📊 Ratios, Rates, Proportions — Topics

Master the core topics and techniques for working with ratios, rates, and proportions in MATHCOUNTS.

Ratio Operations

Simplifying Ratios

Method: Divide by GCF of both numbers Example: $12:18 = 6:9 = 2:3$ (GCF of 12 and 18 is 6)

Mixed numbers: Convert to improper fractions first Example: $2\frac{1}{3}:1\frac{1}{2} = \frac{7}{3}:\frac{3}{2} = \frac{14}{6}:\frac{9}{6} = 14:9$

Pitfall: Not simplifying to lowest terms Fix: Always divide by GCF

Equivalent Ratios

Method: Multiply or divide both terms by same number Example: $3:4 = 6:8 = 9:12 = 15:20$

Cross products: If $a:b = c:d$, then $ad = bc$ Example: $3:4 = 6:8$ because $3 \times 8 = 4 \times 6 = 24$

Pitfall: Forgetting that order matters Fix: Always check that ratios are in same order

Ratio Arithmetic

Addition: $a:b + c:d = (ad + bc):bd$ Subtraction: $a:b - c:d = (ad - bc):bd$ Multiplication: $(a:b) \times (c:d) = ac:bd$ Division: $(a:b) \div (c:d) = ad:bc$

Example: $3:4 + 1:2 = (3 \times 2 + 4 \times 1):(4 \times 2) = 10:8 = 5:4$

Rate Calculations

Unit Rates

Definition: Rate with denominator of 1 Method: Divide numerator by denominator

Examples:

  • 60 miles in 2 hours = 30 mph
  • $15 for 3 pounds = $5 per pound
  • 24 pages in 3 hours = 8 pages per hour

Pitfall: Not simplifying to unit rate Fix: Always divide to get denominator of 1

Rate Conversions

Method: Use conversion factors Example: Convert 60 mph to feet per second

  • $60 \text{ mph} = 60 \times \frac{5280 \text{ ft}}{3600 \text{ s}} = 88 \text{ ft/s}$

Common conversions:

  • 1 mile = 5280 feet
  • 1 hour = 3600 seconds
  • 1 gallon = 4 quarts
  • 1 pound = 16 ounces

Combined Rates

Addition: $r_1 + r_2$ (same direction) Subtraction: $r_1 - r_2$ (opposite direction) Harmonic mean: $\frac{2r_1 r_2}{r_1 + r_2}$ (average rate)

Example: Car A goes 60 mph, Car B goes 40 mph

  • Same direction: 60 - 40 = 20 mph relative speed
  • Opposite direction: 60 + 40 = 100 mph relative speed

Proportion Solving

Cross Multiplication

Method: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ Example: $\frac{3}{4} = \frac{x}{12}$ becomes $3 \times 12 = 4x$, so $x = 9$

Pitfall: Forgetting to cross multiply Fix: Always multiply diagonally

Scale Factor Method

Method: Find scale factor, then multiply Example: $\frac{3}{4} = \frac{x}{12}$

  • Scale factor: $\frac{12}{4} = 3$
  • $x = 3 \times 3 = 9$

Equivalent Ratios

Method: Use equivalent ratios Example: $\frac{3}{4} = \frac{x}{12}$

  • $\frac{3}{4} = \frac{9}{12}$ (multiply by 3)
  • So $x = 9$

Direct and Inverse Proportions

Direct Proportion

Formula: $y = kx$ where $k$ is constant Method: Find constant, then use formula

Example: If $y$ varies directly with $x$ and $y = 6$ when $x = 2$, find $y$ when $x = 5$

  • $6 = k \times 2$, so $k = 3$
  • $y = 3x$, so when $x = 5$, $y = 15$

Pitfall: Confusing direct and inverse Fix: Direct means both increase together

Inverse Proportion

Formula: $y = \frac{k}{x}$ where $k$ is constant Method: Find constant, then use formula

Example: If $y$ varies inversely with $x$ and $y = 6$ when $x = 2$, find $y$ when $x = 3$

  • $6 = \frac{k}{2}$, so $k = 12$
  • $y = \frac{12}{x}$, so when $x = 3$, $y = 4$

Pitfall: Forgetting to use reciprocal Fix: Inverse means one increases as other decreases

Scale and Similarity

Scale Factor

Definition: Ratio of corresponding lengths Method: $\text{Scale factor} = \frac{\text{New length}}{\text{Original length}}$

Example: If a model is $\frac{1}{24}$ scale, then 1 inch on model = 24 inches on actual

Area and Volume Scaling

Area: New area = Original area × (scale factor)² Volume: New volume = Original volume × (scale factor)³

Example: If scale factor is 3, then area is multiplied by 9, volume by 27

Similar Triangles

Properties:

  • Corresponding angles are equal
  • Corresponding sides are proportional
  • Ratio of areas = (scale factor)²

Example: If triangles are similar with scale factor 2:3, then area ratio is 4:9

Word Problems

Part-Whole Problems

Template: $\frac{\text{part}}{\text{whole}} = \frac{\text{part}}{\text{whole}}$ Example: If 3 out of 7 students are boys, how many boys in a class of 28?

  • $\frac{3}{7} = \frac{x}{28}$, so $x = 12$ boys

Rate Problems

Template: $\frac{\text{quantity 1}}{\text{quantity 2}} = \frac{\text{quantity 1}}{\text{quantity 2}}$ Example: If a car travels 60 miles in 2 hours, how far in 5 hours?

  • $\frac{60}{2} = \frac{x}{5}$, so $x = 150$ miles

Work Problems

Template: $\frac{\text{work 1}}{\text{time 1}} = \frac{\text{work 2}}{\text{time 2}}$ Example: If 3 workers can build a wall in 8 days, how long for 6 workers?

  • $\frac{3}{8} = \frac{6}{x}$, so $x = 4$ days

Mixture Problems

Template: Use weighted averages Example: Mix 2 liters of 20% solution with 3 liters of 80% solution

  • $\frac{2 \times 0.20 + 3 \times 0.80}{2 + 3} = \frac{0.4 + 2.4}{5} = 0.56 = 56%$

Common Mistakes

Ratio Mistakes

Mistake: Adding ratios incorrectly: $3:4 + 1:2 = 4:6$ Fix: Use proper formula: $(3 \times 2 + 4 \times 1):(4 \times 2) = 10:8 = 5:4$

Mistake: Forgetting to simplify: $6:8 = 6:8$ Fix: Always simplify: $6:8 = 3:4$

Proportion Mistakes

Mistake: Wrong cross multiplication: $\frac{3}{4} = \frac{x}{12}$ becomes $3 \times 4 = 12x$ Fix: Multiply diagonally: $3 \times 12 = 4x$

Mistake: Forgetting to check units Fix: Always ensure units are consistent

Rate Mistakes

Mistake: Wrong unit conversion Fix: Use conversion factors carefully

Mistake: Confusing direct and inverse proportion Fix: Remember that direct means both increase together

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