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SAT Math Unit 5: Ratios, rates, percentages & units

These questions are about proportional reasoning: setting up ratios, working with percents and converting units.

Covers: Ratios and proportions, unit rates, percent and percent change, unit conversion

Study guide

Ratios compare quantities, and proportions set two ratios equal; solve a proportion by cross-multiplying. A unit rate is a ratio with a denominator of 1, like miles per hour or dollars per item, and it's often the easiest way to compare options.

Percent means "per hundred." To find a percent of a number, multiply by the decimal (30% of 80 = 0.3 × 80). Percent change is (new − old)/old × 100%. Increasing by p% multiplies by (1 + p/100), and decreasing by p% multiplies by (1 − p/100). Successive changes multiply, so a 10% increase followed by a 10% decrease is 1.1 × 0.9 = 0.99, a 1% net loss, not zero. To undo a percent change and find the original, divide by the multiplier.

Unit conversion uses conversion fractions arranged so unwanted units cancel. For example, to change 90 km/h to meters per second, multiply by 1,000 m/1 km and by 1 h/3,600 s. Watch squared and cubed units: 1 square foot is 144 square inches, not 12. Keep units written in every step and the setup usually becomes obvious.

Density is an amount per unit of space: mass density is mass ÷ volume, and population density is number of people ÷ area. Rearrange to amount = density × size when you need the total, and check that the units match before dividing. When a unit is squared in a rate (for example, meters per second²), apply the time conversion twice. To change m/s² to m/min², multiply by 60 twice, which is 3,600.

Key ideas

• Set up proportions with matching units in matching positions.
• Percent change = (new − old)/old.
• Increase by p%: multiply by (1 + p/100). Decrease: multiply by (1 − p/100).
• Successive percent changes multiply; they don't add.
• Convert units by multiplying by fractions equal to 1 (like 60 min/1 hr).

Formulas and rules

Common mistakes

• Adding successive percents (a 25% raise then 20% cut isn't +5%).
• Finding a percent of the new value instead of the original.

Words to know

Percent change formula

(new − old)/old × 100%

Decay multiplier

1 − r (a 15% decrease → × 0.85)

Successive percent changes

Multiply the factors: +10% then −10% = 1.1 × 0.9 = 0.99 (a 1% net drop).

Percent increase

new = original × (1 + r); reverse it by dividing by (1 + r).

Dimensional analysis

Multiplying by conversion fractions so unwanted units cancel.

Ratio

A comparison of two quantities, like 3:4.

Proportion

Two equal ratios; solve by cross-multiplying.

Unit rate

A rate with a denominator of 1, like 60 miles per hour.

Percent

Per hundred: 30% of 80 = 0.3 × 80.

Finding the original amount

Divide the new amount by the percent multiplier.

Converting squared units

Square the conversion factor: 1 ft² = 144 in².

Sample practice questions

167 more questions like these, from level 1 to 5, are in the practice set.