Enter the two values you know out of original value, percent change and new value, leave the one you want to find blank, and press "Calculate". The blank one is calculated.
Enter the percent change with a sign (for a 10% increase, enter "10"; for 10% off, enter "-10"). If all three fields are filled in, nothing can be calculated, so always leave the one you want to find blank.
Result
Enter the two values you know in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter any two of the original value, the percent change and the new value, and the third is calculated on the spot
- "What is $50 increased by 10%?", "200 went up to 250, so what is the percent increase?", "It cost $40 after 20% off, so what was the original price?" This one tool answers all three kinds of questions about going up and going down
- Use it as is for everyday percent increases and decreases such as adding sales tax, sale discounts, pay raises and year-over-year changes
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter the percent change with a sign (for a 10% increase, enter "10"; for 10% off, enter "-10"). Increases over 100% also work as is (for example, tripling is a 200% increase).
What is this calculation used for?
Adding sales tax or a tip in one step (shopping and dining)
With an 8% sales tax, a $40 item costs \(40 \times 1.08 = 43.20\), or $43.20. Remembering that "up 8% = times 1.08" lets you get the total with a single multiplication.
A tip works the same way. A 20% tip on a $65 restaurant bill makes the total \(65 \times 1.2 = 78\), or $78. "Up some percent = times (1 + the rate as a decimal)" is math you can use every day.
The price after a discount, and how to work back to the original price
If a $60 jacket is 30% off, you pay \(60 \times 0.7 = 42\), or $42 (30% off = 0.7 times).
Going the other way, "The jacket was $42 on sale, so what was the regular price?" is answered by dividing by 0.7: \(42 \div 0.7 = 60\), or $60. Adding 30% back onto the sale price gives the wrong answer. Knowing this difference helps you check "compare at" prices and other price claims for yourself.
Checking a pay raise as a percent (work and money)
If your hourly pay goes from $20 to $21, the raise is \((21 - 20) \div 20 \times 100 = 5\%\).
"A 5% raise" is easier to compare with the inflation rate than "a dollar more". It is a common yardstick for judging whether money went up or down, such as raises, Social Security adjustments and rent increases.
Reading "up 15% year over year" in the news (economics)
If a company reports sales up 15% year over year and last year's sales were $2 million, this year's sales are \(2 \times 1.15 = 2.3\), or $2.3 million. You can also check the percent change from the two years' numbers: \((2.3 - 2) \div 2 \times 100 = 15\%\).
Many numbers in business news are percent changes, so knowing this formula lets you check the news for yourself.
Comparing growth between things of different sizes (statistics and data)
Say visitors to Town A went from 200 to 250 (a 25% increase) and visitors to Town B went from 1,000 to 1,050 (a 5% increase). Both towns gained 50 visitors, but the percent change shows that Town A grew much faster.
It is the standard way to compare fairly how things of different sizes go up or down, such as population, website traffic or case counts.
Checking "25% more" and "30% less" on packages (food and household goods)
A 12 oz bag of chips labeled "25% more free" holds \(12 \times 1.25 = 15\) oz. A snack with "30% less sugar" has 0.7 times the sugar of the product it is compared with.
Once you can turn the percents on a package into real amounts, you can decide for yourself which is the better deal, based on both price and amount.
Formula
Formula for the value after a percent increase
Standard notation
(the usual math form)
\(Z\)
\(=\)
\(X\)
\(\times\)
\((\)
\(1\)
\(+\)
\(P\)
\(\div\)
\(100\)
\()\)
In words
(symbols replaced with words)
⑤
\(Z\): value after the increase
\(=\)
④
\(X\): original value
\(\times\)
\((\)
③
\(1\): the original as 1
\(+\)
①
\(P\): percent increase (%)
\(\div\)
②
\(100\): per hundred
\()\)
The formula in words
①
Take the
\(P\): percent increase (%)
②
divide it by
\(100\): per hundred
to turn it into a decimal
③
add it to
\(1\): the original as 1
(the original value itself) to make the multiplier
④
multiply the
\(X\): original value
by that multiplier
⑤
and you get the
\(Z\): value after the increase
Quick example
The price of a $50 item after a 10% price increase is
\(Z\): new price
\(=\)
original price ($50)
\(\times\)
\((\)
\(1\): the original as 1
\(+\)
percent increase (10%)
\(\div\)
per hundred (100)
\()\)
\(10 \div 100 = 0.1\)
\(1 + 0.1 = 1.1\)
\(50 \times 1.1 = 55\)
Key idea
"Up 10%" is the same as "100% of the original + 10% = 110%". So the multiplier is \(1 + 0.1 = 1.1\), and a single multiplication gives the answer.
