How to Find Percentage Change Step by Step

The basic formula is change divided by original value, multiplied by 100. Suppose a price rises from 40 to 50. The change is 10. Divide 10 by the original value, 40, to get 0.25. Multiply by 100, and the percentage change is a 25 percent increase.

The original value is the key part of finding percentage change. If the price instead falls from 50 to 40, the change is 10 in size, but now you compare that 10 with the original 50. Ten divided by 50 is 0.2, so the decrease is 20 percent. A 25 percent increase and a 20 percent decrease can therefore reverse each other between the same two numbers.

You may also write percentage change as new value minus original value, divided by original value, multiplied by 100. This version keeps the sign. A positive result shows an increase and a negative result shows a decrease. In many school questions, you find the size first and then label it clearly as an increase or decrease.

Here is another example. A test score rises from 60 to 72. The increase is 12. Divide 12 by the original score of 60 to get 0.2, then multiply by 100. The score increased by 20 percent. Notice that the answer describes the relative change, not the 12-point difference itself.

A percentage is always a comparison with some reference amount. In percentage change, the reference is where you started. That is why you divide by the original value rather than the new value or the average of the two. Changing the denominator changes the question you are answering.

When finding percentage change, say the comparison aloud: the amount changed by this much, compared with what it was before. That sentence points directly to the original value. It is a useful check when the problem contains several numbers and you are tempted to divide by whichever one appears last.

Percentage change is useful because it lets you compare changes across different starting sizes. An increase of 10 units is huge if the original value was 20, but small if the original was 1,000. Converting both changes to percentages puts them on a common scale and makes the comparison more meaningful.

A steady method is more reliable than trying to remember several separate percentage rules:

• •Identify the original value and the new value.
• •Subtract to find the size of the change.
• •Divide the change by the original value.
• •Multiply by 100 to convert the decimal to a percentage.
• •Label the answer as an increase or a decrease.

Be careful when the original value is zero. The usual percentage change formula asks you to divide by the original value, and division by zero is undefined. In that case, report the absolute change or use another comparison that fits the context rather than forcing the standard formula.

Percentage changes do not simply add up. Suppose a price of 100 rises by 10 percent to 110, then falls by 10 percent. The fall is 10 percent of 110, which is 11, so the final price is 99, not 100. A quick way to handle this is with multipliers. A 10 percent increase means multiplying by 1.1, and a 10 percent decrease means multiplying by 0.9. Multiply them together, 1.1 times 0.9 equals 0.99, and the overall change is a 1 percent decrease.

Another common confusion involves percentage points. If an interest rate rises from 2 percent to 3 percent, it has risen by one percentage point, but by 50 percent in relative terms, because the increase of 1 is half of the original 2. Both statements are true, yet they sound very different. News reports and advertisements sometimes choose whichever version sounds more dramatic, so it is worth checking which kind of change is being described.

A related question asks you to find the original amount after a percentage change has already happened. Suppose a jacket costs 60 pounds after a 20 percent price rise, and you want the price before the rise. The tempting method is to take 20 percent off 60, giving 48, but that is wrong, because the 20 percent was calculated on the old price, not the new one. Taking 20 percent of the larger new figure removes too much.

Multipliers make this kind of problem straightforward. A 20 percent increase means the new price is the original multiplied by 1.2, so divide the new price by 1.2 to reverse it: 60 divided by 1.2 is 50. You can check by increasing 50 by 20 percent, which gives 60 again. Decreases work the same way. If a sale price of 72 follows a 10 percent discount, the original was 72 divided by 0.9, which is 80. Questions about prices before tax are a common real use of this method.

When a quantity changes over several years, the total change is not the yearly rate multiplied by the number of years. Suppose an investment grows from 100 to 150 pounds over five years. That is a 50 percent increase overall, but not 10 percent a year, because each year's growth builds on the last. The steady yearly rate that does the job is about 8.4 percent, found by taking the fifth root of the multiplier 1.5 and subtracting 1. You can check it: 1.084 multiplied by itself five times gives very nearly 1.5.

A handy shortcut for compound growth is the rule of 72. Divide 72 by the yearly percentage rate and you get the approximate number of years needed for a quantity to double. At 6 percent a year, doubling takes about 72 divided by 6, or 12 years, and the exact figure is very close to that. At 8 percent, it takes about 9 years. The rule works well for rates between roughly 2 and 15 percent. It also works for inflation: if prices rise by 3 percent a year, they double in about 24 years, halving what a fixed sum of money can buy.

Health news often reports relative changes, which can sound alarming when the starting figure is small. Suppose a medicine raises the risk of a side effect from 2 in 1,000 people to 3 in 1,000. The percentage change in risk is 1 divided by 2, multiplied by 100, a 50 percent increase. That is accurate, but the absolute change is only 1 extra case per 1,000 people, or 0.1 percentage points. A useful habit is to look for both figures: the relative change tells you how strong the effect is, and the absolute change tells you how many people it actually affects.

Sometimes there is no original value, because you are comparing two things rather than tracking one over time. If one shop sells a game for 45 pounds and another for 55, the difference of 10 is about 22 percent of 45 but only about 18 percent of 55, so the answer depends on which price you treat as the base. When neither value has a better claim, people sometimes use the percentage difference, which divides by the average of the two. Here that is 10 divided by 50, or 20 percent. Whichever you choose, state the base you used so a reader can check the figure.

Finding percentage change follows one consistent idea: compare the change with the original value. Subtract, divide by the starting amount, multiply by 100, and label the direction. The most common mistake is using the wrong denominator, so circle or note the original value before you start calculating.