Proportional Reasoning: Ratios, Rates, Percents, and Proportional Relationships
What this domain tests
Proportional Reasoning asks you to compare ratios, find unit rates, recognize proportional relationships in tables, graphs, and equations, and solve percent problems set in shopping, tax, tip, and simple-interest contexts. The central idea is a constant multiplier.
Core rules
- A rate compares quantities with different units; a unit rate has a denominator of 1. Divide to find it: 12 pens for $4.80 gives $0.40 per pen.
- A relationship is proportional when y/x is constant for every pair. That constant is k, and the equation is y = kx. Its graph is a straight line through the origin.
- Solve proportions by cross multiplication or by finding the scale factor between corresponding quantities.
- Percent means per 100. Use part = percent × whole, writing the percent as a decimal. For percent change, divide the change by the original amount: (new − old)/old × 100.
- For discounts, subtract the markdown from the original price; for tax, tip, or markup, add the extra amount. Apply tax to the discounted price when both appear.
- Simple interest is I = Prt, where P is principal, r is the annual rate as a decimal, and t is time in years.
Worked example
A jacket costs $80 and is marked 25% off. Sales tax is 7%. What is the final cost? Discount: 0.25 × 80 = $20, so the sale price is 80 − 20 = $60. Tax: 0.07 × 60 = $4.20. Apply tax to the sale price, not the original price. Final cost: 60 + 4.20 = $64.20. Check with a single multiplier: 80 × 0.75 × 1.07 = $64.20.
Traps
- Using the new value as the denominator in percent change; the original value is the base.
- Adding a 25% discount and 7% tax as 32% off, which mixes a decrease and an increase.
- Calling a table proportional when y/x changes or when the graph does not pass through (0, 0).
- Mixing units in a proportion, such as putting dollars over dollars on one side and items over dollars on the other.