Percentage Change Calculator

The formula and the base you divide by

Percent change compares two values and reports the gap as a share of the starting point. Subtract the old value from the new one, divide by the old value, then multiply by 100: change = (new - old) / old x 100.

The number under the line is the whole question. Divide by the old value and you get how far the figure moved from where it started. Divide by the new value and you get how big the change is compared with where it ended up. Both are percentages, both are useful in different sentences, and only one of them is percent change.

The same $6.50 sits inside all three rows. Only the base changes, and the answers differ by more than five points.

Worked example: a price that rose

An item goes from $24.99 to $31.49. The change is 31.49 - 24.99 = 6.50, and the percent change is 6.50 / 24.99 x 100 = 26.01, so the price rose 26.01 percent.

The new price is also 126.01 percent of the old one, and the multiplier is 31.49 / 24.99 = 1.26. Reports mix these up constantly. A line that says 'prices are up 126 percent' when the figure moved 26 percent overstates the move by a factor of five, and the arithmetic behind the mistake is usually a multiplier quoted as a percent.

If the rise happened across twelve months, the monthly rate is not 26.01 / 12 = 2.17 percent. Compounding matters even at these sizes: twelve steps of 2.17 percent multiply out to 1.0217 to the twelfth power, which is 29.4 percent, three points above the move that happened. When a page reports a period, keep the period with the number.

Worked example: a price that fell

A price drops from $89 to $71. The change is 71 - 89 = -18, and -18 / 89 x 100 = -20.22, so the fall is 20.22 percent. The minus sign is doing real work: it says the direction, and dropping it turns a discount into a markup.

Dividing by the new price instead gives 18 / 71 = 25.35 percent, which answers a different question: what share of today's price the discount represents. Retailers quote that one because it sounds bigger, and it is not wrong, it is just labelled wrong when it is called percent change.

Reversing the move takes a different percentage than the one that made it. Undoing a 20.22 percent fall needs 18 / 71 = 25.35 percent on the way back up, because the base changed when the price fell. The calculator shows that undo figure next to the change so the asymmetry stays visible.

Percent change versus percentage points

When the thing being measured is already a percentage, there are two honest answers and they are not interchangeable. The difference in the raw numbers is measured in percentage points. The relative move is measured in percent.

The Bureau of Labor Statistics walkthrough of CPI math makes the point with index levels: the CPI moved from 239.261 in April 2016 to 240.236 in May 2016, a change of 0.975 index points, and 0.975 / 239.261 = 0.004075 becomes 0.4 percent once multiplied by 100. The index points are the raw gap; the percent is the gap scaled to the starting level.

The interest rate row is the trap. A rate climbing from 4.2 percent to 5.1 percent rose 0.9 percentage points, which is a 21.4 percent increase in the rate itself (0.9 / 4.2 = 0.2143). A headline that says 'rates jumped 21 percent' is accurate about the rate and misleading about borrowing costs; a headline that says 'rates rose 0.9 percent' is wrong. The BLS FAQ on index numbers explains the same distinction for published statistics.

Several moves: total change and the yearly rate

Percent changes do not add. A price up 20 percent then down 20 percent ends at 1.20 x 0.80 = 0.96, which is 4 percent below where it started. Two rises of 20 percent give 1.20 x 1.20 = 1.44, a 44 percent gain rather than 40. The order does not change the product, which is why a portfolio that drops 50 percent needs a 100 percent gain to recover: 0.50 x 2.00 = 1.00.

A long run needs the same treatment. An index going from 250 to 310 is up 24 percent in total. The yearly rate is the r that solves 250 x (1 + r) to the fourth power = 310, and that works out to 0.0553, or 5.53 percent a year. Dividing 24 by 4 gives 6 percent a year, which compounds to 1.06 to the fourth power = 1.2625, a 26.25 percent gain that never happened.

The gap between the two columns shrinks as the period lengthens, but it never disappears. For anything quoted as an annual return, the compound version is the correct one; the SEC investor site runs the same compounding on savings figures.

Working backwards to the original value

Retail and payroll questions often arrive backwards: the final number and the percent are known, the starting number is not. Divide instead of multiply. After a 15 percent cut the price is $68, so the original was 68 / 0.85 = 80.

A rise works the same way: a bill of $56 after a 12 percent increase came from 56 / 1.12 = 50. Subtracting 12 percent from 56 gives 49.28, which is the wrong answer, and it is the error people make when they apply the percentage in the wrong direction. The base for the 12 percent was $50, not $56.

The same trap shows up with sales tax. If a receipt total includes 8 percent tax, the pre-tax amount is total / 1.08. Multiplying the total by 0.92 understates the tax by about 0.6 percent of the bill, which is small on a coffee and not small on a car.

When percent change misleads

Negative starting values. A loss shrinking from -$20 to -$10 gives (-10 - -20) / -20 = -50 percent, which reads as an improvement in the wrong direction. When the base is zero or negative, report the dollar change and leave the percentage out.

Tiny bases. Going from 2 to 3 is a 50 percent rise, and from 2 to 4 is 100 percent. On a base of two units the percentage amplifies noise, so quote the counts as well.

Averaging percentages. The average of +50 percent and -50 percent is zero, but the money went 1.50 x 0.50 = 0.75, a 25 percent loss. Averaging rates only works on logarithms, and nobody quotes those in a headline.

Mixing periods. A 10 percent annual rise and a 10 percent monthly rise are the same number and different events. State the interval every time, especially for prices, rent and subscriptions.

Checks before you quote a number

Four quick checks catch most errors. First, name the base out loud: 'up 26 percent from $24.99' leaves no room for the new-denominator reading. Second, keep the absolute change beside the percent, since 26 percent of $25 and 26 percent of $2,500 look identical in a headline.

Third, round once, at the end. Doing the arithmetic on $25.00 and $31.50 instead of $24.99 and $31.49 gives 26.00 percent against the true 26.01; the difference is trivial here and can flip a rounded tenth on smaller numbers. Fourth, sanity-check the direction against the sign: a negative change paired with the word 'increase' means one of the two is wrong.

If what you need is the yearly dollar effect of a single raise rather than a general change figure, the percent increase calculator takes that narrower case and adds the pay math. This page is built for the general one: either direction, several periods, index series and the percentage-point distinction.