People type this into search every day: percentage increase calculator.
The usual answer is “subtract, then divide by the old number.” That is the whole method. People still get stuck on the reverse question, and on two discounts that look like they add.
This page is a labeled example. The numbers are the formula. They are not a forecast of prices in your town.
The new value sits over the old value
The Bureau of Labor Statistics percent-change factsheet uses the same steps for the Consumer Price Index. Subtract the earlier figure from the later one. Divide that difference by the earlier figure. Multiply by 100. Their December 2021 to December 2022 CPI-U all-items example is (296.797 − 278.802) ÷ 278.802 × 100 = 6.5%. That 6.5% is BLS’s worked index example. It is not a grocery quote, and it is not this site’s default.
BLS also says a year-over-year change uses the same month in both years. January to December is 11 months, not a year. And the yearly change is not the sum of the monthly changes. Those two warnings are why people search this.
The percentage change and reverse price calculator does both jobs:
change = (new − original) ÷ original
reverse original = changed value ÷ (1 + known percentage)
The two comparison values need the same units. The reverse step applies one percentage, with no compounding.
Fig 1
Forward change, then the reverse
The old value is the baseline. The reverse step divides by the change factor.
80 to 100 is a 25% rise. 120 after a 20% rise reconstructs 100.
If the original value is 0, the tool stops and says there is no baseline. You cannot divide by zero.
The published 80-to-100 example
Here are the calculator defaults:
- original value: 80
- new value: 100
- value after a known change: 120
- known percentage change: 20%
Absolute change: 100 − 80 = 20
Percentage change: 20 ÷ 80 × 100 = 25%
Known change factor: 1 + 20 ÷ 100 = 1.2
Reconstructed original: 120 ÷ 1.2 = 100
Calculator result
Percentage change & reverse price
- Original value
- 80
- New value
- 100
- Value after a known change
- 120
- Known percentage change
- 20%
25%
change from original to new
- Absolute change
- 20
- Reconstructed original value
- 100
- Known change factor
- 1.2
80 to 100 is the forward pair. 120 after a 20% rise is a separate reverse pair. They share a card because people ask both questions on the same afternoon.
Fig 2 · 1/2
The 80-to-100 rise
Calculator defaults for the forward pair
Same units
Original
80
New
100
The original value is 80.
The new value is 100, which is a 25% increase and an absolute change of 20.
A 20% rise on 80 would have been 80 × 1.20 = 96, not 100. 100 is 25% up from 80. That gap is why the reverse box is its own input.
Percentage decrease uses the same fraction
Hold the reverse boxes at the defaults. Change only the forward pair.
- 50 to 60: absolute 10, change 20%
- 80 to 92: absolute 12, change 15%
- 80 to 100 (default): absolute 20, change 25%
- 200 to 230: absolute 30, change 15%
- 100 to 80: absolute −20, change −20%
80 to 92 and 200 to 230 are both 15%. The absolute changes are 12 and 30. Percent change does not care that 30 is larger. It cares that 12 ÷ 80 and 30 ÷ 200 are the same 0.15.
A drop from 100 to 80 is −20%, not −25%. The baseline is 100, so 20 ÷ 100 = 0.20. People look at the 80-to-100 rise, see 25%, and expect the trip back to use 25% too.
Fig 3 · 1/5
Five labeled pairs
Absolute change divided by the original value
- Percent change
| Category | Percent change |
|---|---|
| 50→60 | $20 |
| 80→92 | $15 |
| 80→100 | $25 |
| 200→230 | $15 |
| 100→80 | -$20 |
50 to 60 is 20%.
100 to 80 is −20%. The 80-to-100 default is +25%.
Raise 80 by 25% and you get 100. Cut that 100 by 25% and you get 75, not 80. 1.25 × 0.75 = 0.9375, so the round trip is a 6.25% drop from the first 80. The two percentages do not cancel because they sit on different baselines.
Reverse the percentage when you only have the later number
The reverse box answers “what was this before the 20% rise?”
120 ÷ 1.20 = 100
A 25% fall, later value 90:
Factor: 1 − 0.25 = 0.75
Reconstructed original: 90 ÷ 0.75 = 120
The forward pair on the default card is still 80 to 100, so the primary print stays 25%. Type −25 and 90 in the reverse boxes if you want that 120 on the card.
A 16% rise, later value 87:
Factor: 1.16
Reconstructed original: 87 ÷ 1.16 = 75
Sale copy that says “20% off, now $120” is claiming the old price was 120 ÷ 0.80 = 150, if 20% off is the only change. The tool’s default known change is +20, so 120 reconstructs 100. Flip the sign when the change was a discount.
