2.1 Ratios, proportions & percents

Ratios, Proportions, and Percents

Quantitative reasoning on the TSIA2 leans on three connected ideas: ratios, proportions, and percents. Learn how they link together and you can solve a large share of the math section quickly and confidently.

What a Ratio Is

A ratio compares two quantities by division. The ratio of 12 boys to 18 girls can be written 12:18, 12/18, or "12 to 18." Ratios simplify like fractions: divide both parts by their greatest common factor. Here the GCF of 12 and 18 is 6, so 12:18 = 2:3. That means for every 2 boys there are 3 girls.

Be careful about part-to-part versus part-to-whole. The part-to-part ratio of boys to girls is 2:3. The part-to-whole ratio of boys to all students is 12:30, which simplifies to 2:5, because 12 + 18 = 30 total students.

Rates and Unit Rates

A rate is a ratio of two quantities measured in different units, such as miles per hour or dollars per pound. A unit rate has a denominator of 1. To find a unit rate, divide.

Worked example: A car travels 210 miles on 7 gallons of gas. The unit rate is 210 / 7 = 30 miles per gallon. If gas costs $3.40 per gallon, that 210-mile trip uses 7 gallons and costs 7 * $3.40 = $23.80.

Unit rates let you compare deals. Which is cheaper: 12 ounces for $3.60 or 16 ounces for $4.48? Compute price per ounce. First: 3.60 / 12 = $0.30 per ounce. Second: 4.48 / 16 = $0.28 per ounce. The 16-ounce package is the better buy per ounce.

Proportions and Cross-Multiplication

A proportion states that two ratios are equal, such as a/b = c/d. Solve a proportion by cross-multiplying: a * d = b * c.

Worked example: A recipe uses 3 cups of flour for every 4 cups of milk. How much flour is needed with 10 cups of milk? Set up 3/4 = x/10. Cross-multiply: 4x = 3 * 10 = 30, so x = 30 / 4 = 7.5 cups of flour.

Worked example (scaling a map): 2 inches represents 25 miles. How many miles do 7 inches represent? 2/25 = 7/x, so 2x = 25 * 7 = 175, and x = 87.5 miles.

Understanding Percent

Percent means "per hundred," so 45% = 45/100 = 0.45. To convert a decimal to a percent, multiply by 100 (0.45 becomes 45%); to convert a percent to a decimal, divide by 100.

• Percent of a number: multiply. 20% of 150 = 0.20 * 150 = 30.
• Finding the whole: 18 is 30% of what number? 0.30 * n = 18, so n = 18 / 0.30 = 60.
• Finding the percent: 27 is what percent of 90? 27 / 90 = 0.30 = 30%.

Percent Change

Percent change compares how much a quantity grows or shrinks relative to its starting value: percent change = (new - old) / old * 100.

Worked example (increase): A share price rises from $40 to $50. Change = 50 - 40 = 10. Then 10 / 40 = 0.25 = 25% increase.

Worked example (decrease): Attendance falls from 250 to 200. Change = 200 - 250 = -50. Then -50 / 250 = -0.20 = 20% decrease.

Discount, Markup, Tax, and Tip

These are all percent-of applications.

• Discount: An $80 jacket is 25% off. Discount = 0.25 * 80 = $20, so the sale price = 80 - 20 = $60. Shortcut: you pay 75%, and 0.75 * 80 = $60.
• Markup: A store buys a lamp for $30 and marks it up 40%. Markup = 0.40 * 30 = $12, so the selling price = 30 + 12 = $42.
• Sales tax: An item costs $60 with 8.25% tax. Tax = 0.0825 * 60 = $4.95, so the total = 60 + 4.95 = $64.95.
• Tip: A meal costs $45 and you tip 18%. Tip = 0.18 * 45 = $8.10, so the total = 45 + 8.10 = $53.10.

Simple Interest

Simple interest uses I = P * r * t, where P is the principal, r is the annual rate as a decimal, and t is the time in years.

Worked example: You deposit $2,000 at 3% simple annual interest for 4 years. I = 2000 * 0.03 * 4 = $240, so the balance is 2000 + 240 = $2,240.

Worked example: A $1,500 loan at 6% for 2 years accrues I = 1500 * 0.06 * 2 = $180 in interest.

Quick Reference

The golden rule: keep every percent as a decimal throughout the calculation, then translate back to a percent or a dollar amount only at the very end.