Percentage Calculator: Share, Percent Change and Difference

About this calculator

The percentage calculator works out the three most common “percentage relationships” between two numbers $A$ and $B$ in one go: what percentage $A$ is of $B$ (the share), by what percentage a quantity rises or falls in going from $A$ to $B$ (the percent change), and the symmetric percent difference, which does not care which number came first. Conversely, given a share and one of the numbers, it finds the other.

Almost every everyday “what percent” question falls into one of these three: test scores, discounts, price rises, growth rates, errors, concentrations. Each formula is a single division—but what you divide by decides the answer. That is precisely where percentages go wrong most often, and it is what this page sets out to make clear.

What it does not do: it does not handle growth compounded over several periods (that is the compound interest calculator), it does not compute differences in percentage points (that is plain subtraction), and it does not “add up several percentages”—because most of the time doing so is wrong.

Formula and variables

The three outputs use three different bases:

\text{share}\;p = \frac{A}{B} \times 100\%

\text{percent change}\;c = \frac{B - A}{|A|} \times 100\%

\text{percent difference}\;d = \frac{|A - B|}{(|A| + |B|)/2} \times 100\%

Solving from a known share $p$ : $A = \dfrac{p}{100}\,B$ , $B = \dfrac{A}{p/100}$ .

Symbol	Meaning	Calculator field	Unit	Range and notes
$A$	First number (numerator of the share, start of the change)	Value A	any (same dimension as B)	Any real number; default 25; percent change is undefined when $A = 0$
$B$	Second number (denominator of the share, end of the change)	Value B	same as A	Any real number; default 200; share is undefined when $B = 0$
$p$	A as a percentage of B	A as % of B	%	Output; can be entered to solve for A or B
$c$	Percent change from A to B	A→B change %	%	Output; positive for an increase, negative for a decrease; denominator is $\lvert A\rvert$
$d$	Symmetric percent difference	Difference %	%	Output; $0 \le d \le 200$ ; independent of the order of A and B

$A$ and $B$ must be the same kind of quantity (both scores, both prices, both head counts); the percentage itself has no unit.

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Percentage calculator” (/#/mathtools/formula?calculator=calculator.percentage). It opens with “Value A” = 25 and “Value B” = 200 already filled in and the three percentages computed.
Compute the three percentageschange A and B, and the three fields marked “computed”—A as % of B, A→B change %, Difference %—update live. Decide first which of the three your question is, then read that field.
Find A from a known shareclear “Value A”, enter the share in “A as % of B” and keep B; A switches to “computed” and shows the result.
Find B from a known shareclear “Value B”, enter the share and A; B switches to “computed”.
Read the resultspercentages are shown to at most 8 decimal places with trailing zeros removed, e.g. 22.22222222. In solve mode only the field being solved for is meaningful; the other two percentages are computed with “missing field = 0” and show meaningless values such as ∞ or 200, which you can simply ignore (see Assumptions).

Worked examples

All four examples were recomputed by the engine’s percentage compute routine; the display convention is at most 8 decimal places with trailing zeros removed.

Example 1: 80 and 100

Inputs: $A = 80$ , $B = 100$ .

Share $p = 80/100 \times 100\% = 80\%$ .
Percent change $c = (100 - 80)/80 \times 100\% = 25\%$ .
Difference $d = |80 - 100| / \big((80 + 100)/2\big) \times 100\% = 20/90 \times 100\% = 22.2\%$ .

Interface shows: A as % of B 80, A→B change % 25, Difference % 22.22222222.

Swap the same two numbers— $A = 100$ , $B = 80$ —and the share becomes 125, the change becomes −20, while the difference stays 22.22222222. “Rising from 80 to 100 is +25%, but falling from 100 back to 80 is only −20%”, because the two bases differ; the difference % is the only one unaffected by order.

Figure 1: the same gap of 20 gives asymmetric percent changes up and down; 1.25 × 0.80 = 1.00 shows the two steps cancel exactly

Example 2: panel defaults, 25 and 200

Inputs: $A = 25$ , $B = 200$ .

Share $25/200 = 12.5\%$ .
Percent change $(200 - 25)/25 = 7 = 700\%$ —going from 25 to 200 is 8 times the original, i.e. an increase of 7 times, or 700%.
Difference $175 / 112.5 = 155.6\%$ .

Interface shows: 12.5, 700, 155.55555556. These numbers show that “8 times as much” and “an increase of 700%” are the same thing, and that “an increase of 800%” is wrong.

Example 3: solving from a known share

An item sells for 36 after a 25% discount; what was the original price? “The sale price is 75% of the original”: clear A (it is convenient to treat B as the original and A as the sale price, though the reverse works too).

