### 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?

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

# A Change in Code

##### Stage: 4 Challenge Level:

Good thinking from several people on this problem.
Below is a combination of solutions from Paul at Castleknock College, and Jean in Warwick
Â

The answer to the first part is a 20% increase.

The most it can be is a 22.5% increase because 80 would then become 98, it can't be 99 or else there would be 2 equal letters.

All numbers have one thing in common -  they're divisible by 5

So the only way all of these numbers could produce integers, with no fraction or decimal part, is if their increase is calculated using fifths.

It must be 20% because anything greater, 40% 60% etc., would turn 80 into a three digit number which it clearly isn't.

An alternative explanation could be : 80 to HF is an increase but still only produces a 2 digit number, so HF is 81 to 99.

Trying each percentage, for example 80 to 90 is 12.5%, all give Â results Â involving fractions from some of the start values, except for 80 to 96 (20%).

And 20% checks as consistent across all data.

In Part 2 the answer is 25%

BC to 15 so B is 1 and C is 2, 3, or 4

CG to 35 so C is in fact 2 or 3 Â

C is not zero so BC is either 11, 12, or 13

11 to 15 and 13 to 15 do not give a percentage that keeps all other results integer.

12 to 15 is an increase of 25% and this is consistent for all the other data.

In Part 3. the percentage increase was 40%

Increasing a number is like multiplying with a top-heavy fraction. For example a 5% increase is achieved multiplying the original number by 21/20.

Dividing by 21/20 will reverse that process, and multiplying by 20/21 is the same thing. Â

I notice that all the results contain a factor of 7, so I am looking for a/7 as that multiplier, where a< 7, and choosing a so as to avoid fractions as required.

EHJ goes to 147

So EHJ is between 101 and 146, because it has 3 digits and this is an increase.

If EHJ/147 is equivalent to the fraction a/7 then EHJ is 105 or 126. Giving an a value of 5 or 6

Next I bring in CJH which increases to 1330

1330 multiplied by 6/7 is 1140, which is four digits not three, so 5/7 is the correct multiplier.

The percentage increase was 40%