# The Sausage Rule: Secrets of the Mathematical Ninja

The student stared, blankly, at the sine rule problem in front of him.

$\frac{15}{\sin(A)} = \frac{20}{\sin(50^º)}$

“I don’t know where to st,” he started whining as something flew past his head. He knew better than to turn and look at whatever implement of death and destruction he had dodged. “I suppose I could cross-multiply…” he said.

The Mathematical Ninja put his head to one side. “Go on?”

“So $15 \sin(50º) = 20 \sin(A)$?”

Nod.

“And I can put that in the… smoking wreck of my calculator.”

“Oopsy,” said the Mathematical Ninja.

“Divide by fift… twenty,” he said. “So $\frac{15\sin(50^º)}{20} = \sin(A)$… and then could I inverse-sine it on the calculator?”

The Mathematical Ninja wrinkled his nose. “You could have done,” he admitted. “And you’d get about 35º.”

“Is there a quicker way to get to that sum?”

“I thought you’d never ask! Of course there is: it’s the sausage rule, and you can use it whenever you have an equation between two fractions.”

“Like this one!”

“Like this one. Draw a circle around your unknown to start with.”

“The $\sin(A)$ is the only thing with a letter in.”

“Correct! Then draw a diagonal sausage.”

“Around the $15$ and the $\sin(50^º)$?”

“Yep. Multiply those two together and divide by the thing that isn’t circled, and you get…?”

”$\frac{15 \sin(50^º)}{20}$, which is equal to $\sin(A)$! Is that related to the Table of Joy, by any chance?”

The Mathematical Ninja smirked. “Maybe,” he said. “Just maybe.”