- What is a percentage
- How to calculate percent change
- How to increase or decrease X by Y percent
- How to calculate X is what percent of Y
- How to calculate X percent of Y
- How to calculate percent difference
- Percent Points vs Relative Percentages
- Compounding and averaging percentages
- Many uses for a percentage calculator
What is a percentage
Before explaining how to use a percentage calculator to calculate percent change, percent difference or percentage of one number from another, it is useful to examine the basics of the concept of percentages.
A percentage is a dimensionless number, represented as a fraction of 100, e.g. 50 out of 100 can be written as 50%, and 1 out of 10 can be written as 10%. A percentage is by definition a ratio. The sign for percent is “%”, but the abbreviation “pct” is sometimes used in its place, while in older literature and documents one can encounter “per cent”, where “cent” is an abbreviation of the Latin “centum” which literally means “one hundred”, so the phrase means “per one hundred” – the literal definition of percentage.
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Percentages have a wide array of applications in many disciplines and everyday usage. They are common in statistics, social sciences, economics, finance, accounting. In everyday usage we often encounter percent off coupons. Promotions, sales, and various discounts are often expressed as percentage from a previous reference price of an item or service. Percentage calculations can be used when measuring productivity or load of a person or machine, e.g. “he is working at 100%” (at maximum capacity).
Percentage increase or decrease are used to describe the relative growth or decline of something, e.g. a population, capital, personal wealth, etc. Differences between any two objects can be expressed as ratios or as percentage difference. The measurement error of a tool or process can be described in terms of percent error and can easily be computed using a percentage calculator. Read more on percent vs percentage.
How to calculate percent change
This is what most people mean when they want to know “how to calculate percentage”, but for other possible percentage calculations see below. Percent change calculators are commonly employed when comparing quantities, business metrics, or other measurements from two time periods, the earlier one serving as a baseline. A percentage change calculation is also useful when comparing a new state of things to an old state of things, e.g. using the census to compare the number of people living in villages in a given municipality before versus after industrialization. Our calculator is of great assistance for calculating percent increase / decrease, but you can also find the percentage change on your own.
For example, say you are reviewing the performance of your business on a monthly basis and you see that the past month you had 80 customers while the month before you were able to acquire only 64. To find the growth rate of your business versus prior month’s base value you need to calculate percent change using the equation below.
Percent Change Formula
Percent change = new / old * 100 – 100
where new is the newer quantity or measure, and old is the older quantity or measure. In the above example this would be 80 / 64 * 100 – 100 = 1.25 * 100 – 100 = 125 – 100 = 25%. Your monthly percentage change (percent growth, percent increase) from 64 to 80 was thus 25 percent versus the baseline from the month prior as you can verify by using the percentage change calculator.
In another situation, you might be examining a proposition to increase your salary from $100,000 a year to $120,000 a year to keep you on the payroll and want to find what percent is the new salary versus your old one. If you do the math manually, start by dividing 120,000 by 100,000 to get 1.2. Then multiply by 100 to get 120. Finally subtract 100 which leaves 20%. Therefore, you were offered a 20% increase of your salary and as the new salary is 120% of your current salary.
Similarly, you can use the calculator to calculate change in speed of different modes of transportation. If you compare a car or bus moving at 60 miles per hour to a high-speed train moving at 120 miles per hour, you can obtain the percentage change to be 100% meaning that the railway is twice as fast as the vehicle.
How to increase or decrease X by Y percent
Oftentimes one may want to alter an original number by a percentage of its value. For example, if you want to calculate what a 20% increase to the price of an item due to VAT would result in, you can use the percent increase formula below.
Increase by percentage formula
Increased value = base + base * % increase / 100
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For example, if the current price is $100, increasing it by 20% means calculating: $100 + $100 * 20 / 100 = $100 + $20 = $120 price after the percent increase.
Decrease by percentage formula
Decreased value = base – base * % increase / 100
The formula for decreasing a value by a percentage is almost identical, but the plus has been replaced with a minus. For an example application, say one is calculating a price discount of 50% from an original price of $200, the calculation would be: $200 – $200 * 50 / 100 = $200 – $100 = $100 discounted price.
How to calculate X is what percent of Y
Let’s say you are a car salesman and you have a car originally priced at $50,000, but you have done some calculation and determined that you can take $5,000 off the price of the car and still be ahead after the sale. How can you determine what percentage is $5,000 from $50,000? Obviously, just plugging in the numbers in the percent calculator above is the fastest way, but to do the math manually use the following formula:
X is What Percent of Y Formula
x is x / y * 100 % of y
so in this case that would be 5,000 / 50,000 * 100 = 0.1 * 100 = 10%. If you were to offer a $5,000 discount on a $50,000 car, that would be a 10% discount.
In another example you might want to find what percentage of your total yearly income you have to pay in taxes. If your yearly income is $80,000 and you have calculated that your total tax amount is $36,000, then your tax rate is 36,000 / 80,000 / 100 = 0.45 * 100 = 45%, since $36,000 is 45 percent of $80,000.
