Expert: Sherry Wallin - 11/26/2008

Question
Hello:

A few days ago I asked how to determine the solution to the following by using a weighted average:

An investor has $30,000 to invest at simple interest in two accounts. One account earns 2% and the second earns 5%. How much must he invest in each account if his total return is 4% or $1,200?

You replied with the following:
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Answer

I tell my students to set up a table of values and to label the variable let x = amount invested at 2% and 30,000-x is the amount invested at 5%.

x = 10,000 is the amount invested at 2% and 30,000-10,000 = 20,000 is the amount invested at 5%

Note that 10,000/30,000 = 1/3 and 20,000/30,000 = 2/3 these are the weights
(1/3)(.02) + (2/3)(.05) = .12/3 = .04 = 4%

(You may prefer to think of 1/3 as .33 and 2/3 as .67
then you have .33(.02) + .67(.05) = .04 = 4%

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I discovered how to determine the fractions of 1/3 and 2/3 without first solving the equation:

2%----3%----4%----5%

From 2% to 5%, 2% is 2/3 of the distance to 4%.
From 5% to 2%, 5% is 1/3 of the distance to 4%.

However, which fraction goes with which percentage, 2% and 5%, without using trial and error?

(2/3)(.02) + (1/3)(.05) does not equal 4%

I thank you for your reply.


Answer
Hi Kenneth~
    There are actually a number of ways to obtain this information but not knowing what your mathematics background is I didn't share the following way with you which requires you to have a knowledge of solving a system of linear equations.

Let x = the weighted average for the amount invested at 2
Let y = the weighted average for the amount invested at 5

You will obtain two relationships using these two variables.
Notice that 1/3   2/3 = 1. In weighted averages you will have then add up to 1, thus relationship 1: x   y = 1  
the 2nd relationship comes about fro knowing one of the rates is 2
nd the other is 5 so relationship 2: .02x   .05 y =.04

equation 1:     x plus  y = 1
equation 2:  .02x plus  .05 y =.04  to make things simpler multiply this equation by 100 getting
2x plus 5y = 4
x plus y = 1 multiply this equation by -2 so when you add the two x's you get 0.

-2x plus -2y = -2
2x plus  5y = 4 adding vertically you have 3y = 2, now divide by 3 by sides
y = 2/3. You have found the weighted average for the
amount at 5 percent substitute
y = 2/3 into either original equation, I will use x plus  y = 1
x plus  2/3 = 1, subtract 2/3 from both sides getting x = 1/3 which is the weighted average for the amount at 2 percent


I hope this is clear for you if not ask again and I will give more detail. It would help if you would tell me what math you have had or what class this is for :)

Math Prof

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