Percent Decrease Calculator

Background.

A share that opened the week at $150 and closed at $126 fell $24, and $24 is 16% of $150. That is all a percent decrease is: the drop, measured as a fraction of where you started, times 100. Enter the old value and the new value and the calculator returns both numbers, the percentage and the plain drop in whatever units you used, so you can say "down 16%" and "down $24" in the same breath.

The starting value is the yardstick, and that single choice explains most of the surprises percentages cause. A price cut from $100 to $80 is a 20% decrease, but climbing back from $80 to $100 is a 25% increase, because the second change is measured against the smaller number. The same asymmetry is why a portfolio that halves needs to double to get back to even, and why two successive 10% cuts take off 19%, not 20%.

The result is signed so that a fall reads as a positive number. If the new value turns out to be larger than the old one, the answer comes back negative: −25% here means the quantity rose by 25%, and the percent increase calculator will show the same change as +25%. A decrease can also exceed 100%, which happens when a quantity falls past zero, as when a company's profit of 40 million turns into a loss of 10 million.

The old value cannot be zero, since no fraction of nothing can describe a change. Negative old values are allowed and handled by dividing by the size of the old value, so a reading that moves from −50 to −80 counts as a 60% decrease: it went down.

What is percent decrease calculator?

Percent decrease expresses how far a quantity has fallen as a share of its original value. In symbols it is (O − N) ÷ |O| × 100, where O is the old value and N the new one. It is a relative measure: a $24 drop is 16% of a $150 share but 48% of a $50 one, and the percentage is what lets you compare declines in things of very different size, such as two share prices, the populations of a village and a city, or a body weight in kilograms against one in pounds. Because it has no units, the old and new values must be in the same units as each other, but which units does not matter. Percent decrease belongs to the family of percent change measures that the Common Core standards introduce in seventh grade, alongside percent increase, markdowns and percent error. It answers a different question from a percentage-point change, which is the plain difference between two values that are themselves percentages, and from percent difference, which divides by the average of two values when neither is the natural starting point: 150 and 126 differ by 24 ÷ 138 × 100 = 17.3913% on that measure. A percent decrease is also nominal unless you adjust it: a price that stays flat for a year while general prices rise 3% has fallen 2.9126% in real terms, because 1 − 1 ÷ 1.03 = 0.029126.

How to use this calculator.

1. 01Enter the earlier or starting figure as Old value: the original price, last year's count, the weight before the diet.
2. 02Enter the later figure as New value, in the same units as the old one and measured the same way: both prices with tax or both without, both weights taken at the same time of day.
3. 03Read Percent decrease for the drop as a share of the old value, and Absolute change for the same drop in the original units. A negative result in either means the value rose.
4. 04If the two numbers are themselves rates, such as interest rates or unemployment rates, also note their plain difference in percentage points; the percent decrease alone will look much larger.
5. 05If you swapped the two values, the answer is not simply reversed in sign: a fall from 100 to 80 is 20%, but 80 entered as old and 100 as new gives −25%.

The formula.

The calculator computes two things. The absolute change is O − N, kept in your units, positive when the value fell. The percent decrease divides that drop by the size of the old value and scales it to a percentage: (O − N) ÷ |O| × 100.

The absolute value in the denominator only matters when the old value is negative. Without it, a move from −50 to −80 would give 30 ÷ (−50) × 100 = −60%, which would read as a rise even though the number went down. With it, the result is 30 ÷ 50 × 100 = +60%. For ordinary positive quantities |O| is just O.

A few properties follow directly from the formula. A fall to exactly zero is always a 100% decrease. A fall below zero gives more than 100%: from 40 to −10 the drop is 50, and 50 ÷ 40 × 100 = 125%. Percent decrease and percent increase for the same pair of numbers are mirror images, so the decrease from O to N always equals minus the increase from O to N. And decreases do not add: a fall of d% needs a rise of d ÷ (100 − d) × 100 percent to get back, which is 25% after a 20% fall and 100% after a 50% fall.

Economists sometimes avoid that asymmetry by using log changes instead: 100 × ln(N ÷ O). A fall from 150 to 126 is −17.4353 log points and the climb back is +17.4353, so the two cancel exactly, and log changes over successive periods simply add. For small changes the two measures nearly agree; for large ones they part company, with a 20% decrease equal to about −22.3144 log points. The calculator reports the ordinary percent decrease, which is what prices, budgets and news reports use.

The result is an exact ratio of the two values you enter. When it does not terminate, as with a fall from 3 to 2, the calculator shows it rounded, here as 33.3333%.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Common Core State Standards for Mathematics, Grade 7, Ratios & Proportional Relationships, Standard 7.RP.A.3 — proportional relationships applied to multistep ratio and percent problems, explicitly naming percent increase and decrease.
2. [2]Common Core State Standards for Mathematics, Grade 6, Ratios & Proportional Relationships, Standard 6.RP.A.3(c) — finding a percent of a quantity as a rate per 100, the foundational skill percent decrease builds on.
3. [3]NIST Guide to the SI, Special Publication 811, §7.10.2 — rules and conventions for expressing relative differences as dimensionless percentages.
4. [4]Board of Governors of the Federal Reserve System — FAQ explaining the distinction between percentage points and percent change when describing rate movements.
5. [5]NIST Digital Library of Mathematical Functions, §4.2 — formal definitions of logarithm, exponential, and power relationships underlying ratio and proportional-decline arithmetic.