Key takeaway
Percent change between two numbers = ((new − old) ÷ old) × 100. Positive means an increase, negative a decrease. Between 50 and 65 the change is ((65 − 50) ÷ 50) × 100 = +30%. The base is the first number you name as 'old' — pick it deliberately, because swapping the two inverts both sign and size.
Scope and assumptions
General arithmetic. This measures change between two states, not one number as a share of another. This site is informational and not official.
Short answer: Percent change between two numbers = ((new − old) ÷ old) × 100. Positive means an increase, negative a decrease. Between 50 and 65 the change is ((65 − 50) ÷ 50) × 100 = +30%. The base is the first number you name as "old" — pick it deliberately, because swapping the two inverts both sign and size. This page is the focused signed-change article (order matters); for the combined up-and-down treatment see Percentage Increase and Decrease, and for a single number as a share of another see Calculate Percentage of Two Numbers.
Quick navigation
- Increase & decrease → percentage-increase-and-decrease
- Net change → net-percentage-change
- Two-number share → calculate-percentage-of-two-numbers
- The calculator → Percentage Calculator
1. The formula
% change = ((new − old) ÷ old) × 100
The first number you name is "old" (the base); the second is "new".
2. Worked pairs
| Old | New | ((new−old)÷old)×100 | Read |
|---|---|---|---|
| 50 | 65 | +30% | increase |
| 65 | 50 | −23.1% | decrease |
| 100 | 120 | +20% | increase |
| 120 | 100 | −16.7% | decrease |
3. Why order flips the answer
From 50 → 65 the base is 50, so +30%. From 65 → 50 the base is 65, so −23.1%. The same two numbers give different percents depending on which is "old" — always state the direction.
4. A population example
A town grew from 8,000 to 9,200 residents.
((9200 − 8000) ÷ 8000) × 100 = (1200 ÷ 8000) × 100 = +15%
5. Share vs change (don't mix them)
- Share: 50 is what % of 65? → (50÷65)×100 = 76.9% (a part of a whole).
- Change: 50 → 65? → +30% (a move between states).
Different questions, different formulas.
6. Common mistakes
- Wrong base — using the new number as the denominator.
- Ignoring the sign — a drop is negative; report it.
- Swapping old/new without rethinking — sign and size both change.
- Confusing with a share — change compares states; a share compares part to whole.
7. Naming the move
Always state the order: "from 50 to 65 is +30%". If you only write "30%", a reader cannot tell whether 50 or 65 was the start. When the two numbers are close, the percent is large (50→65 = +30%); when they are far apart in ratio, even a big absolute gap is a small percent. The wording — old first, then new — removes every ambiguity about the base, which is the number you divide by.
8. Old first, then new
The cleanest way to report a change is "from old to new", with the old value stated first so the base is explicit. From 50 to 65 is +30%; if you only write 30%, a reader cannot recover whether 50 or 65 was the start. Stating the endpoints also makes the sign obvious — a fall reads as negative, a rise as positive — and removes the ambiguity that leads people to quote a percent with the wrong direction. The two numbers, in order, are the whole answer, and the base is named by that order.
9. The order names the base
Reporting a change as "from old to new" does more than sound tidy — it names the base in the word order. From 50 to 65 uses 50 as the base and returns +30%; reverse the order and the base becomes 65, returning −23%. The numbers are identical, the answers differ, and the order is the only thing that decided. Whenever you state a percent change, lead with the old value so the base is unmistakable and the sign and size are both fixed by the sentence you wrote.
FAQ
Q: What is the percent change between two numbers? ((new − old) ÷ old) × 100. Positive = increase, negative = decrease.
Q: What is the change from 50 to 65? ((65 − 50) ÷ 50) × 100 = +30%.
Q: Does the order matter? Yes — from 65 to 50 it is −23.1%, a different sign and size, because the base flips.
Q: Is this the same as one number as a percent of another? No. That is a share (see calculate-percentage-of-two-numbers); this is a before/after move.
Q: What base do I use? The earlier (old) number is the base; the later number is new.
Sources: Investopedia — Percentage Change; Khan Academy percentages.
Disclaimer: General arithmetic. This measures change between two states, not one number as a share of another. This site is informational and not official.
Sources and references
- Investopedia — Percentage Changehttps://www.investopedia.com/terms/p/percentage-change.asp
- Khan Academy percentageshttps://www.khanacademy.org/math/pre-algebra/x31947de81e3f6453:percentages