Expert: Josh - 8/20/2009

Question
QUESTION: Hello:

An investor invests $1000.00 in two accounts, one account of $200 earning 2% at simple interest and $800 at 5% at simple interest. The 2% account earned $4.00 and the 5% account earned $40.00. If these two amounts are added and divided by $1000, he earned a total percentage of 4.4%.  

I want to determine this percentage, 4.4% by using 2% and 5%, but when I add 2% and 5% and divide by 2, the result is an average of 3.5%. How can I get the result of 4.4% by using the percentages of 2% and 5%?

I can add $4.00 and $40.00 and divide by $1000 and get the result, but why does adding 2% and 5% divided by two fail to produce the 4.4%?

If possible, use a simple calculation!

I thank you for your reply.

ANSWER: Kenneth,

Re: second paragraph
We need to take a weighted average of 2% and 5% to get 4.4%. Since the money is not evenly split, but divided in the ratio of 1:4 (i.e., the account with $200 represents 20% of $1000, and the account with $800 represents 80% of $1000), we have to take 2*0.2 + 5*0.8 to get 4.4.

Re: third paragraph
Unweighted average (e.g., 2*0.5 + 5*0.5) works when the same amount is deposited into each account. This is not the case here. So, we need to take a weighted average to obtain the equivalent interest rate.

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

QUESTION: Hello:

I want to thank you for the reply and answer. I did find it very helpful; however, I am used to using division the find averages. Can division be used to find the 4.4%?

I thank you once again for your reply and assistance!

Answer
Well, you can look at "2*0.2 + 5*0.8" in terms of [2*200+5*800]/(200+800). The division is known as the "normalization" step. It obtains the weights used in calculating the weighted average. [You can see, for example, that 200/(200+800)=0.2]

This step ensures that the weights (viz., 0.2 and 0.8) will add up to 1. This is generally the way to go if the weighted average involves more than 2 terms (accounts).