Percentages: from meaning to the return to the original value

Percentages become much easier when the % sign is treated as an operation rather than a label added at the end. We will start from two identities, % = 1/100 and 100% = 1, and rebuild every formula. No proportions and no memorized tricks: just elementary algebra.

1. What % means

The percent sign literally means one hundredth. Applied to a number, it divides that number by 100:

% = 1 / 100
a% = a · 1 / 100 = a / 100
25% = 25 / 100 = 0.25

Percentages, fractions, and decimals are therefore not different objects: they are different representations of the same number. For example, 1/4 = 0.25 = 25%.

2. Why 100% = 1

100% = 100 · 1 / 100 = 100 / 100 = 1

One hundred percent of a quantity is the whole quantity. Multiplying by 100% does not change a value because it is the same as multiplying by 1. Percentages above 100 are valid too: 150% = 1.5.

Common mistake. The decimal number corresponding to 100% is not 100 but 1. The % sign already contains the division by 100.

3. Converting a number to a percentage

For any number x, multiply by the identity 100% = 1:

x = x · 1 = x · 100% = (100x)%

Thus 0.37 = 37%, 1.2 = 120%, and 3/5 = 0.6 = 60%. The coefficient before % does not have to be an integer: 1/3 = 33.333…%.

When asking how much a part represents relative to a total, “relative to” means division:

percentage = part / total · 100%

If 18 out of 24 elements have a property, then 18/24 · 100% = 75%. The total, which is the reference value, goes in the denominator.

4. Two successive increases or discounts

An increase of v% multiplies by 1 + v/100; a discount of v% multiplies by 1 − v/100. Using signed changes v₁ and v₂, positive for increases and negative for discounts:

F = I · (1 + v₁/100) · (1 + v₂/100)

• +20%, then +10%: 100 · 1.20 · 1.10 = 132, so the total increase is 32%.
• −20%, then −10%: 100 · 0.80 · 0.90 = 72, so the total discount is 28%.
• +20%, then −10%: 100 · 1.20 · 0.90 = 108, so the total change is +8%.

Successive percentages are not simply added because the second change acts on the value that has already changed.

To derive the formula for an a% change followed by a b% change, write the whole as 100% and subtract the original whole:

(100% + a%) · (100% + b%) − 100%
= 100% · 100% + 100% · b% + a% · 100% + a% · b% − 100%
= a% + b% + a% · b%
= (a + b + ab/100)%

Indeed, 100% = 1, while a% · b% = (ab/100)%. The cross term ab/100 explains why addition alone is not enough.

5. Calculating percentage change

Let I be the initial value and F the final value, with I different from zero. First find the difference F − I, then compare it with the initial value:

percentage change = (F − I) / I · 100%

From 80 to 100, the change is (100 − 80)/80 · 100% = 25%. From 100 to 80, it is (80 − 100)/100 · 100% = −20%. The outward and return changes differ because the reference in the denominator changes.

If the initial value is zero, the usual percentage change is undefined. With a negative initial value, the result can also be counterintuitive, so the chosen convention must be stated. Ordinary school and economic examples generally assume a positive initial value.

Finally, moving from 20% to 25% is an increase of 5 percentage points, while relative to 20% the increase is (25 − 20)/20 · 100% = 25%.

6. The case +a% followed by −a%

An increase and a decrease with the same coefficient do not cancel. Set b = −a in the general formula:

(a + b + ab/100)%
= (a − a − a²/100)%
= (−a²/100)%

The same result follows by multiplying the factors:

(1 + a/100)(1 − a/100)
= 1 − (a/100)²

For every nonzero a in the usual discount range, the result is below the starting value. The total loss as a percentage is (a²/100)%.

For a = 20, starting from 100 gives 100 → 120 → 96: the loss is 4%, because 20²/100 = 4. An increase of 50% followed by a decrease of 50% produces a 25% loss.

7. Returning to the original value

To find the change that truly reverses an initial signed change v%, require the product of the two factors to equal 1:

(1 + v/100)(1 + r/100) = 1
r = −100v / (100 + v)

After a 25% increase, a 20% decrease is needed: 1.25 · 0.80 = 1. After a 20% decrease, a 25% increase is needed: 0.80 · 1.25 = 1.

In particular, after an increase of a%, the required reduction is 100a/(100+a)%. After a reduction of a%, with a less than 100, the required increase is 100a/(100−a)%. The formula does not apply to a −100% change, meaning v = −100: a quantity that has reached zero cannot be restored by a finite percentage increase.

Formula summary

The same thread runs through every case: identify the reference value and work with multiplicative factors. These formulas are not isolated rules; they all follow from the meaning of %.