Percentages, Discounts, Markups, Taxes, and Tips

Objective: Multiply by the retained-price factor.

Prerequisite skills: percentage conversion and money arithmetic. A discount of p% removes p/100 of the original price, leaving factor 1−p/100. The discount amount and final selling price are different requested quantities. Common error: reporting the discount amount as the sale price. Restrictions and interpretation: Apply each rate to its stated base; round currency only at the requested final stage.

Objective: Use an increase factor for a stated base.

A markup or tax adds a fraction of the stated base. The total multiplier is 1+r. A markup on cost uses acquisition cost as the base, unlike profit margin as a share of selling price. When tax applies after a discount, the discounted price becomes the tax base unless the problem states otherwise. Common error: multiplying by the increase rate alone when asked for the new total. Restrictions and interpretation: Apply each rate to its stated base; round currency only at the requested final stage.

Objective: Apply tax and tip to the specified amounts.

Different percentage charges need not share a base. Read whether a tip uses the pretax bill and whether tax applies before or after a discount. Build separate amounts before adding them. Common error: applying a tip to the taxed total when the prompt says pretax. Restrictions and interpretation: Apply each rate to its stated base; round currency only at the requested final stage. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.