Solve multistep percent price models with explicit calculation rules

Learning goal: Solve hypothetical price problems involving markdowns, markups, tax, gratuity and fees. Identify which amount each percentage uses, keep intermediate values accurate and explain the order of operations stated in the problem.

These examples are mathematical models, not current store prices or tax rules. Each problem supplies its own rate, calculation base and rounding instruction. Real bills can follow different rules, so do not import an unstated convention. Begin by naming the original amount, every change and the reference amount for each percentage.

Suppose a model item costs 80 budget units before a 25% markdown. The discount is 0.25 × 80 = 20 units, leaving 60. Equivalently, multiplying by 1 − 0.25 = 0.75 gives 80 × 0.75 = 60. The 25% is the amount removed relative to the original price; it is not the fraction of the original price that remains.

Now apply a hypothetical 10% tax to the discounted 60, as explicitly required by the model. The tax is six and the subtotal including tax is 66. Add a fixed four-unit delivery fee after tax, with the fee excluded from the tax base by the problem's rule. The final total is 70. A compact expression is 80(0.75)(1.10) + 4. Naming each intermediate amount makes this expression interpretable.

The order and base matter. Charging 10% tax on the original 80 would produce eight, which is not the tax specified here. Applying the markdown to the fixed fee would also change a rule. Percent calculations are not a list of rates to combine indiscriminately. Even when two multipliers can be exchanged algebraically, their labeled intermediate amounts may have different meanings and a real problem may specify rounding at a particular stage.

A second markdown is based on the amount the problem identifies. If an item marked 100 receives a 20% markdown followed by another 10% off the reduced amount, it becomes 100 × 0.80 × 0.90 = 72. The total markdown is 28%, not 30%. The second ten-percent reduction removes eight units because its base is eighty. Adding percentages would only be valid under a different rule in which both discounts are computed from the same original amount and then subtracted.

A markup increases a stated cost to determine a model selling price. If a shop's item cost is 40 and the markup is 30% of that cost, the increase is 12 and the price becomes 52. This does not mean the twelve-unit difference is 30% of the selling price; 12/52 is about 23.08%. Markup on cost and a share of selling price use different denominators. Use the denominator actually named by the question.

Gratuity problems also require a stated base. A hypothetical meal has a pre-tax amount of 50, a tax of 8% of that amount and a gratuity of 18% of the pre-tax amount. Tax is four and gratuity is nine, giving 63 total. Because both percentages use fifty in this specified model, the expression 50(1 + 0.08 + 0.18) is appropriate. If gratuity were instead based on the tax-inclusive amount, the computation would differ.

A fixed fee is an amount, not a percentage. A three-unit service charge adds three regardless of the current subtotal unless the rule states otherwise. A five-percent service charge scales with its named base. Reading the units distinguishes the operations: budget units suggest addition, while a percentage of a subtotal suggests multiplication by a fraction of that subtotal.

Rounding should follow an explicit convention. In a cents-based model, keep enough precision through intermediate percentage calculations and round where directed. For example, 12.50 with a 15% markdown becomes 10.625 before rounding. If the problem directs rounding the discounted price to the nearest cent using the usual school half-up convention, it becomes 10.63. If tax must be computed before rounding, retain 10.625 for that step. State the policy rather than silently switching methods.

Estimate before calculating. The 80-unit item after a quarter discount should be near 60, then somewhat higher after tax and delivery. A final answer of 24 or 700 signals a likely base or decimal error. Estimates do not replace exact calculations, but they provide a useful independent check.

Worked bill: Apply a 20% markdown to 50, then a model 5% tax to the discounted price, then a two-unit untaxed fee. The discounted amount is 40, tax is two and the total is 44. The expression is 50(0.80)(1.05) + 2. Each rate acts on the quantity stated by the problem.

Worked gratuity comparison: Eighteen percent of a 50-unit pre-tax bill is nine. Eighteen percent of the 54-unit tax-inclusive amount is 9.72. The difference is 0.72. Neither base is automatically the one a mathematical problem intends; use its explicit condition and label the choice.

Worked markup check: A 40-unit cost marked up 30% becomes 52. Reversing that markup requires dividing 52 by 1.30 to recover 40, not reducing 52 by 30%. A 30% reduction of 52 would remove 15.60 and produce 36.40 because the percentage would use a new whole.

Bill reconstruction: For the 80-unit example, write a line for original price, markdown amount, discounted price, tax, fixed fee and total. Beside each percentage, identify its base. Then alter one rule, such as making the fee taxable, and recompute the result while explaining which line changes.

Independent performance: Create three hypothetical bills: one with discount and tax, one with a gratuity and fixed fee, and one with markup followed by a markdown. Supply every rate, base, fee rule and rounding instruction. Solve each with both a sequence of labeled amounts and a compact expression. A partner audits the bases and identifies whether any percentage has been added to another despite applying to a different whole. Revise ambiguous wording before checking the final arithmetic.