Percentage Calculator

Introduction

Percentages appear all around us; they're used in discounts at the cashier, raises on paychecks, test scores and percentages, and a stock that has declined by 12% over the past month — the math behind them is the one area of everyday math that steals the show. This is not a problem with percentages—there are in fact four or five types of percentage problems, all of which resemble each other on the surface, but require different formulas. This tool does all of them and displays the steps, so you're certain you got the correct answer and understand how. Useful for checking bills, a stack of tests, or just to make sure there was a discount when looking at that "sale"!

The Four Percentage Problems (and which one you might be in actually have)

The majority of the confusion with percentages is that people do not realize what they are solving. The following is how the two are distinguished.

1. To find a percentage of a quantity "What is 20% of 150"? Formula: (Percentage ÷ 100) × Number. Example: 20% of 150 = (20 ÷ 100) × 150 = 30
2. Determine what percent one number is of another "45 is what percent of 180?". Formula: (Part ÷ Whole) × 100. Example: (45 ÷ 180) × 100 = 25%
3. Percentage increase or decrease: "A price increases from $80 to $92 — what is the percentage increase?" Formula: ((New − Original) ÷ Original) × 100. Example: ((92 − 80) ÷ 80) × 100 = 15% increase
4. Percentage difference (two values with no clear "original") "What is the percentage difference between $40 and $50? (Store A charges $40, Store B charges $50)" Formula: (|A − B| ÷ ((A + B) ÷ 2)) × 100. Example: (|40 − 50| ÷ 45) × 100 ≈ 22.2%

Problems 3 and 4 seem to be very similar but they ask different questions; increase/decrease has one number as the "before" and one number as the "after," whereas percentage difference has two numbers that are the same and being compared, but none as "before" or "after. A wrong use of one of these two is the most frequent error with percentages.

Why 10% Up Then 10% Down Doesn't Get You Back to Where You Started

This has caught everybody at least once: Take a number and add 10% to it, then subtract 10% from it, and you don't end up with the original number.

• $100 increased by 10% = $110
• $110 decreased by 10% = $99

It is very easy to understand why this is the case: the second 10% is based on the new, larger base ($110), and not the original ($100). When you are calculating percentage increase and decrease, you always have to compare with a number at that moment and not a set number. That is why it is important to read the sale prices, investment losses, and salary changes in percentages carefully, not literally.

The terms percentage points and percentage are not synonymous

This is something news headlines get wrong all the time; it's how the actual meaning of a statistic changes:

When an interest rate changes from 5% to 7%, this is a change of 2 percentage points — but this is also a change of 40% in the interest rate (2/5 × 100 = 40%).

While both sentences are grammatically correct and mean the same, they actually refer to very different sized changes in interest rates: "interest rates rose 2%" and "interest rates rose 2 percentage points".

When you are reading about a percentage change in the news, such as the rate of inflation or the unemployment rate, ask yourself if the change is given in percentage points or as a percentage of a base. The two numbers may be the same, yet have completely different meanings.

Reverse Percentages

Once you're able to write and do some basic operations with percentages, you can apply that knowledge to solve problems similar to the following: Occasionally you have an answer after a percentage has been applied, and want to obtain the original number. Occurs frequently when dealing with sale prices and totals including taxes.

"An item costs $76.50 after a 15% discount. What is the cost of the item before tax?

Formula: Result ÷ (1 − Discount as decimal)
$76.50 ÷ (1 − 0.15) = $76.50 ÷ 0.85 = $90

Many people make the mistake of just putting the 15% back on the $76.50 — this is not correct since it is not 15% of the original price, but rather 15% of the discounted price. Notice that you always divide by (1 − percentage) with reverse percentage problems, and never multiply by (1 + percentage).

Real world percentage examples

• Shop discounts: Discount on a jacket is 30%, the original price of the jacket is $45. Discount = 45 × 0.30 = $13.50. Final price = $31.50.
• Tips: A $67 dinner bill, 18% tip. Tip = 67 × 0.18 = $12.06. Total = $79.06.
• Grades: On a test you got 42 out of 50 correct. Percentage = (42 ÷ 50) × 100 = 84%.
• Investment returns: The value of a stock increases from $120 to $138. Percentage gain = ((138 − 120) ÷ 120) × 100 = 15%.
• Sales tax: The sales tax on an item is $80 for an item that costs $100. Tax = 80 × 0.08 = $6.40. Total = $86.40.
• Population change: The population of a town increases from 24,000 to 26,880. Percentage growth = ((26,880 − 24,000) ÷ 24,000) × 100 = 12%.

Common Percentage Mistakes to Avoid

Frequently Asked Questions