# Secrets of the Mathematical Pirate: Switcheroos

“Don’t tell the Mathematical Ninja,” said the Mathematical Pirate.

The student shook his head enthusiastically. “Narr!”

“You’ve got $ \frac {7}{x} = 14$. Ask yourself: what would the Mathematical Ninja do?”

“The Mathematical Ninja would do something that looked extremely dangerous and terrifying, but was completely under control.”

“Correct!” said the Mathematical Pirate. “And after that?”

“Um… he’d multiply both sides by $x$ to get $7 = 14x$!”

“Yarr! Then?”

“Then he’d divide by 14 to get $\frac{7}{14} = x$ and simplify to say it’s a half!”

The Mathematical Pirate danced a little reel. “And how would a Mathematical Pirate deal with it?”

“A Mathematical Pirate would say it’s the same shape as a formula triangle, so you can swap the bottom with the other side, to get $\frac 7{14} = x$ directly.”

The Mathematical Pirate made to offer the student some rum, but remembered in time. “It doesn’t just work with $a \div b = c$, either,” he said. “You can do the switcheroo with $a - b = c$, too.”

“So $15 -x = 12$ turns into $15 - 12 = x$, and $x = 3$!”

“Ahoy! So you can always swap the denominator with the other side, and you can always swap the subtrahend with the other side.”

“The who to the what now?”

“$a \div b = c \iff a \div c = b$ and $a - b = c \iff a - c = b$”, said the Mathematical Pirate, patiently.

“Now you’re talking,” said the student.