The reverse step is one factor. Two stacked changes need two divisions, in reverse order.
A grocery basket turns the percent into a yearly bill
The grocery price change calculator uses the same percent, then multiplies the dollar gap by how often you shop:
(current basket − earlier basket) ÷ earlier basket × 100
The baskets need equivalent quantities and products. The trip count stays constant.
Defaults:
- earlier basket: $145
- current basket: $158
- shopping trips per month: 4
Change per trip: 158 − 145 = $13.00
Basket price change: 13 ÷ 145 × 100 = 9%
The raw fraction is 8.9655…%. The tool prints one decimal, which lands on 9%.
Annual spending change: 13 × 4 × 12 = $624.00
Calculator result
Grocery price change
- Earlier basket price
- $145.00
- Current basket price
- $158.00
- Shopping trips per month
- 4 trips
9%
basket price change
- Change per trip
- $13.00
- Annual spending change
- $624.00
Two trips a month, same $13 gap: annual change $312.00. The percent is still 9%. Frequency moved the yearly cash, not the percent.
A labeled 20% basket, $145 to $174, still 4 trips:
Change per trip: $29.00
Annual change: $1,392.00
$145 × 1.20 = $174. That is a clean 20% so you can see the yearly line without the 8.9655 rounding.
Put the 80-to-100 pair in this tool as $80 to $100 with 4 trips.
Change per trip: $20.00
Basket change: 25%
Annual change: $960.00
The 25% matches the percentage card. The $960 is what four $20 gaps do over 12 months.
Fig 4 · 1/2
The same $13 gap, two trip counts
$145 to $158. The percent stays 9%.
Annual spending change
2 trips / mo
$312
4 trips / mo
$624
Two trips a month is $312.00 a year.
Four trips a month is $624.00 a year.
The rent increase post is this idea on a monthly rent. A percent there is still (new − old) ÷ old. The unit price post is a different fraction, price divided by quantity.
Two discounts multiply. They do not add.
“20% off, then 10% off” looks like 30% off. The discount stack and final price calculator applies them in order:
price × (1 − discount 1) × (1 − discount 2) × (1 + tax) + shipping
Tax applies after the discounts and before shipping.
Defaults:
- original price: $180
- first discount: 20%
- second discount: 10%
- sales tax: 8.5%
- shipping: $12
After the 20%: 180 × 0.80 = $144.00
After the 10%: 144 × 0.90 = $129.60
That pair is 1 − 0.80 × 0.90 = 28% off the $180, not 30%.
Tax: 129.60 × 0.085 = $11.02
Final price: 129.60 × 1.085 + 12 = $152.62
Saved versus original plus tax plus shipping: 180 × 1.085 + 12 = $207.30, then 207.30 − 152.62 = $54.68
Calculator result
Discount stack and final price
- Original price
- $180.00
- First discount
- 20%
- Second discount
- 10%
- Sales tax
- 8.5%
- Shipping
- $12.00
$152.62
final price
- After discounts
- $129.60
- Tax
- $11.02
- Saved vs original + tax + shipping
- $54.68
Fig 5 · 1/3
The $180 ticket after both discounts
20% then 10%, before tax
First 20%
$36.00
Next 10%
$14.40
After both
$129.60
The first 20% leaves $144.00.
The second 10% leaves $129.60, which is 28% off, not 30%.
One 30% discount and no second discount, same tax and shipping:
After discounts: $126.00
Final price: $148.71
That is 152.62 − 148.71 = $3.91 cheaper than the stacked 20-then-10 path. Adding the percents overstated the sale by $3.60 on the merchandise (129.60 − 126.00) and by $3.91 after tax.
Turn the second discount off and keep the 20%:
After discounts: $144.00
Final price: $168.24
Fig 6 · 1/3
Three ways to leave the $180
Tax 8.5% and $12 shipping stay on every bar
- After discounts
- Tax
- Shipping
| Category | After discounts | Tax | Shipping |
|---|---|---|---|
| 20% only | $144 | $12 | $12 |
| 20% then 10% | $130 | $11 | $12 |
| 30% only | $126 | $11 | $12 |
20% only finishes at $168.24.
20% then 10% finishes at $152.62. A single 30% finishes at $148.71.
Markup is a different percent. The markup vs margin post starts from cost and asks what selling price you need. A 25% markup on 80 is 100. A 25% margin on 100 is 25 of selling price, which is a different split.
Open the percentage change and reverse price calculator with the two numbers you want to compare, or with the later number and the one percentage you know. Use the grocery price change calculator when the percent should become a yearly grocery gap. Use the discount stack calculator when two percentages land on the same ticket.