Percentages, discounts, and VAT

Percentages turn up in tips, sale stickers, score reports, and tax invoices. The wording changes, but the algebra is short. This guide ties together the ideas behind our Percentage, Discount, VAT / Sales Tax, and Tip calculators.

Core percentage patterns

• What is X% of N? → N × X ÷ 100
• A is what % of B? → A ÷ B × 100
• Percent change A→B → (B − A) ÷ A × 100

Percent change always divides by the starting value. A rise from 4 to 5 is +25% relatively, but only +1 percentage point on a rate scale.

Retail discounts

A 25% discount on £40 saves £10 (sale price £30). Stacked discounts multiply: 20% off then another 20% off leaves 64% of the original (36% total reduction), not 40% off. Quantity simply multiplies the per-item saving — see the Discount calculator. Compare pack sizes with Unit Price so a “bigger discount” does not hide a worse price per gram.

VAT and sales tax: net vs gross

When an amount is tax-exclusive (net): tax = net × rate ÷ 100; gross = net + tax. When an amount already includes tax (gross), reverse with division: net = gross ÷ (1 + rate÷100). Subtracting rate% of the gross is the classic mistake. Worked numbers at 20%: £100 net → £20 tax → £120 gross; £120 gross → £100 net.

Tips

Tip amount = bill × tip% ÷ 100. Split by headcount after you decide whether the tip sits on pre-tax or post-tax totals — customs differ. The Tip calculator shows per-person shares when you set party size.

Worked shopping story

A jumper priced £50 with 20% off sells at £40. If VAT at 20% was already in the £50 ticket, the discount usually applies to the shelved gross under store rules — always read the till. Separately estimating tax-exclusive content uses the VAT reverse formula above.

Common mistakes

• Adding percentage points as if they were percent-of-price discounts.
• Removing VAT with × (1 − rate) instead of ÷ (1 + rate).
• Double-tipping when service is already included.

FAQs

Is “£10 off” a percentage?

Only once you compute 10 ÷ original × 100 for that specific price.

Do discounts happen before or after tax?

Jurisdiction and till software vary — convert with the order your receipt uses.

Where do loan percentages fit?

See APR vs interest rate for borrowing costs.

Unit price still wins

A steeper percent off on a small pack can still lose to a larger pack with a milder discount. Convert the shelf price you will actually pay into price per kg, litre, or item with the Unit Price calculator, including multi-buy maths when the till requires a minimum quantity.

Tips across borders

In some countries service is included; in others tipping is customary on the pre-tax subtotal. Travellers who tip both the included service line and an extra cash amount pay twice. Use the Tip calculator after you decide the base: tax-inclusive total, pre-tax subtotal, or amount after an employer-paid voucher.

Sources

Standard arithmetic of percentages; VAT-inclusive reverse formula used in retail tax education materials.