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Advanced Math/Weighted Average

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Expert: - 11/24/2008


Question
QUESTION: Hello:

Can you solve the following by using a weighted average calculation?

An investor has $30,000 to invest at simple interest for one year. One portion at 2% and the other portion at 5%.
What amounts must be invested at these percentages if the total return is 4% or $1,200?

Answers: $20,000 and $10,000

I thank you for your reply.


ANSWER: You're welcome?

%5 of $20K is $1,000.
+ 2% of $10K is $200.

Add them up and you get $1,200.

Looks right to me.

---------- FOLLOW-UP ----------

QUESTION: Thanks for the reply!

How would you solve this without knowing the answers, $20,000 and $10,000?

5% of $?K is $1,000 + 2% of $?K is $200


ANSWER: Let A be the amount of one of the investments.

30,000-A is what we have left for the other investment.

The equation would then be 0.05A + 0.02(30,000-A) = 0.04(30,000).

This can be reduced by doing the multiplication, givin
0.05A + 600 - 0.02A = 1,200.

Subtracting the 600 from both sides and combining the A terms gives
0.03A = 600.

---------- FOLLOW-UP ----------

QUESTION: I Thank you for your replies:

I do have this idea, but I cannot seem to understand whether or not it will work.

4% is in between 2% and 5%.

|---|---|---|
2%, 3%, 4%, 5%

2% is 2/3 from 4% and 4% is 1/3 from 5%.

As you can see the line above in divided into thirds, but the problem 2/3 of 2% + 1/3 of 5 does not equal 4%. The opposite is true 1/3 of 2% and 2/3 of 5% equal (12/3)% or 4%. It seems to me that 2/3 of 2% + 1/3 of 5 should equal the percentage of 4%, but it does not. Can you explain?

I thank you for your help and assistance.  

Answer
The number is 2/3 from 2% and 1/3 from 5%, so you take 1/3 of 2% and 2/3 or 5% giving 2(1 - 2/3)% + 5(1 - 1/3)% = 2/3 + 10/3 = 12/3 = 4%.

Hopefully this clarifies the problem.  So if you had functional values at 2% and at 5%, you would do the same.  The approximation  of f(3%) would be (2/3)f(2%) + (1/3)f(5%).

The approsimation of f(2.6%) would be 0.8f(2%) + 0.2f(5%) since 2.6% is 1/5 of the way from 2% to 5% and the value of 1/5 is 0.2.  The value of 4/5 is 0.8.  

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