The Missing Inverse in Percentage Calculations
“An increase of 20% followed by a decrease of 20% does not bring you back to where you started.”
Most students know this fact. Many teachers demonstrate it with examples. Students memorize it, use it in examinations, and then move on.
But there is a more interesting question.
If a 20% decrease is not the inverse of a 20% increase, then what is?
Surprisingly, school mathematics rarely asks this question.
Yet the answer connects percentage calculations with one of the most fundamental ideas in mathematics—the concept of an inverse.
We already understand inverses
When students learn algebra, they encounter two important inverses.
Additive inverse
A number and its additive inverse cancel each other.
a+(-a)=0
For example,
8+(-8)=0.
Also, if you add 8 to any number and then subtract 8 from the result you get back to the original number. That's why these two are inverse of each other.
Multiplicative inverse
A number and its multiplicative inverse return us to the multiplicative identity.
Multiply a and 1/a, we get 1. For example, 5*(1/5) = 1.
Also, multiply any number by 5 and then divide the result by 5, we get back to the same number.
Students become comfortable with these ideas because each operation has another operation that exactly reverses its effect.
Shouldn’t percentage changes have the same property?
Looking differently at percentage changes
Suppose a quantity increases by (m%).
Instead of thinking in terms of percentages, think in terms of the multiplication factor.
An increase of (m%) means multiplying by
1+(m/100).
Similarly, a decrease of (n%) means multiplying by
1-(n/100).
For these two operations to cancel each other and bring back the value to original
(1+(m/100)) * (1-(n/100)) = 1,
This is nothing more than the condition for two multiplication factors to be reciprocals. Something like x * (1/x) = 1.
Solving this equation gives
n = 100m / (100+m).
This is the inverse percentage decrease corresponding to an increase of (m%).
Likewise,
m = 100n / (100-n).
Every percentage increase has exactly one inverse percentage decrease, and every percentage decrease has exactly one inverse percentage increase.
A familiar example
Suppose a price increases by 25%.
The multiplication factor 1.25.
Its reciprocal is
1/ 1.25 = 0.8.
Multiplying by 0.8 means decreasing by 20%.
Therefore,
- Increase by 25%
- Decrease by 20%
are inverse percentage changes.
Notice that the percentages themselves are different.
The multiplication factors are inverses.
That is the real mathematical idea.
Here are some inverse pairs.
Increase by 10% and decrease by 9.09%,
Increase by 20% and decrease by 16.67%,
Increase by 25% and decrease by 20%,
Increase by 50% and decrease by 33.33%,
Increase by 100% and decrease by 50%.
It's interesting to observe that a 100% increase is wiped out by just 50% decrease.
Another Situation
While exploring this idea, another interesting relationship appears.
Suppose the length of a rectangle increases by (m%) while its width decreases by (n%).
The new area becomes
(1+(m/100)) * (1-(n/100))
times the original area.
So, the percentage increase in area is
m-n-(mn /100).
Exactly the same expression appears in profit and loss.
Suppose a shopkeeper marks an article up by (m%) and later offers a discount of (n%).
The profit percentage is
m-n-(mn /100).
Two completely different situations—geometry and commerce—are governed by the same mathematical structure.
Once we focus on multiplication factors, the connection becomes obvious.
A small idea with a big lesson
Mathematics becomes powerful when different topics reveal the same underlying pattern.
Profit and loss.
Area of rectangles.
Successive percentage changes.
Reciprocals.
Inverse operations.
At first glance, these seem different topics from different chapters of a Mathematics book.
A closer look shows that they are all manifestations of one simple idea:
Percentage changes are multiplicative transformations.
And like every multiplicative transformation, each has its own inverse.
Now try this question. You get instant feedback when you select an option.
Concept Check: Successive Percentage Changes
When a number is increased by m% and subsequently decreased by n%, the final value will equal the original value if and only if their multiplication factors are reciprocals:
Now, suppose a quantity is increased by 40%. We want to apply a subsequent percentage decrease n% such that the final number is 15% higher than the original number.
To find the required percentage decrease, which of the following approaches should be taken?
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