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%.

5.1. Inverse formulas: finding the new or initial value

So far, we knew the initial and final values and wanted the percentage change. But if we know the percentage and either value, we can find the other. Use the letters in the example: V is the old or initial value, N is the new or final value, and P is the signed number expressing the percentage. Thus V corresponds to I and N corresponds to F in the previous sections. We assume V is positive.

For a 15% increase, write P = 15; for a 3.8% decrease, write P = −3.8. Notice that P is the number −3.8, whereas P% = −3.8/100 = −0.038.

Where do the inverse formulas come from?

Start with the change formula. Multiplication by 100% explicitly shows the conversion of the ratio into a percentage:

Percentage change
P% = (N − V) / V × 100%

Since P% = P/100 and 100% = 1, the same equality without percentage signs is:

P / 100 = (N − V) / V
P × V = 100 × (N − V)
P × V = 100 × N − 100 × V
100 × V + P × V = 100 × N
V × (100 + P) = 100 × N

In the second line, both sides were multiplied by 100 × V; then we moved 100 × V to the left and factored out V. Dividing by 100 gives the new value:

New value
N = V × (100 + P) / 100 = V × (1 + P/100)

Dividing instead by 100 + P gives the initial value:

Initial value
V = 100 × N / (100 + P) = N / (1 + P/100)

For example, with V = 200 and a 15% increase: N = 200 × (100 + 15) / 100 = 230. Conversely, knowing that the value after the increase is 230: V = 100 × 230 / 115 = 200. The percentage still refers to the initial value, even when that value is unknown.

Problem: recover the previous year’s births

Use the rounded numbers in the proposed news item as an exercise: “There were 355 thousand births in 2025, 3.8% fewer than in 2024.” What number of births for 2024 follows from these figures? Before opening the solution, identify which year provides the reference value.

V = 100 × N / (100 + P)
V = 100 × 355000 / (100 − 3.8)
V = 35500000 / 96.2
V = 355000 / 0.962
V = 369022.869… ≈ 369023

(N − V) / V = 0.962 − 1 = −0.038
P% = −0.038 × 100% = −3.8%

When can these formulas be used?

• The change formula requires a nonzero V; here we assumed V is positive.
• Recovering V requires 100 + P ≠ 0, or P ≠ −100. A 100% decrease gives N = 0 whatever V was: zero alone cannot reveal the initial value.
• For nonnegative quantities such as prices and counts, a decrease cannot exceed 100%. A positive new value requires P > −100.

Two different problems: here we find the initial value from the final value and the original percentage. In “Returning to the original value”, we will instead find the new percentage that cancels an earlier change.

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

We revisit an argument developed in 2014 and checked in Excel: first find the reciprocal factor, then distinguish the percentage value to reach from the change required.

A share price falls by 10%. What change restores its original price? More generally, given a change of a%, positive, negative or zero, what change r% brings us back? Here a is the signed number: −10 means a 10% fall and +5 means a 5% rise.

7.1. A 10% fall leaves 90%

After a 10% fall, 90% of the initial value remains, or 90/100. To undo this multiplication, take its reciprocal: multiply by 100/90.

(100 − 10)% = 90% = 90/100
(90/100) × (100/90) = 1 = 100%

Convert the reciprocal to a percentage by explicitly multiplying by 100%. About 111.11% is the value to reach relative to the current value: it is not yet the change.

100/90 × 100% ≈ 111.11%

To measure how much the current value must increase, subtract its whole, 100%. About +11.11% is required, not +10%. Keep the exact ratio for verification instead of rounding first.

100/90 × 100% − 100%
= (100/90 − 1) × 100% ≈ +11.11%
100 → 90 → 90 × (100/90) = 100

7.2. A 5% rise reaches 105%

After a 5% rise we have 105% of the original value, or 105/100. The restoring factor is its reciprocal, 100/105. To obtain the change, convert this ratio to a percentage and subtract 100%.

(100 + 5)% = 105% = 105/100
(105/100) × (100/105) = 1 = 100%
100/105 × 100% ≈ 95.24%
100/105 × 100% − 100%
= (100/105 − 1) × 100% ≈ −4.76%
100 → 105 → 105 × (100/105) = 100

7.3. The general formula: reciprocal, percentage, difference

For an initial change of a%, the first factor is (100+a)/100; the return factor is 100/(100+a). Converting it to a percentage and subtracting the whole gives:

r% = (100/(100 + a) − 1) × 100%

100/(100 + a) × 100% − 100%
= (100/(100 + a) − 1) × 100%

r is the number before %. The following expression is equivalent, but its explanation remains the reciprocal argument:

r = 100 × (100/(100 + a) − 1)
  = −100a/(100 + a)

The denominator requires a ≠ −100. With a positive initial value and nonnegative quantities, a finite return requires a > −100: a may be negative, positive or zero. When a = 0, r = 0 too. A 100% fall leaves zero, and no finite percentage increase can restore the initial value.

7.4. Checking with Excel

Enter the number −10 in A2 without %, and 5 in A3. In B2 enter =100/(100+A2)-1 and copy it to B3. Apply Percentage formatting to column B, for example with two decimal places.

The formula returns decimals, about 0.111111 and −0.047619. Percentage formatting displays them as 11.11% and −4.76%: it changes their representation, not their stored value. Do not multiply by 100 again in the formula, or this format would display about 1111.11% and −476.19%.

Check the input convention: if A2 already contains −10% (numeric value −0.10), use =1/(1+A2)-1 instead, with Percentage formatting for the result. Do not mix the two conventions.

Try the reciprocal method in the lab: −10%, +5% and your own values

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 %.

Open the percentages laboratory: examples, charts and worked steps

Percentages of what? Groups, different bases and rate ratios