21 Aug 2013

Unequal Slices of Pizza: Middle Secondary Mathematics Competition Question

Alice and Brenda stood transfixed as a tower of pizza boxes was delivered and laid out on the kitchen table. The smell of freshly baked dough wafted through the kitchen, enveloping the girls in a kind of spell. “Let me! Let me!” cried Alice, as she opened the first box.

“Hey, Brenda! Look! They forgot to cut this one into slices.” Before Brenda had a chance to register the seriousness of the crime, Alice had found the pizza slicer and was ready to commence surgery. With two clean cuts, Alice divided the pizza into four slices.

“Alice... that’s not really in the centre, is it?!” Brenda was most disappointed with her friend’s eye for geometric precision.

“Mmm... you’re right. Never mind; I’ll take the big slice and the small one. That looks fair!” Alice waltzed off with her plate full of pizza. Brenda stared at her two pieces, certain that she’d been diddled.

Was the division fair? If not, how much more pizza did Alice take?

The diagram shows Alice’s off-centre cuts. The lines AB and CD are perpendicular and cross at point P. Let AP=a, BP=b, CP=c and DP=d, with d > b > a > c. Alice takes the slices BPD and APC while Brenda has the other two. Find an expression for how much extra pizza Alice has compared to Brenda.

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