Working it out in two steps, "original + original × 0.1", gives the same answer (\(50 + 5 = 55\), so $55). Once you are used to the multiplier form, you can add an 8% sales tax in one step by multiplying by 1.08.
Formula for the value after a percent decrease
Standard notation
(the usual math form)
\(Z\)
\(=\)
\(X\)
\(\times\)
\((\)
\(1\)
\(-\)
\(P\)
\(\div\)
\(100\)
\()\)
In words
(symbols replaced with words)
⑤
\(Z\): value after the decrease
\(=\)
④
\(X\): original value
\(\times\)
\((\)
③
\(1\): the original as 1
\(-\)
①
\(P\): percent decrease (%)
\(\div\)
②
\(100\): per hundred
\()\)
The formula in words
①
Take the
\(P\): percent decrease (%)
②
divide it by
\(100\): per hundred
to turn it into a decimal
③
subtract it from
\(1\): the original as 1
(the original value itself) to make the multiplier
④
multiply the
\(X\): original value
by that multiplier
⑤
and you get the
\(Z\): value after the decrease
Quick example
The price of a $50 item at 10% off is
\(Z\): sale price
\(=\)
original price ($50)
\(\times\)
\((\)
\(1\): the original as 1
\(-\)
percent off (10%)
\(\div\)
per hundred (100)
\()\)
\(10 \div 100 = 0.1\)
\(1 - 0.1 = 0.9\)
\(50 \times 0.9 = 45\)
Key idea
"10% off" is the same as "100% of the original − 10% = 90%", so the multiplier is \(1 - 0.1 = 0.9\). Finding the discount first (\(50 \times 0.1 = 5\), so $5) and subtracting it gives the same answer (\(50 - 5 = 45\), so $45).
In the calculator on this page, a decrease is entered as a negative percent change (10% off → "-10"). Putting a negative number for \(P\) in the first formula gives the same calculation as this second formula.
Formula for the percent change (how many percent up or down?)
Standard notation
(the usual math form)
\(P\)
\(=\)
\((\)
\(Z\)
\(-\)
\(X\)
\()\)
\(\div\)
\(X\)
\(\times\)
\(100\)
In words
(symbols replaced with words)
⑤
\(P\): percent change (%)
\(=\)
\((\)
①
\(Z\): new value
\(-\)
②
\(X\): original value
\()\)
\(\div\)
③
\(X\): original value
\(\times\)
④
\(100\): per hundred
The formula in words
①
From the
\(Z\): new value
②
subtract the
\(X\): original value
to get the amount of change
③
divide that by the
\(X\): original value
to get the rate as a decimal
④
multiply it by
\(100\): per hundred
to turn it into a percent
⑤
and you get the
\(P\): percent change (%)
Quick example
If a store's customers grow from 200 to 250, the percent change is
\(P\): percent change (%)
\(=\)
\((\)
customers after (250)
\(-\)
customers before (200)
\()\)
\(\div\)
customers before (200)
\(\times\)
per hundred (100)
\(250 - 200 = 50\)
\(50 \div 200 = 0.25\)
\(0.25 \times 100 = 25\ \ (25\%)\)
Key idea
The sign of the answer tells you the direction. A plus is an increase (\(+25\) is a 25% increase), and a minus is a decrease (\(-20\) is a 20% decrease).
The most important point is to divide by the original value. The same increase of 50 people is a 25% increase from 200, but only a 5% increase from 1,000. The percent change compares the change with the original value, so you always divide by the value before the change.
Do not confuse this with percentage points. If an interest rate goes from 4% to 5%, it rose by 1 percentage point, but its percent change is \((5 - 4) \div 4 \times 100 = 25\%\).
