Discounts, Sales Tax and Tips

One calculation, three names

A discount, a sales tax and a tip look like three separate topics in a textbook. They are one calculation: apply a percent change to an amount of money.

Situation	You end up with	Multiply by
$30\%$ off	$70\%$ of the price	$0.7$
$8\%$ sales tax	$108\%$ of the price	$1.08$
$20\%$ tip	$120\%$ of the bill	$1.2$
$15\%$ off then $6\%$ tax	—	$0.85 \times 1.06$

Learn the pattern once and all three follow.

Discounts: pay for what is left

”$30\%$ off” tells you what you do not pay. What matters is the rest.

$$100\% - 30\% = 70\% \quad\Rightarrow\quad \text{pay } 0.7 \times \text{price}$$

A $80 coat at $30\%$ off:

$$80 \times 0.7 = \$56$$

The two-step route — find $30\%$ of $80$, which is $24$, then subtract — gives the same $56. The one-step version is faster and removes the chance of subtracting when you meant to add.

Tax and tips: keep everything, then add

Tax and tips run the other way. You still owe the whole amount, plus more.

$$100\% + 8\% = 108\% \quad\Rightarrow\quad \text{pay } 1.08 \times \text{price}$$

The $1$ carries the original price and the $0.08$ carries the tax, which is why one multiplication does both jobs.

Tips without a calculator

$10\%$ is one decimal-point move, and everything else is built from it.

Tip	How
$10\%$	move the point one place left
$20\%$	double the $10\%$
$15\%$	$10\%$ plus half of $10\%$
$18\%$	$20\%$ minus $2\%$

On a $45 bill: $10\%$ is $4.50, so a $20\%$ tip is $9. A $15\%$ tip adds half of the $10\%$ to itself, giving $4.50 plus $2.25, which is $6.75.

Why the order does not matter

A shop takes $15\%$ off, then adds $6\%$ tax. Would doing it the other way round change your bill?

$$100 \times 0.85 \times 1.06 = 90.10$$ $$100 \times 1.06 \times 0.85 = 90.10$$

Identical, because both are multiplications and multiplication does not care about order. The combined multiplier is $0.85 \times 1.06 = 0.901$ either way.

Shops still apply the discount first, but for a legal reason rather than a mathematical one: tax is charged on the amount you actually pay.

Why two discounts do not add

”$25\%$ off, then a further $25\%$ off” is not $50\%$ off.

$$0.75 \times 0.75 = 0.5625$$

You pay $56.25\%$, so the true discount is $43.75\%$. The second cut is taken from the already-reduced price, so it is worth less than the first.

This is the same reason a rise and an equal fall do not cancel — each percent is measured against a different amount.

Worked examples

Common mistakes

Practice problems