Inputs: $p = 15$ , $B = 240$ (A cleared) → interface shows A 36: 15% of 240 is 36.

Inputs: $p = 12.5$ , $A = 30$ (B cleared) → interface shows B 240: 30 is 12.5% of 240.

The discount example: $p = 75$ , $A = 36$ , clear B → $B = 36 / 0.75 = 48$ .

Example 4: scores, negative numbers and decimals

$A = 45$ , $B = 60$ (45 out of 60): share 75, change 33.33333333, difference 28.57142857.
$A = 3.5$ , $B = 4.2$ (an interest rate rising from 3.5% to 4.2%): change 20—the rate rose by 20%, but only by 0.7 percentage points. Both statements are correct; they mean different things.
$A = -50$ , $B = 50$ (a temperature or a profit turning from negative to positive): share −100, change 200 (denominator $|A| = 50$ ), difference 200 (the maximum of the symmetric difference).

The inputs of Example 1 can be entered directly into the panel to reproduce it; swap A and B and watch the change go from 25 to −20 while the difference % stays the same.

Principles and derivation

A percentage is just a fraction with denominator 100

“ $p$ percent” means $p/100$ . The ratio of $A$ to $B$ is $A/B$ ; written with denominator 100 it is $A/B = p/100$ , hence $p = 100\,A/B$ . That is the share formula; view it as the proportion $A : B = p : 100$ , solve for the fourth quantity from the other three, and you get the two inverse solutions. The international standard ISO 80000-1 defines % as the pure number 0.01, so “multiplying by 100%” is mathematically just multiplying by 1—rewriting 0.8 as 80% does not change its value.

Percent change: change relative to the starting point

Going from $A$ to $B$ , the absolute change is $B - A$ ; comparing it with the starting point $A$ gives the relative change $(B - A)/A$ . Choosing the start as the denominator is a convention: “up 25%” is understood relative to the price before the rise. The engine uses $|A|$ rather than $A$ as the denominator, so that when the start is negative (a loss of 50 becoming a profit of 50) the sign still indicates “improved” (+200%) instead of being flipped by the minus sign.

Two useful equivalences follow:

“ $B$ is $k$ times $A$ ” ⟺ “from $A$ to $B$ is an increase of $(k - 1) \times 100\%$ ”. 8 times = up 700%, 2 times = up 100%, 1.5 times = up 50%.
After a “decrease of $c\%$ ”, $(1 - c/100)$ of the original remains; a 100% decrease is zero, and a decrease cannot exceed 100% (unless the end value is negative).

Why rises and falls are asymmetric, and why successive changes multiply

+25% followed by −20% returns exactly to the start, because $1.25 \times 0.80 = 1.00$ . Successive percent changes combine by multiplication, not addition: up 10% then down 10% gives $1.10 \times 0.90 = 0.99$ , a net fall of 1%; up 50% then down 50% gives $1.5 \times 0.5 = 0.75$ , a net fall of 25%. The larger the changes, the more the “additive intuition” misleads. This is also the essence of compound interest: $n$ periods each growing by $r$ do not grow by $nr$ but to $(1+r)^n$ times the original.

Symmetric percent difference

When two numbers have no natural order (two measurements of the same quantity, prices at two shops), using either as the denominator is unfair, so the average of their absolute values is used instead: $|A - B| / \big((|A|+|B|)/2\big)$ . Its properties: it is unchanged when $A$ and $B$ are swapped, and for two numbers of the same sign its maximum is 200% (reached when one of them is 0). The “percent difference” in physics lab reports is this quantity; the “percent error” used when comparing with a theoretical value is the percent change with the theoretical value as denominator.

Percentage points versus percent

Subtracting two percentages gives percentage points, not a percentage. An interest rate rising from 3.5% to 4.2% is up 0.7 percentage points and, at the same time, up $0.7/3.5 = 20\%$ . When the news says “approval rose 5%”, it usually means 5 percentage points; confusing the two turns 20% into 0.7%, a factor of 30.

The base effect

For the same absolute change, the smaller the base the larger the percentage: a rise of 20 is 25% on an item costing 80 but only 1% on one costing 2,000. Conversely, a huge percentage on a tiny base may be a very small absolute amount—“300% year-on-year growth” on a base of 1 only means it became 4. Whenever you see a percentage, ask what the base is.

Assumptions

Same dimension: $A$ and $B$ are the same kind of quantity in the same unit. A “percentage” between 25 marks and 200 dollars is meaningless.
Non-zero denominators: the share requires $B \ne 0$ , the percent change requires $A \ne 0$ ; when either is zero the corresponding output shows ∞.
Percent change uses the absolute value of the start: $c = (B - A)/|A|$ . This is the convention of this calculator; some textbooks use $A$ without the absolute value, which reverses the sign when the start is negative.
Solving affects only the solved field: when solving for $A$ or $B$ , the other two percentages are computed with “missing field = 0” and show ∞ or 200; ignore them.
Single-step change: the percent change describes one step from $A$ to $B$ ; for several successive changes, multiply step by step or use the compound interest calculator.

