##### Calculus Help Please

*label*Mathematics

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*schedule*1 Day

*account_balance_wallet*$5

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?

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 :)

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!

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