Tax
In math, tax is a percentage of a purchase price or income added to or subtracted from a base amount, calculated using percent concepts.
Formula
\text{tax amount} = \text{price} \times \text{tax rate}; \quad \text{total} = \text{price} \times (1 + \text{tax rate})
Definition
Tax is an extra amount you pay on top of a price, calculated as a percentage of the price; sales tax is added at checkout, found by multiplying the price by the tax rate. Formally, sales tax $T = Pr$, where $P$ is the pre-tax price and $r$ the tax rate as a decimal, giving a total cost of $P(1 + r)$; income tax often uses a progressive (bracketed) structure where different portions of income are taxed at different rates. Tax calculations are applications of linear and piecewise-linear functions: each bracket applies a rate only to the income within that bracket, and the effective tax rate (total tax divided by total income) is a weighted average of the marginal rates.
Example
A $\$40$ video game with a $7\%$ sales tax rate costs $\$40 \times 0.07 = \$2.80$ in tax, for a total of $\$42.80$, or in one step, $\$40 \times 1.07$. A $\$250$ jacket with $8.25\%$ tax costs $250 \times 0.0825 \approx \$20.63$ in tax, for a total of $\$270.63$; working backward from a tax-inclusive total requires dividing by $(1+r)$, not subtracting the percent, so $\$270.63/1.0825 \approx \$250.00$ recovers the original price. A three-bracket system taxing $10\%$ on the first $\$10{,}000$, $20\%$ on the next $\$30{,}000$, and $30\%$ above $\$40{,}000$, applied to a $\$55{,}000$ income, gives tax of $1000 + 6000 + 4500 = \$11{,}500$, an effective rate of about $20.9\%$ even though the marginal rate on the last dollar is $30\%$.
Key Insight
Tax is always a percent of the price, so to find the total quickly, add the tax rate to $100\%$ and multiply in one step. A common error is trying to reverse this by subtracting the tax percent from a tax-inclusive total, "the total is $162 with 8% tax, so the original price is $162 minus 8%," which is wrong; the correct method divides by $(1+r)$: $\$162/1.08 = \$150.00$. The distinction between the marginal tax rate (the rate on the next dollar) and the effective tax rate (the average rate overall) is widely misunderstood: a raise that pushes income into a higher bracket does not subject all income to the higher rate, only the income within that new bracket, a classic application of piecewise-linear functions with real policy consequences.