Calculus Help Please

label Mathematics
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Inverse mutual funds, sometimes referred to as "bear market" or "short" funds, seek to deliver the opposite of the performance of the index or category they track, and can thus be used by traders to bet against the stock market. This question is based on the following table, which shows the performance of three such funds as of August 12, 2011.
Year-to-date Loss
SHPIX (Short Smallcap Profund)6%
RYURX (Rydex Inverse S&P 500)5%
RYIHX (Rydex Inverse High Yield)7%

You invested a total of $11,000 in the three funds at the beginning of 2011, including an equal amount in RYURX and RYIHX. Your year-to-date loss from the first two funds amounted to $450. How much did you invest in each of the three funds? 

Jul 7th, 2015

Thank you for the opportunity to help you with your question!

1) Write the equations. Let's set: x = Amount of money invested in SHPIX

y = Amount of money invested in RYURX

z = Amount of money invested in RYIHX

Since it was invested a total of $11,000 then our first equation is: x + y + z = 11,000

Then it says that it was invested the same amount in RYURX and RYIHX which means our second equation is:     y = z

Finally, it says that year-to-date loss from the first two funds amounted to $450 (this comes from the interest of the first two investment). So its interest rate times its invested amount from the first two funds would be equal to the $450. So we have:  0.06x + 0.05y = 450

x + y + z = 11000

y = z

0.06x + 0.05y = 450

2) Then we solve the sytem of equations. Let's substitute the z by y into the first equation. So we have:

x + y + z = 11000     ------------->  x + y + y = 11000 --------------> x + 2y = 11000

Now we solve for x in the last obtained equation.

x + 2y = 11000  ------------> x + 2y - 2y = 11000 - 2y  --------------->  x = 11000 - 2y

Now we enter (11000 - 2y) in place of x into the equation: 0.06x + 0.05y = 450. So we have:

0.06(11000 - 2y) + 0.05y = 450  --------> 0.06*11000 - 0.06*2y + 0.05y = 450

660 - 0.12y + 0.05y = 450  --------> 660 - 0.07y = 450  ----------> 660 - 660 - 0.07y = 450 - 660

-0.07y = -210 --------> -0.07y/-0.07 = -210/-0.07 ---------> y = $3000

Since y = z  then z = $3000. And finally, we have:

x + y + z = 11000  -----------> x + 3000 + 3000 = 11000 --------> x + 6000 = 11000

x + 6000 - 6000 = 11000 - 6000 ------------> x = $5000

x = $5000 , y = $3000, z = $3000

Please let me know if you have any doubt or question :)

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 7th, 2015

Thanks for taking the time to walk me through this problem. It did end up making much more sense once I saw the work.

Thanks Again!

Jul 9th, 2015

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