A Change in Code

Why do this problem?

Code-breaking is often about partial conclusions gradually adding up to possibilities. This problem is unlikely to be done instantly by most students, so discussion should bring up lots of helpful thoughts to share around a group, energising explanation and stimulating individuals into new reasoning and strategy.

Possible approach

Ask students to look at the code and the column of original values and to share their first thoughts. Hopefully including the insight that whole numbers have stayed as whole numbers after the increase.
The three codings make progressively more demanding challenges.
Maintain an emphasis on the deductive process that establishes the solution rather than merely confirming that a particular multiplier works, though verification should of course be part of the process.
 
A teacher comments:
After some initial thought and discussion all (Year 9 set 1) made good progress and found a number of different ways into the problem. The second part of the problem raised points which led neatly into reverse percentages.

Key questions

What could all the original numbers be divided by without producing a decimal anywhere in the results column ?

Possible extension

Possible support

Encourage exploration to discover the multipliers that tend not to produce many decimal options, and then pick the original numbers so that not even these decimal residuals appear.