Greek Anthology Book XIV: Metrodorus: 138

Arithmetical Epigram of Metrodorus

Nicarete, playing with five companions of her own age,

gave a third of the nuts she had to Cleis,

the quarter to Sappho,

and the fifth to Aristodice,

the twentieth and again the twelfth to Theano,

and the twenty-fourth to Philinnis.

Fifty nuts were left for herself.

Solution

Let $n$ be the number of nuts in total.

$\dfrac n 3$ went to Cleis

$\dfrac n 4$ went to Sappho

$\dfrac n 5$ went to Aristodice

$\dfrac n {20} + \dfrac n {12}$ went to Theano

$\dfrac n {24}$ went to Philinnis

$50$ stayed with Nicarete.

We have:

\(\ds n\)

\(=\)

\(\ds \dfrac n 3 + \dfrac n 4 + \dfrac n 5 + \dfrac n {20} + \dfrac n {12} + \dfrac n {24} + 50\)

\(\ds \leadsto \ \ \)

\(\ds 120 n\)

\(=\)

\(\ds 40 n + 30 n + 24 n + 6 n + 10 n + 5 n + 120 \times 50\)

multiplying through by $120 = \lcm \set {3, 4, 5, 20, 12, 24}$

\(\ds \leadsto \ \ \)

\(\ds \paren {120 - 40 - 30 - 24 - 6 - 10 - 5} n\)

\(=\)

\(\ds 120 \times 50\)

\(\ds \leadsto \ \ \)

\(\ds 5 n\)

\(=\)

\(\ds 120 \times 50\)

\(\ds \leadsto \ \ \)

\(\ds n\)

\(=\)

\(\ds \frac {120 \times 50} {10}\)

\(\ds \)

\(=\)

\(\ds 1200\)

So there were $1200$ nuts.

$400$ went to Cleis

$300$ went to Sappho

$240$ went to Aristodice

$60 + 100 = 160$ went to Theano

$50$ went to Philinnis

and we know that Nicarete kept $50$.

$\blacksquare$

Source of Name

This entry was named for Metrodorus.

Sources

• 1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $138$