1510.51 – Stella's Musical Scores

As is well-known, Stella studied piano with the renowned Nadia Boulanger in Paris after World War II. Later, when settled in Amsterdam, she had three shelves of four-hand piano scores from which she choose music to play with her husband Graziano (who needed to practice more). The top shelf contained $\dfrac{1}{6}$ of the scores. The middle shelf contained more than $\dfrac{2}{5}$ of the scores, and the bottom shelf contained only four scores (the thickest ones).

What is the smallest possible number of scores in total that Stella might have kept on these shelves?

Solution

If $x$ is the total number of scores, then $x$ is a multiple of $6$ .

Note that the top and bottom shelves together hold less than $\dfrac{3}{5}$ of the scores:

$\begin{aligned}\dfrac{x}{6} + 4 &< \dfrac{3x}{5} \\4 &< \left(\dfrac{3}{5} - \dfrac{1}{6}\right)x = \dfrac{(18 - 5)x}{30} = \dfrac{13x}{30} \\120 &< 13x \\x &> 9.23\end{aligned}$

So $x$ can be as small as $12$ , the first multiple of $6$ after $9.23$ . Then Stella's shelves, from top to bottom, would hold 2, 6, and 4 scores. Note that

$\dfrac{6}{12} > \dfrac{2}{5}$

That’s a decent collection of four-hand piano scores! (Stella wanted Graziano to practice more, but she didn't want to nag. You can check her biography for further information.)