Weighted average of percentages - where is the error?

I have a hard time understanding the following Problem:

I want to calculate the weighted average of two ratios. The Ratio is based on percentages.

As example: 30% of 1,15 and 70% of 1,45. What is the weighted Ratio? My understanding was that I would calculate like this:

(0,3 x 1,15) +(0,7 x 1,45) = 1,36

However I have seen a different calculation that is doing the following:

1 / ((30%/1,15) + (70%/1,45)) = 1,344758

I don't understand why there is a difference between the two. As for example with an even Distribution of 50% or 0% and 100% both ways Show the same result. However for all other ratios there is a slight difference.

I am lost.

Thank you very much.

You have found two different ways of finding the "mean" ratio. Which one is most appropriate to use depends entirely on what $1.15$ and $1.45$ are. The guiding principle is "If I replace all of the different ratios and percentages with $100\%$ of one single number, what would that number be to give the same effect?" where how your numbers relate to the desired effect is the key thing that dictates which mean is appropriate.

For instance, if I have milk with different fat percentages (say, for the sake of argument, $1.15\%$ and $1.45\%$), and I mix a new milk which consists of $30\%$ from the milk with less fat, and $70\%$ from the milk with more fat, then $1.36\%$ correctly describes the new fat percentage of the mix.

On the other hand, say I am running some kind of production in my garage, and I have two machines that I can use to make stuff: On one I can spend $1.15$ hours making one item of product $A$, and on the other machine I spend $1.45$ hours making one item of product $B$, and I can only operate one of them at a time. If I use the first machine $30\%$ of the time, and use the second machine $70\%$ of the time, then on average, I will spend $1.344758$ hours on each item I make.

• Thank you for your thorough answer. The 1.15 and 1.45 are cover ratios used in finance. (cash flow available for debt service divided by debt service) In order to calculate a specific target repayment profile both of these ratios need to be weighted in order to calculate the maximum amount of debt repayment possible. – Günther Nov 10 '17 at 10:11
• In this specific case there are two different cover ratios that need to be weighted to calculate the Maximum debt service per period given a certain amount of cash flow available for debt service. (e.g. 1000 EUR (cash flow) / 1.15 (cover Ratio) = 870 EUR max debt service. – Günther Nov 10 '17 at 10:19

What you did first is the weighted arithmetic mean. The other solution you saw is the weighted harmonic mean.