3.3 Percentages, Ratios, and Proportions

Percent: A Special Fraction Out of 100

Percentages, ratios, and proportions are three lenses on the same idea — comparing quantities — and they appear throughout the ParaPro because they show up everywhere in real life. Percent literally means "per hundred," so any percent is a fraction with a denominator of 100. That single fact lets you move freely among percent, decimal, and fraction forms.

Converting Among the Three Forms

• Percent → decimal: divide by 100 (move the decimal two places left). 45% = 0.45.
• Decimal → percent: multiply by 100 (move two places right). 0.08 = 8%.
• Percent → fraction: write over 100 and simplify. 75% = 75/100 = ¾.
• Fraction → percent: convert to a decimal, then to a percent. ⅖ = 0.4 = 40%.

Finding a Percent of a Number

The core formula is Part = Percent × Whole, using the decimal form of the percent.

Worked Example: A jacket costs $80 and is 25% off. The discount is 0.25 × 80 = $20, so the sale price is 80 − 20 = $60. Notice the two-step nature of discount problems: find the part, then subtract.

Finding What Percent One Number Is of Another

Rearrange the formula: Percent = Part ÷ Whole × 100. If a student answers 18 of 20 questions correctly, the score is 18 ÷ 20 = 0.9 = 90%.

Ratios and Unit Rates

A ratio compares two quantities and can be written three ways: 3 to 4, 3:4, or as the fraction ¾. Order matters — the ratio of boys to girls is not the same as girls to boys.

Simplifying a ratio works like simplifying a fraction: divide both terms by their greatest common factor. With 12 boys and 8 girls, the ratio 12:8 reduces to 3:2 (divide both by 4).

A unit rate is a ratio expressed per one unit of something, found by dividing so the second quantity becomes 1. Unit rates power smart-shopping and speed problems.

Worked Example: Which is the better buy — 12 oz of juice for $3.00 or 20 oz for $4.40? Compute unit prices: 3.00 ÷ 12 = $0.25/oz versus 4.40 ÷ 20 = $0.22/oz. The 20-oz bottle is cheaper per ounce, so it is the better value.

Proportions and Cross-Multiplication

A proportion states that two ratios are equal: a/b = c/d. When one term is unknown, cross-multiply (multiply each numerator by the other denominator) and solve.

Worked Example: A map scale reads 1 inch = 10 miles. Two cities are 3.5 inches apart on the map. Set up 1/10 = 3.5/x. Cross-multiply: 1 × x = 10 × 3.5, so x = 35 miles.

Proportions also solve recipe scaling, similar-figure, and currency problems — any time two quantities change at the same rate.

Mental-Math Shortcuts

The ParaPro is partly a timed test (90 questions in 150 minutes), so fast percent estimates pay off. These also model efficient thinking for students.

To find 15% of 80, take 10% (= 8) plus 5% (= 4) for 12. Building percents from 10% and 1% chunks is a teachable, error-resistant method.

Classroom Application

Percents and ratios are intuitive when tied to contexts students already understand:

1. Shopping — discounts, tax, and tips make percent change meaningful.
2. Cooking — doubling or halving a recipe is proportional reasoning.
3. Sports — batting averages and free-throw percentages are real-world percents.
4. Ratio tables lay out equivalent ratios in columns so patterns are visible.
5. Bar and pie models show a percent as part of a whole before any formula.

The strongest application answers teach why cross-multiplication works (equal ratios stay equal when scaled) rather than presenting it as a magic trick, and they connect new percent work to fraction knowledge students already have.

Recap

Percent is a fraction out of 100, so the three forms convert by moving the decimal; Part = Percent × Whole anchors percent problems; ratios compare and simplify like fractions; unit rates divide to a per-one comparison; proportions are solved by cross-multiplication; and real contexts plus ratio tables make all of this teachable.