Formula to work back to the original value
Standard notation
(the usual math form)
\(X\)
\(=\)
\(Z\)
\(\div\)
\((\)
\(1\)
\(+\)
\(P\)
\(\div\)
\(100\)
\()\)
In words
(symbols replaced with words)
⑤
\(X\): original value
\(=\)
④
\(Z\): new value
\(\div\)
\((\)
③
\(1\): the original as 1
\(+\)
①
\(P\): percent change (%)
\(\div\)
②
\(100\): per hundred
\()\)
The formula in words
①
Take the
\(P\): percent change (%, + for up, − for down)
②
divide it by
\(100\): per hundred
to turn it into a decimal
③
add it to
\(1\): the original as 1
to make the multiplier
④
divide the
\(Z\): new value
by that multiplier
⑤
and you get the
\(X\): original value
Quick example
If an item cost $40 in a 20%-off sale, its original price is
\(X\): original price
\(=\)
sale price ($40)
\(\div\)
\((\)
\(1\): the original as 1
\(+\)
percent change (−20%)
\(\div\)
per hundred (100)
\()\)
\(-20 \div 100 = -0.2\)
\(1 + (-0.2) = 0.8\)
\(40 \div 0.8 = 50\)
Key idea
"20% off" is a percent change of \(-20\)%, so the multiplier is \(1 + (-0.2) = 0.8\) (0.8 times the original value). To undo an increase, keep the plus sign (for a 10% increase, divide by 1.1).
A common mistake here is to add 20% back onto the $40 (\(40 \times 1.2 = 48\), so $48). The 20% discount was 20% of the original price, not 20% of the sale price, so the right way back is to divide by the multiplier, not to multiply.
Every percent change calculation starts from one formula, "original value × multiplier = new value". The multiplier is "1 ± the rate as a decimal" (up 10% → 1.1, 10% off → 0.9). To find how many percent it changed, use "(new − original) ÷ original × 100". To get back from the new value to the original, divide by the multiplier.
Symbols and terms
Symbols
| \(X\) | X | The original value, before it goes up or down. It is what you compare against, like the $50 in "$50 increased by 10%". |
| \(P\) | P | The percent change. It shows how many percent the value went up or down compared with the original. On this page it has a sign: \(+10\) is a 10% increase and \(-10\) is 10% off. |
| \(Z\) | Z | The new value, after it goes up or down. It is the $55 in "$50 increased by 10% is $55". |
| \(Z - X\) | Z minus X | The amount of change, how much the value actually went up or down. A minus sign shows a decrease. It is not a percent but an amount in real units, such as dollars or people. |
| \(1 + P \div 100\) | 1 plus P divided by 100 | The multiplier. It is the size of the new value when the original counts as 1 (up 10% → 1.1 times, 10% off → 0.9 times). |
| \(\%\) | percent | The percent sign. It shows how many out of 100. The word comes from the Latin per centum, "for each hundred". |
Terms
| percent change | How much a value went up or down compared with the original, written as a percent. It is called a percent increase when it goes up and a percent decrease when it goes down, but the calculation is the same. |
| percent increase | Adding a set percent on top of the original value. "Up 10%" or "10% more" makes the new value 1.1 times the original. |
| percent off | Taking a set percent away from the original price, as in "20% off". The sale price is 0.8 times the original. It is also called a percent decrease or a discount. |
| multiplier | The size of the new value when the original counts as 1. Up 10% gives 1.1 times, 10% off gives 0.9 times. Seeing every increase and decrease as "multiply or divide by the multiplier" is the key idea behind the formulas on this page. |
| original value | The amount you compare against in a percent calculation (also called the base). For a percent change, it is always the value before it went up or down. It is easy to divide by the wrong number, so take care. |
| percent | A rate written with the base counted as 100. The sign is %. Multiply a decimal rate by 100 to get the percent (0.25 → 25%). |
| year over year | A comparison with the same period one year earlier, often shortened to YoY. "Sales up 15% year over year" gives the percent change from last year. News and company reports use it all the time. |
| percentage point | The plain difference between two percents. A rate that goes from 4% to 5% rose by 1 percentage point, which is a 25% increase in the rate itself. News reports often mix the two up. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| Percents (Grade 6) |
|
| Percent increase and decrease (Grade 7) |
|
| Positive and negative numbers (Grades 6–7) |
|
| Multiplying and dividing decimals (Grades 5–6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for the value after an increase (500 increased by 10%?)
| Original value | 500 |
| Percent increase (%) | 10 |
| Value after increase | =B1*(1+B2/100) |
Table for the value after a decrease (500 with 10% off?)
| Original value | 500 |
| Percent off (%) | 10 |
| Value after decrease | =B1*(1-B2/100) |
Table for the percent change (200 to 250 is what percent increase?)
| Original value | 200 |
| New value | 250 |
| Percent change (%) | =(B2-B1)/B1*100 |
Table for the original value (400 after 20% off, so what was it before?)
| New value | 400 |
| Percent change (%, minus for off) | -20 |
| Original value | =B1/(1+B2/100) |
After pasting, B1 and B2 are your inputs and B3 is calculated automatically.
