement account
y: Amount invested in savings account
z: Amount invested in certificate of depositWe know that the total invested is $26,300 so:x + y + z = 26300We also know that the interest on each account is as follows:x: 5.1\% = 0.051
y: 2.8\% = 0.028
z: 3\% = 0.03Additionally we know that the total interest desired is 1133.60 giving us:0.051x + 0.028y + 0.03z = 1133.60Finally we are told that the amount of the savings account is twice that of the cd so:y = 2zWe can use this last equation to modify our other equations as follows:x + (2z) + z = 26300
x + 3z = 26300and0.051x + 0.028(2z) + 0.03z = 1133.60
0.051x + 0.056z + 0.03z = 1133.60
0.051x + 0.086z = 1133.60Now our system of equations is:x + 3z = 26300
0.051x + 0.086z = 1133.60If we set the top equation equal to x we can solve using substitution:x = -3z + 26300Substitute this into the second equation for x and solve for z:0.051(-3z + 26300) + 0.086z = 1133.60
-0.153z + 1341.3 + 0.086z = 1133.6
-0.067z + 1341.3 = 1133.6
-0.067z = -207.7
z = 3100This means that $3,100 should be invested in the certificate of deposit. From this we also know that the savings account should have twice that amount so the savings account should have $6,200 invested in it. Lastly, going back to our initial equation all three investments need to add up to $26,300. To find the missing value we will just use what we know and solve for x:x + 6200 + 3100 = 26300
x + 9300 = 26300
x = 17,000So the amount invested in the retirement account should be $17,000. A quick check with our other equation should show that the interest will work out too:0.051x + 0.028y + 0.03z = 1133.60
0.051(17,000) + 0.028(6,200) + 0.03(3,100) = 1133.60
867 + 173.6 + 93 = 1133.60
1133.60
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