Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
A pair of jeans goes up by 25%, then the shop announces "back to the original price". What discount should it display? If you answered 25%, this lesson is for you: the right answer is 20%! Percentage changes do not cancel out by simply switching the sign, and that is exactly what this chapter is about.
You already know the right tool: the multiplier. A 25% increase multiplies by 1.25. To get back to the starting point, you must multiply by the reciprocal multiplier, that is the inverse: 1/1.25 = 0.8. And 0.8 corresponds to a 20% decrease. That is why increase and decrease are not symmetric: the return percentage is computed on the NEW value, which is larger.
Second big idea: the average rate of change. If a value grows by 44% over two years, it did not grow by 22% per year! Changes combine by multiplication, not by addition. So we look for the multiplier c which, applied twice, gives the overall change: c × c = 1.44, so c = 1.2, meaning an average rate of 20% per year. For n periods, we look for c such that c to the power n equals the overall multiplier.
Remember the three-step routine: translate each change into a multiplier, work on the multipliers (inverse for the reciprocal, root for the average rate), then translate back into a percentage. With this reflex, no sales pitch or statistic will ever trick you again.
An item goes up by 25%. What percentage decrease brings it exactly back to its original price?
CM = 1 + 25/100 = 1,25
I translate the 25% increase into a multiplier.
CM réciproque = 1/1,25 = 0,8
To undo the change, I take the inverse of the multiplier.
0,8 = 1 - 20/100
I translate back: multiplying by 0.8 means a 20% decrease. Check: 100 euros becomes 125 euros, and 125 × 0.8 = 100 euros.
Result: A 20% decrease is needed (not 25%).
Over two years, a subscription increased by 44% overall. What is the average annual rate of change?
CM global = 1,44
I translate the overall 44% increase into a multiplier.
c × c = 1,44
The average rate is the one that, applied twice, produces the overall change.
c = √1,44 = 1,2
I take the square root: 1.2 × 1.2 = 1.44, that checks out.
1,2 = 1 + 20/100
Multiplier 1.2, so an average rate of 20% per year. Note: 22% per year would give 1.22 × 1.22 = 1.4884, too much!
Result: The average rate is 20% per year.