Greek Anthology Book XIV: Metrodorus: 146
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Arithmetical Epigram of Metrodorus
- $A$. Give me two minas and I become twice as much as you.
- $B$. And if I got the same from you I am four times as much as you.
Solution
Let $a$ minas be the quantity of $A$.
Let $b$ minas be the quantity of $B$.
We have:
| \(\text {(1)}: \quad\) | \(\ds a + 2\) | \(=\) | \(\ds 2 \paren {b - 2}\) | |||||||||||
| \(\ds a + 2\) | \(=\) | \(\ds 2 b - 4\) | ||||||||||||
| \(\ds a\) | \(=\) | \(\ds 2 b - 6\) | ||||||||||||
| \(\text {(2)}: \quad\) | \(\ds b + 2\) | \(=\) | \(\ds 4 \paren {a - 2}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + 2\) | \(=\) | \(\ds 4 a - 8\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 4 a - 10\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 2 \paren {4 a - 10} - 6\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 8 a - 20 - 6\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 7 a\) | \(=\) | \(\ds 26\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds \frac {26} 7\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \frac 5 7\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 4 \times \frac {26} 7 - 10\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {104 - 70} 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {34} 7\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \frac 6 7\) |
So:
- $A$ is $3 \frac 5 7$
- $B$ is $4 \frac 6 7$.
$\blacksquare$
Source of Name
This entry was named for Metrodorus.
Sources
- 1918: W.R. Paton: The Greek Anthology Book XIV ... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $146$