SAT Problem-Solving: Ratios, Rates, and Percentages

The three question types for percentages, the part-to-whole ratio trap, and the multiplier stacking trick that turns "20% off then 15% off" into one fast multiplication.

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Ratios, rates, and percentages account for roughly half of the Problem-Solving & Data Analysis domain — about 3–4 questions per test. The math is middle-school easy; the trap is the wording.

Ratios: parts of a whole

A ratio compares two quantities. The SAT writes it three ways: a : b, a to b, or the fraction a/b. All three mean the same thing.

Part-to-part vs. part-to-whole

If the ratio of boys to girls is 3:5, the parts add to 8. So 3/8 are boys and 5/8 are girls — not 3/5.

A class of 40 students has a boy-to-girl ratio of 3 : 5. Total parts: 3 + 5 = 8 Boys: (3/8) · 40 = 15 Girls: (5/8) · 40 = 25 Always divide the whole into "total parts" before computing.

Proportions: setting up the cross-multiply

Cross-multiplication: the diagonals are equal whenever two fractions are equal.

A recipe calls for 3 cups flour per 2 cups sugar. For 8 cups sugar, how much flour? 3/2 = x/8 2x = 24 x = 12 cups Keep matching units on top and bottom: flour/sugar on both sides.

Rates: unit analysis

A rate is a ratio with units — miles per hour, dollars per gallon, words per minute. SAT rate problems hinge on unit cancellation.

A car travels 240 miles in 4 hours. At this rate, how far in 7 hours? Rate = 240 mi / 4 hr = 60 mi/hr Distance = 60 · 7 = 420 mi Always compute the unit rate first — it makes the rest trivial.

Percent: three question types

Type 1: What is x% of N? → multiply: (x/100) · N Type 2: x is what % of N? → divide: (x/N) · 100 Type 3: x is y% of what? → divide: x / (y/100) Translate "is" to "=" and "of" to "·" — every percent question writes itself.

Percent change

Percent change = (new − old) / old · 100% Always divide by the original value. Example: $80 → $100 is +25% (20/80), not +20%.

The reversal trap

A 20% increase followed by a 20% decrease does not return to the original. Start with $100 → $120 → $96. Always compute step by step.

Percent multipliers — the speed trick

Convert percent changes to multipliers and stack them.

"30% off, then 10% off again" = 0.70 · 0.90 = 0.63 → final price is 63% of original "15% increase then 8% increase" = 1.15 · 1.08 = 1.242 → up 24.2% Stacking multipliers is faster than computing each change.

Unit conversion and dimensional analysis

The SAT loves multi-step rate problems that demand unit conversions. The trick: chain fractions so units cancel.

Convert 90 km/hr to meters per second. 90 km/hr · (1000 m / 1 km) · (1 hr / 3600 s) = 90 · 1000 / 3600 = 25 m/s Write every unit. Cancel diagonally. The remaining unit confirms you set up the math right.

Simple vs. compound interest

Both appear on the Digital SAT; compound interest is the harder version.

Simple: A = P(1 + rt) Compound (annually): A = P(1 + r)^t Compound (n times per year): A = P(1 + r/n)^nt P = principal, r = annual rate (decimal), t = years.

Common mistakes

Treating a 3:5 ratio as 3/5 of the whole (it's 3/8)
Dividing by the new value instead of the original on percent change
Assuming reversibility — "up 30%, down 30%" is not back to start
Forgetting unit cancellation on multi-step rate problems
Using percent (5) instead of decimal (0.05) inside an exponential formula

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