Scope and limitations

Can calculate: the share, single-step percent change and symmetric difference of any two real numbers of the same dimension, and either number from a known share.
Approximate only: results show at most 8 decimal places; when the inputs are themselves rounded data, the last digits of the percentage carry no meaning.
Cannot calculate: compound annual growth rate (CAGR), weighted average percentages, differences in percentage points (just subtract), or comparisons across mixed bases.
Does not: judge whether the inputs are sensible (for example a percentage entered as a decimal), or convert units.
Convention differences: other tools’ “percent difference” and “percent error” may use different denominators; check definitions before comparing results.

Common mistakes

Dividing by the wrong base: “what percent of B is A” divides by B; “by what percent did A change to B” divides by A. For the same 80 and 100 the answers are 80% and 25%.
Adding rises and falls: up 50% then down 50% is not back to the start but a net fall of 25%. Successive changes multiply.
Percentage points as percent: a rate going from 3.5% to 4.2% is up 0.7 percentage points and up 20%, not “up 0.7%”.
Confusing multiples with increases: “tripled” = up 200%, not up 300%.
Entering a percentage as a decimal: when solving, the “A as % of B” field takes 15, not 0.15; entering 0.15 gives A = 0.36 rather than 36.
Ignoring the base: a high percentage on a small base may be negligible; check both denominators before comparing two percentages.
Using the formula without absolute value on negatives: from −50 to 50, $(B-A)/A$ gives −200%, as if things got “worse”; this calculator uses $|A|$ and gives +200%.

Typical use cases

Discounts and sale prices

A 240 item at 15% off (sold for 85% of list): $p = 85$ , $B = 240$ , solve for A to get a sale price of 204. Conversely, “a sale price of 36 is 75% of the original”: $p = 75$ , $A = 36$ , solve for B to get an original price of 48.

Scores and pass rates

45 out of 60: share 75%; 40 last time and 45 this time: change +12.5%. A class average rising from 72 to 75: +4.17%.

Relative error in experimental data

Measured 9.65 m/s², theoretical 9.81: with the theoretical value as A and the measurement as B, the change of −1.63% is the relative error (percent error). For the difference between two independent measurements, 9.65 and 9.78, use “Difference %”: 1.34%.

Financial and statistical reporting

Revenue rising from 250,000 to 2,000,000 is +700% (8 times); a gross margin going from 3.5% to 4.2% is up 0.7 percentage points, or up 20%. State clearly which one you mean in a report so readers are not misled.

FAQ

What is the difference between “A as % of B” and “A→B change %”?

The first answers “how large a part of B is A”, with B as denominator; the second answers “by how much did A rise or fall in becoming B”, with A as denominator. For 80 and 100: share 80%, change +25%.

Why isn’t the percent change simply negated when I swap A and B?

Because the denominator changes: 80→100 divides by 80 and gives +25%; 100→80 divides by 100 and gives −20%. Only “Difference %” is invariant under the swap (22.22%).

When should I use “Difference %”?

When the two numbers have no natural order or base—two measurements, two shops’ quotes, two sample means. It uses their average as denominator and has a maximum of 200%.

Why do the other two percentages become ∞ or 200 when I solve?

In solve mode the engine treats the cleared field as 0 when computing the other outputs, which yields meaningless values. Read only the field you solved for.

What does “an increase of 100%” mean?

Doubling. After “an increase of $c\%$ ” the value is $(1 + c/100)$ times the original; “becoming $k$ times” equals an increase of $(k-1)\times 100\%$ .

How do I calculate a growth rate over several periods?

Convert each period’s change to a multiplier $(1 + c_i/100)$ , multiply them together and subtract 1. Or use the compound interest calculator—it is exactly the repeated product of “the same percentage per period”.

References and further reading

The percentage definition in the CalcX engine source src/data/formulas.ts (targets pct, change and diff; change uses $|A|$ as denominator; supports solving for a / b from pct); every figure on this page was recomputed by that engine.
Wikipedia, Percentage （访问日期：2026-09-08）—definition, history, and the ISO 80000-1 treatment of %.
Wikipedia, Relative change and difference （访问日期：2026-09-08）—comparison of the definitions of relative change, percent error and symmetric percent difference.
Wikipedia, Percentage point （访问日期：2026-09-08）—the distinction between percentage points and percent.

Privacy

All inputs and calculations run inside your browser and are never uploaded to a server. The text on this page is static content; it neither contains nor records any user input.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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