"B1" and "B2" in a formula tell Excel to use the number in that cell. "*" is multiplication and "/" is division.
For example, B3 shows 550 in the first table, 450 in the second, 25 in the third and 500 in the fourth. In the fourth table, enter the percent change with a sign (-20 for 20% off). Just replace B1 and B2 with your own numbers.
"B1" and "B2" in a formula tell Excel to use the number in that cell. "*" is multiplication and "/" is division.
For example, B3 shows 550 in the first table, 450 in the second, 25 in the third and 500 in the fourth. In the fourth table, enter the percent change with a sign (-20 for 20% off). Just replace B1 and B2 with your own numbers.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for the value after an increase (500 increased by 10%?)
| Original value | 500 |
| Percent increase (%) | 10 |
| Value after increase | =B1*(1+B2/100) |
Table for the value after a decrease (500 with 10% off?)
| Original value | 500 |
| Percent off (%) | 10 |
| Value after decrease | =B1*(1-B2/100) |
Table for the percent change (200 to 250 is what percent increase?)
| Original value | 200 |
| New value | 250 |
| Percent change (%) | =(B2-B1)/B1*100 |
Table for the original value (400 after 20% off, so what was it before?)
| New value | 400 |
| Percent change (%, minus for off) | -20 |
| Original value | =B1/(1+B2/100) |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace B1 and B2 with your own numbers.
How to calculate it in Python
# Find the new value (500 increased by 10%? For 10% off, set percent = -10)
before = 500 # original value
percent = 10 # percent change (%): + for increase, - for decrease
after = before * (1 + percent / 100)
print(f"{before} changed by {percent}% is {after}")
# Find the percent change (200 to 250 is what percent change?)
before = 200 # original value
after = 250 # new value
percent = (after - before) / before * 100
print(f"The percent change from {before} to {after} is {percent}%")
# Work back to the original value (it is 400 after 20% off, so what was it before?)
after = 400 # new value
percent = -20 # percent change (%): -20 because it is 20% off
before = after / (1 + percent / 100)
print(f"If a {percent}% change gives {after}, the original value is {before}")
Runs with the standard library only. Write the percent change with a sign (10 for a 10% increase, -10 for 10% off). Pick the block for the value you want to find, change the numbers at the top, and run it.
How to write it in LaTeX and other math languages (copy and paste)
Formula for the value after a percent increase
Z = X × (1 + P ÷ 100)
Z = X \times \left(1 + \frac{P}{100}\right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Z</mi>
<mo>=</mo>
<mi>X</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac><mi>P</mi><mn>100</mn></mfrac>
<mo>)</mo>
</mrow>
</mrow>
</math>
Z = X xx (1 + P/100)
before*(1 + percent/100)
after := before*(1 + percent/100);
after = before*(1 + percent/100);
Z = X × (1 + P/100)
Formula for the value after a percent decrease
Z = X × (1 − P ÷ 100)
Z = X \times \left(1 - \frac{P}{100}\right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Z</mi>
<mo>=</mo>
<mi>X</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac><mi>P</mi><mn>100</mn></mfrac>
<mo>)</mo>
</mrow>
</mrow>
</math>
Z = X xx (1 - P/100)
before*(1 - percent/100)
after := before*(1 - percent/100);
after = before*(1 - percent/100);
Z = X × (1 - P/100)
Formula for the percent change (how many percent up or down?)
P = (Z − X) ÷ X × 100
P = \frac{Z - X}{X} \times 100
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>Z</mi><mo>−</mo><mi>X</mi></mrow>
<mi>X</mi>
</mfrac>
<mo>×</mo>
<mn>100</mn>
</mrow>
</math>
P = (Z - X)/X xx 100
(after - before)/before*100
percent := (after - before)/before*100;
percent = (after - before)/before*100;
P = (Z - X)/X × 100
Formula to work back to the original value
X = Z ÷ (1 + P ÷ 100)
X = \frac{Z}{1 + \frac{P}{100}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>X</mi>
<mo>=</mo>
<mfrac>
<mi>Z</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac><mi>P</mi><mn>100</mn></mfrac>
</mrow>
</mfrac>
</mrow>
</math>
X = Z/(1 + P/100)
after/(1 + percent/100)
before := after/(1 + percent/100);
before = after/(1 + percent/100);
X = Z/(1 + P/100)
How to have ChatGPT do the calculation
You are a calculation assistant for percent increase and decrease. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 1. $50 increased by 10% 2. The percent change when 200 people grow to 250 3. The original price of an item that cost $40 in a 20%-off sale Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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