### Percentage Unchanged

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

### CD Heaven

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at each price?

### Dating Made Easier

If a sum invested gains 10% each year how long before it has doubled its value?

# 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

Able students may like to design similar coded columns for each other to crack.
Discussion may include an exploration of how many values need to be seen coded before the solution multiplier is known for sure.

### Possible support

Students who are not ready for this problem without some preliminary activity should generate simple two-digit 'codings' for themselves and swap these around the group for others to crack.

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.