Discount
Fractions & DecimalsA discount is a reduction in price, usually expressed as a percentage of the original price.
Formula
\text{discount amount} = \text{original price} \times \text{discount rate}; \quad \text{sale price} = \text{original price} \times (1 - \text{discount rate})
Definition
A discount is an amount taken off the original price. A $20\%$ discount means you pay $20\%$ less than the original price. To find the discount amount, multiply the original price by the discount percent.
Example
A $\$60$ sweater is $25\%$ off. Discount $= \$60 \times 0.25 = \$15$. Sale price $= \$60 - \$15 = \$45$. Or directly: $\$60 \times 0.75 = \$45$.
Key Insight
A discount reduces the price by a percentage of the original. To find the sale price in one step, subtract the discount rate from $100\%$ and multiply: $(100\% - 25\%) = 75\%$, so $\$60 \times 0.75 = \$45$.
Definition
Discount $D = Pd$, where $P$ is the original price and $d$ is the discount rate (as a decimal). Sale price $S = P - D = P(1 - d)$. Sequential discounts do not simply add: a $20\%$ discount followed by a $10\%$ discount gives a multiplier of $0.8 \times 0.9 = 0.72$, a total $28\%$ discount (not $30\%$).
Example
A $\$200$ coat: first marked down $30\%$, then an additional $15\%$ off for a sale. Prices: $\$200 \times 0.70 = \$140$, then $\$140 \times 0.85 = \$119$. Total discount $= \$81$, which is $40.5\%$ (not $45\%$). The combined multiplier is $0.70 \times 0.85 = 0.595$.
Key Insight
Sequential discounts multiply rather than add because each successive discount applies to the already-reduced price. Retailers sometimes present stacked discounts to make sales seem larger than they are - the mathematical reality is always the product of the individual multipliers.
Definition
Discounts are percentage decreases applied to price. A sequence of $n$ discounts $d_1, d_2, \ldots, d_n$ yields a combined multiplier of $\prod (1-d_i)$. The equivalent single discount is $1 - \prod(1-d_i)$. In present value (PV) analysis, a discount rate is applied to future cash flows: $PV = FV/(1+r)^t$, a different use of "discount" meaning to reduce the value of future amounts to their present equivalents.
Example
Three discounts of $10\%$, $20\%$, and $15\%$: combined multiplier $= 0.9 \times 0.8 \times 0.85 = 0.612$. Equivalent single discount $= 1 - 0.612 = 38.8\%$. For PV: $\$1000$ in $5$ years at $6\%$ discount rate: $PV = 1000/(1.06)^5 = \$747.26$.
Key Insight
Present value discounting and price discounting share the same mathematical structure: both multiply a base value by $(1-r)^n$ or $1/(1+r)^n$ to reduce it. Financial discounting reveals the "time value of money" - the principle that a dollar today is worth more than a dollar in the future, a foundational concept in all of corporate finance and investment analysis.