📊 Ratios, Rates, Proportions — Formulas

Essential formulas and shortcuts for working with ratios, rates, and proportions in MATHCOUNTS.

Ratio Formulas

Basic Operations

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: $(a:b) \div (c:d) = ad:bc$

Simplifying Ratios

GCF method: $a:b = \frac{a \div \text{GCF}(a,b)}{b \div \text{GCF}(a,b)}$ Prime factorization: Cancel common prime factors

Example: Simplify $24:36$

• GCF(24, 36) = 12
• $24:36 = \frac{24 \div 12}{36 \div 12} = 2:3$

Equivalent Ratios

Cross products: If $a:b = c:d$, then $ad = bc$ Scale factor: If $a:b = c:d$, then $c = \frac{ad}{b}$

Example: If $3:4 = x:12$, then $3 \times 12 = 4x$, so $x = 9$

Rate Formulas

Basic Rate

Formula: Rate = $\frac{\text{Quantity 1}}{\text{Quantity 2}}$ Unit rate: Rate with denominator of 1

Examples:

• Speed: $\frac{\text{Distance}}{\text{Time}}$
• Price: $\frac{\text{Cost}}{\text{Quantity}}$
• Work: $\frac{\text{Work Done}}{\text{Time}}$

Rate Conversions

Length: 1 mile = 5280 feet, 1 km = 1000 meters Time: 1 hour = 3600 seconds, 1 day = 86400 seconds Volume: 1 gallon = 4 quarts, 1 liter = 1000 milliliters

Conversion formula: $\text{New rate} = \text{Old rate} \times \frac{\text{Conversion factor}}{\text{Conversion factor}}$

Example: Convert 60 mph to ft/s

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

Combined Rates

Same direction: $r_{\text{combined}} = r_1 + r_2$ Opposite direction: $r_{\text{relative}} = |r_1 - r_2|$ Harmonic mean: $r_{\text{average}} = \frac{2r_1 r_2}{r_1 + r_2}$

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 Formulas

Direct Proportion

Formula: $y = kx$ where $k$ is constant of proportionality Cross products: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$

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

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

Inverse Proportion

Formula: $y = \frac{k}{x}$ where $k$ is constant of proportionality Product rule: If $y$ varies inversely with $x$, then $xy = k$

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

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

Solving Proportions

Method 1: Cross multiplication

• $\frac{a}{b} = \frac{c}{d}$ becomes $ad = bc$

Method 2: Scale factor

• If $\frac{a}{b} = \frac{c}{d}$, then $c = \frac{ad}{b}$

Method 3: Equivalent ratios

• $\frac{a}{b} = \frac{c}{d}$ becomes $\frac{a}{c} = \frac{b}{d}$

Scale and Similarity Formulas

Scale Factor

Definition: $\text{Scale factor} = \frac{\text{New length}}{\text{Original length}}$ Area scaling: New area = Original area × (scale factor)² Volume scaling: New volume = Original volume × (scale factor)³

Example: If scale factor is 3

• Area is multiplied by 9
• Volume is multiplied by 27

Similar Figures

Corresponding sides: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k$ Area ratio: $\frac{A_1}{A_2} = k^2$ Volume ratio: $\frac{V_1}{V_2} = k^3$

Example: Two similar triangles with scale factor 2:3

• Area ratio: 4:9
• Volume ratio: 8:27

Word Problem Formulas

Part-Whole Problems

Formula: $\frac{\text{part}}{\text{whole}} = \frac{\text{part}}{\text{whole}}$ 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

Mixture Problems

Weighted average: $\frac{a_1 \times p_1 + a_2 \times p_2 + … + a_n \times p_n}{a_1 + a_2 + … + a_n}$

Example: Mix 2 liters of 20% solution with 3 liters of 80% solution

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

Work Problems

Individual rate: $r = \frac{\text{work}}{\text{time}}$ Combined rate: $r_{\text{combined}} = r_1 + r_2 + … + r_n$ Time to complete: $t = \frac{\text{work}}{r_{\text{combined}}}$

Example: Pipe A fills tank in 3 hours, Pipe B in 6 hours

• Combined rate: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ tank/hour
• Time together: 2 hours

Distance Problems

Basic formula: $d = rt$ Same direction: $d_1 - d_2 = (r_1 - r_2)t$ Opposite direction: $d_1 + d_2 = (r_1 + r_2)t$

Example: Two cars start 100 miles apart, one goes 60 mph, other 40 mph

• Relative speed: 60 + 40 = 100 mph
• Time to meet: $\frac{100}{100} = 1$ hour

Unit Conversion Formulas

Length Conversions

Time Conversions

Volume Conversions

Mental Math Shortcuts

Common Ratios

1:2: Half 1:3: Third 1:4: Quarter 2:3: Two-thirds 3:4: Three-quarters 1:5: Fifth 1:10: Tenth

Rate Shortcuts

10% of $n$: $n \div 10$ 20% of $n$: $n \div 5$ 25% of $n$: $n \div 4$ 50% of $n$: $n \div 2$ 75% of $n$: $n \div 4 \times 3$

Proportion Shortcuts

If $a:b = c:d$, then:

• $a:c = b:d$
• $b:a = d:c$
• $c:a = d:b$

Example: If $3:4 = 6:8$, then:

• $3:6 = 4:8$ (1:2 = 1:2)
• $4:3 = 8:6$ (4:3 = 4:3)
• $6:3 = 8:4$ (2:1 = 2:1)

Common Applications

Business Problems

Markup: Selling price = Cost + (Markup % × Cost) Discount: Sale price = Original price - (Discount % × Original price) Profit: Profit = Revenue - Cost Profit margin: Profit margin = $\frac{\text{Profit}}{\text{Revenue}} \times 100%$

Science Problems

Density: Density = $\frac{\text{Mass}}{\text{Volume}}$ Pressure: Pressure = $\frac{\text{Force}}{\text{Area}}$ Speed: Speed = $\frac{\text{Distance}}{\text{Time}}$ Acceleration: Acceleration = $\frac{\text{Change in velocity}}{\text{Time}}$

Geometry Problems

Scale factor: $\frac{\text{New length}}{\text{Original length}}$ Area ratio: (Scale factor)² Volume ratio: (Scale factor)³ Perimeter ratio: Scale factor

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