ToolboxKit

Percentages Explained: 12 Everyday Problems Solved Step by Step

Percentages show up in almost every decision that involves money: a sale price, a pay rise, a loan rate, a tax bill, a tip. Most of us can do the easy cases in our heads and then freeze when the question gets slightly different. This guide walks through twelve of the problems people actually run into, each with the formula, a worked example and the common mistake to avoid. You can check any of them with the percentage calculator.

The one idea behind every percentage problem

"Percent" means "per hundred". So 15% is 15 out of 100, which is the decimal 0.15. Every percentage problem is one of three questions about three numbers: the part, the whole, and the percentage. If you know two, you can find the third.

  • Part = whole × percentage ÷ 100
  • Percentage = part ÷ whole × 100
  • Whole = part ÷ percentage × 100

Keep these three lines in mind and the twelve problems below become variations on a theme.

1. Finding a percentage of a number

Question: What is 15% of 240?

Multiply 240 by 0.15. 240 × 0.15 = 36.

Mental shortcut: 10% of 240 is 24 (move the decimal one place), 5% is half of that, 12, and 15% is 24 + 12 = 36.

2. What percentage one number is of another

Question: You answered 18 of 60 questions correctly. What is your score?

Divide the part by the whole and multiply by 100: 18 ÷ 60 = 0.3, so 30%.

The most common error here is dividing the wrong way round. The whole always goes underneath.

3. Increasing a number by a percentage

Question: A price of 85 goes up by 20%. What is the new price?

Multiply by 1 plus the percentage: 85 × 1.20 = 102.

Why this works: the original is 100% of itself, the rise adds another 20%, so the new price is 120% of the original. Adding the percentage back in one step is faster than working out the increase and adding it separately.

4. Decreasing a number by a percentage

Question: A 120 jacket is 30% off. What do you pay?

Multiply by 1 minus the percentage: 120 × 0.70 = 84. The saving is 120 − 84 = 36.

5. Percentage change between two values

Question: A salary goes from 2,000 to 2,300. What is the percentage increase?

Use (new − old) ÷ old × 100. The change is 300, so 300 ÷ 2,000 = 0.15, which is +15%.

Always divide by the old value. Dividing by the new one is the classic mistake, and it gives 13% instead of 15% in this case.

6. Why a rise and the same fall do not cancel out

Question: A share price rises 25%, then falls 25%. Is it back where it started?

Start at 100. After +25% it is 125. A 25% fall takes off 31.25, leaving 93.75. You are 6.25% below where you began.

The reason is that the second percentage is taken from a different, bigger number. To get back from 125 to 100 you need a fall of 20%, not 25%. This asymmetry matters for investments: a 50% loss needs a 100% gain to recover.

7. Reverse percentages: finding the original

Question: A price including 20% tax is 120. What was the price before tax?

The 120 is 120% of the original, so divide by 1.20: 120 ÷ 1.20 = 100.

The same logic works for discounts. If a sale price of 75 is after a 25% discount, the original is 75 ÷ 0.75 = 100.

The mistake to avoid is subtracting 20% from 120, which gives 96 and is simply wrong. The tax was added to the original, not to the total. If you work with VAT often, the VAT calculator does the reverse step for you.

8. Stacked discounts

Question: A shop offers 20% off, and then an extra 10% off the reduced price. What is the total discount on an item that costs 100?

Apply them one after another: 100 × 0.80 × 0.90 = 72. The total discount is 28%, not 30%.

Stacked percentages multiply, they do not add. The same applies to two successive pay rises: 5% then 5% is 1.05 × 1.05 = 1.1025, or 10.25%.

9. Percentage points versus percent

Question: An interest rate moves from 4% to 5%. How big is the change?

It is 1 percentage point. In relative terms it is a rise of 25%, since 1 ÷ 4 = 0.25.

Headlines mix these up constantly. "Rates up 25%" and "rates up one point" can describe the same event. When you read a percentage change of a percentage, check which one is meant.

10. Compound percentage growth

Question: Something grows 10% a year for three years. What is the total growth?

Multiply the growth factors: 1.10 × 1.10 × 1.10 = 1.331, so 33.1%, not 30%.

This is compound growth, the engine behind savings and debt. A quick check: 7% a year for 20 years multiplies money by about 3.87 (1.07 to the power of 20), so 10,000 becomes roughly 38,700. See how it plays out with your own numbers in the compound interest calculator.

11. Markup versus margin

Question: You buy an item for 60 and sell it for 100. What are the markup and the margin?

  • Markup is profit relative to cost: (100 − 60) ÷ 60 = 66.7%.
  • Margin is profit relative to the selling price: (100 − 60) ÷ 100 = 40%.

Same sale, two very different numbers. Confusing them is how sellers end up underpricing. If you want a 40% margin, you cannot simply add 40% to the cost. You need price = cost ÷ (1 − margin) = 60 ÷ 0.60 = 100. Online sellers face this every day when fees take a share of the price too; our marketplace profit calculator works it out including the fees.

12. Tips and sales tax in one go

Question: A restaurant bill is 64.00. You want to leave 18%. What is the total?

18% of 64 is 11.52, so the total is 75.52. A quick estimate: 20% of 64 is 12.80, so a bit under that. For more on tipping customs and mental shortcuts, see how to calculate a tip.

A quick checklist before you trust any answer

  1. Is the percentage applied to the right base (old value, original price, pre-tax amount)?
  2. Are you adding percentages that should be multiplied?
  3. Did you mix up "percent" with "percentage points"?
  4. Does the answer make sense? A discount cannot exceed 100%, and a price after a 20% rise must be larger than the original.

Percentages are not hard once you name the part, the whole and the percentage. Write them down before you calculate, and most of the traps disappear.