John, Bill, and George sold apples next to each other one day. In the morning, they sold 2 apples for 5 cents - that is, 1 apple for $5 / 2$ cents, and in the afternoon, 3 apples for 5 cents - that is, 1 apple for $5 / 3$ cents. John sold the same amount in the afternoon as in the morning. Bill's revenue was the same in the morning and in the afternoon. The ratio of the number of apples George sold in the morning to the number he sold in the afternoon was the same as the ratio of $5 / 3$ to $5 / 2$. At what average price did each boy sell one apple? - What formulas yield the average unit price if the morning unit price is denoted by $A$ and the afternoon unit price by $B$?
Let the number of apples sold in the morning be denoted by $a$, and the number sold in the afternoon by $b$. The average price per unit, which is the ratio of the total revenue to the total number of units, can be expressed as

$$
d=\frac{a A+b B}{a+b}
$$

For John, $b=a$, so

$$
d=\frac{a(A+B)}{2 a}=\frac{A+B}{2}
$$

For Bill, $a A=b B$, thus $b=a A / B$,

$$
d=\frac{2 a A}{a+\frac{a A}{B}}=\frac{2 A B}{A+B}
$$

Finally, for George, $a: b=\frac{5}{3}: \frac{5}{2}$, and this ratio is the same as the ratio of $B$ and $A$ given in the problem; $a: b=B: A$.

From this, it is clear that the average price per unit is the same as in Bill's case.

Numerically, John sold his apples for 25/12 cents each, while Bill and George sold theirs for 2 cents each, which is $1 / 25$ less, or 4% cheaper.

Judit Bod (Bp., Apáczai Csere J. Gymn., I. o. t.)

Note. The relationship found for the average price per unit in the last two cases can be written more simply if we switch to the reciprocal of $d$ and divide term by term:

$$
\frac{1}{d}=\frac{A+B}{2 A B}=\frac{\frac{1}{A}+\frac{1}{B}}{2}
$$

In this case, $d$ is called the harmonic mean of the numbers $A$ and $B$. The relationship can also be stated as: the reciprocal of the harmonic mean of two numbers is equal to the arithmetic mean of the reciprocals of the numbers (of course, neither $A$ nor $B$ can be 0).