Exercise 26

 

Assessment started: undefined.

Item 1

Rachel, Chan, and Sue work at a grocery store that sells pens, erasers, and pencils. 

Rachel says, "Seven pens and five erasers cost the same as six pencils.

Chan says, "Four pens and nine pencils cost the same as five erasers.

Sue says, "Six pencils and three erasers cost the same as four pens.

Only one is lying. Can you tell us who?

• Rachel
• Chan
• Solution

  Let A$A$ equal the price of pens, B$B$ equal the price of erasers, and O$O$ equal the price of pencils.
  Rachel is saying that 7A+5B=6O$7A+5B=6O$. 
  Chan is saying that 4A+9O=5B$4A+9O=5B$. 
  Sue is saying that 6O+3B=4A$6O+3B=4A$.

  Combining Rachel's and Chan's statements, we get 11A=−3O$11A=-3O$. This is impossible.
  Combining Chan's and Sue's statements, we get 15O=2B$15O=2B$. This is possible.
  Combining Rachel's and Sue's statements, we get 8B=−3A$8B=-3A$. This is impossible.
  This means that either Rachel is lying, or Chan and Sue are both lying. But there can be only one liar.

  Hence, Rachel is the liar.