Other applications of this equation can be found in percent error calculations. Say a measurement of width, height or weight is off by a given absolute value, then the absolute error can be translated into percentage error by the X is what percent of Y formula.
How to calculate X percent of Y
Let’s say you are told you are eligible to get a 20% discount on an item costing $500. How can you determine what the discount value is at this percentage? The formula to use is:
X Percent of Y Formula
x% of y is y * (x / 100)
In the example above that would be calculated as 500 * (20 / 100) = 500 * 0.2 = 100. If you were to purchase the $500 item with $20 off, you would be getting a discount of $100.
Another example of using a percent calculator would be if you wonder how many minutes are 75% of a 60-minute video. The answer is 60 * (75 / 100) = 45 minutes as you can verify with our tool.
How to calculate percent difference
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The percentage difference of two numbers (quantities) a and b is the relative difference, expressed as a percent. It should be calculated using the formula:
Percentage Difference Formula
Percent Difference = |a – b| / ((a + b) / 2) * 100 percent
For example, if one item costs $5 and another costs $6 the percent difference between them is: |5 – 6| / ((5 + 6) / 2) * 100 = 1 / (11 / 2) * 100 = 1 / 5.5 * 100 = 18.18%. Please, note that this doesn’t mean that 5 is 18.18% smaller than 6, or that 6 is 18.18% larger than 5. The correct percentages if you are asking the question of “what percent is a from b” would be 16.66% and 20%, respectively, as explained above.
Similarly, a percentage difference calculator might be useful if computing the difference in elevation of two mountains. If one is 6,000 feet high and the other 3,000 feet, the absolute difference is 2000 ft while the percent difference is 40%.
Percentage difference is useful in a few situations, so it should be used with care. For example, one should not use percentage difference when comparing time periods, as the first metric is another state of the second metric, so percent change is the appropriate calculation. Similarly, finding a price changed by some percentage should not be done using percentage difference.
Percent Points vs Relative Percentages
You might have noted that often when newspapers are talking about USA elections for members of parliament, premier, or president of a town, municipality, or a whole country, there would be polls tracking the favorability of each candidate. Differences in sentiment and changes in said polling measures will often be communicated in percent points instead of percentages. A percent point is like absolute difference, but expressed in percentages.
Note that such calculations only apply when comparing differences in percentage measures – something not supported by our percentage difference calculator mode. For example, the percent in favor of candidate A might be 40% before a certain political event and only 35% after. The percent change is simply 40% minus 35% which equals 5 p.p. (but is a percentage change of 12.5 percent).
Compounding and averaging percentages
Percentages should not be added up (compounded) or averaged like simple numbers, as this will result in an incorrect end result. Compounding is often encountered in finance, e.g. when calculating compound interest or multi-year return of a financial portfolio. Averaging percentages is often encountered in business calculations, for example to determine the average growth of a company, but also in finance and banking where average growth of an asset or asset portfolio may be calculated.
Here is an example of adding percentages: say you have a $100,000 bank deposit at a 2% interest rate, applied yearly at the end of the year. If you keep it for 5 years, you might think that the way to calculate your deposit’s value at the end of the 5-year period is to simply multiply 2% x 5 = 10% (or, equivalently, 2% + 2% + 2% + 2% + 2% = 10%), add 100% and then use our percentage calculator to calculate 110% of $100,000. By this calculation you would expect to have $110,000 at the end of the period (10% of 100,000 is $10,000). However, you will have $110,408, since at the end of each year you will get your interest but then in each of the following years you will accrue interest over the interest from the first year. And so on for the second, third…
To average percentage growth a year, it would be incorrect to just sum up the growth % in each year and then divide by the number of years. Let us say you are the founder of an organization which has an asset that grew 5% the first year, 6% the second year, 10% the third year, and then lost 10% the fourth year. The growth of the principal value is not 5% + 6% + 10% – 10% = 11%, but it is instead the geometric mean: 2.4549% times the number of years = 2.4549 x 4 = 9.82%. Again, this is not something you can solve for using the above calculator.
The above peculiarities are also the reason why if you lose 25% of some asset, you need to grow it 33.33% to recoup the loss. A quick calculation shows that $10,000 * 25% = $7,500, while $10,000 of $7,500 = 133.33% (or $10,000 – $7,500 = $2,500 to return to zero, and $2.500 is 33.33% of $7,500).
Many uses for a percentage calculator
As the numerous examples above demonstrate, a percentage calculator can be useful in so many practical scenarios. Changes in electoral sentiment in democratic processes, industrial production, stocks of material goods in warehouses, albums sold by an artist / musician, publications in a scientific journal, personnel changes in an organization, as well as natural things such as the amount of flora or fauna in given island, etc. can be computed using such a tool. Another application of percentage calculations is for differences in achievements such as records at Olympic games and other tournaments and championships.
A specialized calculator can be used to measure the relative difference in physical quantities such as speed, density, luminosity, reflectivity, strength, tensile strength, and so on. However, a calculator is most often used for financial assets and benchmarks and for everyday things such as discounts, commissions, and tipping at a restaurant.
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