Ratios, Rates, Percents, and UnitsRatios, Rates, Percents, and Units

SATMATH

Ratios, Rates, Percents, and Units

SAT Math Guide 5

Goal: Handle percent change, proportions, unit conversion, and rates in word problems. Original study notes on standard SAT Math topics, with original examples. Not copied from any prep book. Big Picture

These word-problem skills show up constantly. Define what the percent is of, track units, and be careful with averages.

Percents

Part = percent/100 × whole. 15% of 80 is 12.

Percent change = (new − old)/old × 100. From 40 to 50 is +25%.

To increase by p%, multiply by (1 + p/100). To decrease by p%, multiply by (1 − p/100). Successive changes multiply. A 20% increase then a 20% decrease: 1.2 × 0.8 = 0.96, a net 4% decrease. Discount and tax: price × (1 − d) × (1 + t).

"150% of" vs. "50% more than": both mean 1.5 times the original.

Percent vs. percentage points: going from 10% to 12% is +2 percentage points but a 20% increase. Ratios and Proportions

A ratio compares parts. Part-to-part (boys to girls 3:2) vs. part-to-whole (boys to students 3:5). Example: 30 students in the ratio 3 boys : 2 girls. Total parts = 5, one part = 6, so 18 boys and 12 girls. Proportion: a/b = c/d means ad = bc. Set up so matching units sit in the same position. Rates and Unit Conversion

Multiply by conversion factors so unwanted units cancel.

Example: 60 mph in ft/s: 60 mi/h × 5280 ft/mi ÷ 3600 s/h = 88 ft/s.

Average speed = total distance ÷ total time, not the average of the speeds. Driving 60 miles at 30 mph (2 h) and 60 miles at 60 mph (1 h) averages 120/3 = 40 mph.

Density = mass/volume. Rate × time = amount.

Check Yourself

Q: A $50 shirt is 20% off. What is the sale price? A: 50 × 0.8 = $40.

Q: What is the percent increase from 80 to 100? A: 20/80 = 25%.

Q: If 3 pencils cost $1.20, how much do 10 cost? A: $0.40 each, so $4.00.

Key Terms

Term / Actor What it is and why it matters

Percent change (new − old)/old × 100.

Proportion Two equal ratios.

Conversion factor Fraction equal to 1 used to change units.

Rate Quantity per unit of another (miles per hour).

Percentage point Difference between two percents.

Making This Essay-Ready

Habit: Write units next to every number in a rate problem.

Trap: Percent "of what?": the base changes after each change.

SAT Math Guide 5 · Original SAT Math study notes