Ratios, Proportional Relationships, and Percents: Unit Rates, Percent Change, and Interest
This domain tests unit rates, the constant of proportionality, and percent applications including percent change, percent error, discounts, markups, tax, tips, and simple interest. Most lost marks come from choosing the wrong base, not from arithmetic.
Proportional relationships
Two quantities are proportional when one is a constant multiple of the other: y = kx. The constant of proportionality k is the unit rate, and it equals y/x for every corresponding pair. The graph is a straight line through the origin, so a line that misses (0, 0) does not represent a proportional relationship.
Percent change and percent error
Percent change = (new − old)/old × 100. The original value is always the denominator. Equivalently, multiply by (1 + r) for an increase or (1 − r) for a decrease, where r is the decimal form of the percent.
Percent error = |approximate − exact|/exact × 100.
Multi-step percent problems
Apply successive percents multiplicatively, never additively. A 25% markup followed by a 20% discount is 1.25 × 0.80 = 1.00, so the final price equals the original rather than showing a 5% change. Tax and tips are computed on the price after any discount.
Simple interest: I = Prt, where P is the principal, r the annual rate as a decimal, and t the time in years. The total amount is P + I.
Worked example
A jacket costs $80. The store marks it up 25%, then offers 20% off the marked price. What is the final price? Markup: 80 × 1.25 = $100. Discount: 100 × 0.80 = $80. The final price is $80, the same as the original, because the 20% discount applies to the larger marked price, not to $80.
Traps
- Taking a percent of the new value instead of the original when computing percent change.
- Adding percents across successive changes, treating 25% then 20% as 45%.
- Inverting the unit rate, writing 3 hours per task as 3 tasks per hour.
- Assuming any linear graph is proportional; it must pass through the origin.
- Using time in months with an annual interest rate; convert to years first.
- Computing a tip on the post-tax total when the question specifies the pre-tax price.