Use percent models for simple interest, commissions and unknown bases

Learning goal: Solve hypothetical multistep percent problems with clearly stated bases and time units. Distinguish simple interest from interest on changing balances and separate a commission component from fixed pay.

All rates and amounts in this lesson are invented for mathematical practice. A percent model describes a relationship only under its stated rules. Define each variable, identify the reference amount and convert time units before calculating. The task is to reason about the supplied model rather than infer current financial or employment terms.

Simple interest is computed on a fixed principal under the model. Let P be the initial principal, r the rate per year written as a decimal, t the number of years and I the interest. Then I = Prt. If P and r remain fixed, interest is proportional to time. The model does not add earlier interest to the principal for later calculations.

For P = 200 budget units and r = 5% per year, one year's interest is 200 × 0.05 = ten. Half a year earns five, two years earns twenty and three years earns thirty. The graph of interest against time passes through (0,0) and (1,10). The point (1,10) describes ten interest units per year for this fixed principal and rate, not a five-unit interest amount simply because the rate is five percent.

The final amount A is principal plus interest: A = P + I. At two years, A = 220. Although I is proportional to t, A is not proportional to t because A starts at 200 when t is zero. The distinction resembles a fixed-fee cost model: adding a constant changes the relation between the complete quantities even when the changing component has a constant rate.

Convert months to years when the rate is annual. Eighteen months is 18/12 = 3/2 years. Under the same simple-interest model, I = 200 × 0.05 × 1.5 = 15, and the final amount is 215. Multiplying by 18 without conversion would treat eighteen months as eighteen years and produce a much larger incorrect result. Units are part of the reasoning, not a label to attach after calculating.

A different rule can produce a different result. If interest were added to principal after each year before the next year's calculation, the second year's base would change. With the same starting 200 and annual 5%, that hypothetical compounded rule gives ten in year one and 10.50 in year two, reaching 220.50 after two years. The simple-interest model reaches 220. The purpose of this comparison is to identify the model's assumption, not to treat all interest as interchangeable.

A commission is a percentage of a stated sales amount in the examples here. If commission is 6% of 500 budget units in qualifying sales, the commission component is 0.06 × 500 = 30. If a fixed base payment of 40 is also supplied, total pay is 70. The commission component is proportional to qualifying sales; total pay including the fixed forty is not proportional to sales.

The qualifying sales base may differ from all recorded sales if the problem specifies returns or exclusions. Suppose gross sales are 600, qualifying returns remove 100 and the model pays 6% on the net 500. The commission remains 30. Applying 6% to 600 would ignore the supplied rule. Read net, gross and qualifying as mathematical definitions within the scenario, and state which amount you used.

Unknown percent bases can be recovered through division. If a 6% commission equals 30, solve 0.06S = 30, giving S = 30/0.06 = 500. Multiplying thirty by 0.06 would find six percent of the commission, a different question. Estimate: six percent is a small part of the sales base, so the base should be much larger than thirty.

An unknown rate or time can also be found in a simple-interest model with appropriate nonzero quantities. If 300 earns 18 over two years under fixed-principal simple interest, r = 18/(300 × 2) = 0.03, or 3% annually. Substitution checks the result: 300 × 0.03 × 2 = 18. The check verifies both arithmetic and the intended annual interpretation.

Worked partial year: A model principal of 240 at 4% annual simple interest for nine months uses t = 9/12 = 3/4. Interest is 240 × 0.04 × 0.75 = 7.20, and the final amount is 247.20. One full year's interest would be 9.60, so a nine-month amount of 7.20 is reasonable.

Worked commission: Gross qualifying sales are 800 with 50 in deductible returns. Net commissionable sales are 750. At 4%, commission is 30. Add a fixed 25-unit payment for total pay of 55. Keep the subtraction, percent and fixed addition distinct; taking 4% of the gross 800 (which gives 32) or adding the fixed 25 before applying the percentage would each represent a different rule.

Worked inverse check: A 12% commission component is 36 units. The sales base is 36 ÷ 0.12 = 300. Check that 10% of 300 is 30 and 2% is six, so 12% is 36. This decomposition independently supports the equation solution.

Representation task: Make a table of simple interest for times 0, 1/2, 1, 3/2, 2 and 3 years using the 200-unit model. Graph interest with accurate scales and identify the unit-rate point. Write a second table for final amount, explaining why adding the principal changes proportionality.

Independent performance: Create one partial-year simple-interest problem, one commission problem with a fixed component and one problem with an unknown rate or base. State all hypothetical rules, units and permitted rounding. Solve with equations and an independent estimate or substitution. Explain which component is proportional to which quantity and identify one changed rule that would require a different model.