Greek Anthology Book XIV: Metrodorus: 145
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Arithmetical Epigram of Metrodorus
- $A$. Give me ten minas and I become three times as much as you.
- $B$. And if I got the same from you I am five times as much as you.
Solution
Let $a$ minas be the quantity of $A$.
Let $b$ minas be the quantity of $B$.
We have:
| \(\text {(1)}: \quad\) | \(\ds a + 10\) | \(=\) | \(\ds 3 \paren {b - 10}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + 10\) | \(=\) | \(\ds 3 b - 30\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 3 b - 40\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds b + 10\) | \(=\) | \(\ds 5 \paren {a - 10}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + 10\) | \(=\) | \(\ds 5 a - 50\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 5 a - 60\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 3 \paren {5 a - 60} - 40\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 15 a - 180 - 40\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 14 a\) | \(=\) | \(\ds 220\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds \frac {110} 7\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 15 \frac 5 7\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 5 \times \frac {110} 7 - 60\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {550 - 420} 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {130} 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 18 \frac 4 7\) |
So:
- $A$ is $15 \dfrac 5 7$
- $B$ is $18 \dfrac 4 7$.
$\blacksquare$
Source of Name
This entry was named for Metrodorus.
Sources
- 1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $145$