Tax
Fractions & DecimalsIn 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 the checkout. To find the tax, multiply the price by the tax rate.
Example
A video game costs $\$40$. The sales tax rate is $7\%$. Tax $= \$40 \times 0.07 = \$2.80$. Total $= \$40 + \$2.80 = \$42.80$.
Key Insight
Tax is always a percent of the price. To find the total price quickly, add the tax rate to $100\%$ and multiply: $\$40 \times 1.07 = \$42.80$. This one-step method saves time.
Definition
Sales tax $T = Pr$, where $P$ is the pre-tax price and $r$ is the tax rate as a decimal. Total cost $= P(1 + r)$. Income tax may use a progressive (bracketed) rate structure where different portions of income are taxed at different rates. Property tax is calculated as assessed value $\times$ mill rate.
Example
A $\$250$ jacket with $8.25\%$ tax: $T = 250 \times 0.0825 = \$20.625 \approx \$20.63$. Total $= \$270.63$. Using the multiplier: $250 \times 1.0825 = \$270.625$. Finding pre-tax price from total: $\$270.63/1.0825 \approx \$250.00$.
Key Insight
Working backward from a total that includes tax requires dividing by $(1 + r)$, not subtracting the percent. A common error: "the total is $\$162$ with $8\%$ tax, so the original price is $\$162 - 8\% = \$149.04$" is wrong. Correct: $\$162/1.08 = \$150.00$.
Definition
Tax calculations are applications of linear functions: $T(P) = rP$ and $A(P) = P(1+r)$. Progressive tax systems apply piecewise-linear functions: each bracket has a different rate applied only to the income within that bracket. Total effective tax rate $=$ total tax$/$total income, a weighted average of marginal rates. Tax incidence theory analyzes how the economic burden of a tax is distributed between buyers and sellers using supply-demand elasticity.
Example
A three-bracket system: $10\%$ on first $\$10{,}000$; $20\%$ on next $\$30{,}000$; $30\%$ on income above $\$40{,}000$. For $\$55{,}000$ income: tax $= 0.10 \times 10000 + 0.20 \times 30000 + 0.30 \times 15000 = 1000 + 6000 + 4500 = \$11{,}500$. Effective rate $= 11500/55000 \approx 20.9\%$. Marginal rate (on the last dollar) $= 30\%$.
Key Insight
The distinction between marginal tax rate (rate on the next dollar) and effective tax rate (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 in the new bracket. This is a classic application of piecewise-linear functions with real policy implications.