Where is the missing ratio? On a day at the Fruits Exchange, an apple can be exchanged for 2 bananas, and a banana exchanged for 2 oranges. The market is perfectly efficient, so an apple can be traded for 4 oranges.
On another day at the Fruits Exchange, there are some shortages of apples and bananas, so the exchange rate of apples to bananas rises to 1 apple for 3 bananas, and 1 banana for 3 oranges.
The average exchange rate over the two days for apples to bananas is $\frac{(1:2 + 1:3)}{2} = 1:2.5$,
And the average exchange rate over the two days for bananas to oranges is the same.
Then the average exchange rate for apples to oranges should be $1:2.5^2 = 1:6.25$
However, if you exchanged an apple on each of the two days, you would have received $4+9=13$ oranges, hence setting the average exchange rate to $2:13=1:6.5$.
Where did the missing 0.25 ratio go ?
 A: Let's simplify the situation. In store A a toothbrush costs $\$10$, and in store B it costs $\$20$. If I buy one toothbrush at each store, how much did I pay on average? $\$15$.
But in store A a single dollar gets me $\frac{1}{10}$ of a toothbrush, and in store B a single dollar gets me $\frac{1}{20}$ of a toothbrush. So a single dollar seemingly gets me an average of $\frac{\frac{1}{10}+\frac{1}{20}}{2}$ toothbrushes which equals $\frac{3}{40}$, which is different from $\frac{1}{15}$.
You can't take the average of ratios (rates) then take the reciprocal of that and expect to get the average of the quantities themselves. This will give false results. That is, $\frac{1}{\frac{\frac{1}{a}+\frac{1}{b}}{2}}$ does not in general equal $\frac{a+b}{2}$. The first quantity is called the harmonic mean of $a$ and $b$ and the second quantity is called the arithmetic mean of $a$ and $b$, and they are not in general equal.
For another example, imagine a car which passes 1 km between 8am and 9am, and then on a slower road it passes 1 km between 9am and 11am. It passed 2 km in 3 hours so its average speed was $\frac{2}{3}$ km/h. On the other hand averaging its two individual speeds on the two separate roads we get $\frac{1+\frac{1}{2}}{2}=\frac{3}{4}$, which is wrong. The reason it is wrong is that the car spent more time on the second road. This same intuition is the reason for why the arithmetic and harmonic means don't in general equal one another.
Specifically for your question, your second calculation is the correct one since instead of averaging the exchange ratios it considered the amount of oranges paid per apple, on the first day we get $4$ oranges per apple, and on the second day we get $9$ oranges per apple, and so on average we get $\frac{9+4}{2}=6.5$ oranges per apple. The first calculation is incorrect for the same reason taking the average speed of the cars in the two separate roads is incorrect.
