Solving Proportions & Cross-Multiplication

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✖️ 1. Defining a proportion as an equality of two ratios

⚖️ What Is a Proportion?

A proportion states that two ratios are equal.
Written as $\frac{a}{b} = \frac{c}{d}$ where $b \neq 0$ and $d \neq 0$ .
Read as "a is to b as c is to d".
The four numbers $a, b, c, d$ are called the terms of the proportion.
Extremes are the outer terms ( $a$ and $d$ ), means are the inner terms ( $b$ and $c$ ).

Example: $\frac{2}{3} = \frac{4}{6}$ is a proportion because both ratios simplify to the same value.

💡 Think of a proportion as a balanced scale: both sides must have the same ratio weight.

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1. Defining a proportion as an equality of two ratios

Defining a Proportion as an Equality of Two Ratios

A proportion is an equation stating that two ratios are equal, written as $\frac{a}{b} = \frac{c}{d}$ where $b \neq 0$ and $d \neq 0$ . This expresses that the multiplicative relationship between $a$ and $b$ is identical to that between $c$ and $d$ .

Intuition: If two quantities scale at the same rate, their ratios remain constant across different measurements.

Core Rules:

Both denominators must be nonzero to ensure the ratios are defined
The terms $a$ and $d$ are called the extremes; $b$ and $c$ are the means
A proportion represents a fundamental equivalence, not merely an approximation
Alternative notation: $a:b = c:d$ (colon form)

Consequence: Proportions preserve the relative size relationship between paired quantities, making them essential for scaling and comparison tasks.

Example: If $\frac{3}{5} = \frac{6}{10}$ , then 3 relates to 5 exactly as 6 relates to 10, since both simplify to the same multiplicative factor of $0.6$ .

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✖️ 2. Deriving the cross-multiplication rule directly from fraction equivalence

✖️ Cross-Multiplication Rule

Start with $\frac{a}{b} = \frac{c}{d}$ .
Multiply both sides by $bd$ to clear denominators: $bd \cdot \frac{a}{b} = bd \cdot \frac{c}{d}$ .
Simplify to get $ad = bc$ .
This is cross-multiplication: multiply diagonally across the equals sign.
Use this to solve for any unknown term in a proportion.

Example: Solve $\frac{x}{5} = \frac{3}{15}$ . Cross-multiply: $15x = 15$ , so $x = 1$ .

💡 Draw an X across the proportion to remember which terms to multiply together.

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2. Deriving the cross-multiplication rule directly from fraction equivalence

Deriving the Cross-Multiplication Rule from Fraction Equivalence

The cross-multiplication rule states that if $\frac{a}{b} = \frac{c}{d}$ , then $ad = bc$ . This follows directly from multiplying both sides of the proportion by the common denominator $bd$ .

Intuition: Clearing denominators transforms the proportion into a simpler linear equation without fractions.

Core Rules:

Multiply both sides by $bd$ : $bd \cdot \frac{a}{b} = bd \cdot \frac{c}{d}$
Simplify to obtain $ad = bc$ (the cross-product equation)
This operation is valid only when $b \neq 0$ and $d \neq 0$
The converse holds: if $ad = bc$ with nonzero denominators, then $\frac{a}{b} = \frac{c}{d}$

Consequence: Cross-multiplication converts proportion problems into standard algebraic equations, enabling straightforward solution of unknown terms.

Example: Given $\frac{x}{7} = \frac{4}{14}$ , cross-multiply to get $14x = 28$ , yielding $x = 2$ .

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✖️ 3. Direct and inverse proportional relationships and their identification

🔄 Direct vs Inverse Proportions

Direct proportion: when one quantity increases, the other increases at the same rate ( $y = kx$ ).
Inverse proportion: when one quantity increases, the other decreases ( $xy = k$ or $y = \frac{k}{x}$ ).
Direct proportion uses $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ .
Inverse proportion uses $x_1 y_1 = x_2 y_2$ .
Check the relationship: if doubling $x$ doubles $y$ , it is direct; if doubling $x$ halves $y$ , it is inverse.

Example: 3 workers finish a job in 8 hours. For 6 workers (inverse): $3 \times 8 = 6 \times t$ , so $t = 4$ hours.

💡 Direct = same direction; Inverse = opposite direction.

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3. Direct and inverse proportional relationships and their identification

Direct and Inverse Proportional Relationships

Direct proportion occurs when $y = kx$ for constant $k > 0$ , meaning $y$ increases as $x$ increases. Inverse proportion occurs when $y = \frac{k}{x}$ , meaning $y$ decreases as $x$ increases.

Intuition: Direct proportion maintains constant ratio $\frac{y}{x}$ ; inverse proportion maintains constant product $xy$ .

Core Rules:

Direct: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ for any two pairs; doubling $x$ doubles $y$
Inverse: $x_1 y_1 = x_2 y_2$ ; doubling $x$ halves $y$
Identification test: Check if ratio or product remains constant across data points
Graph of direct proportion is a line through the origin; inverse proportion is a hyperbola

Consequence: Recognizing the relationship type determines the correct equation form for modeling and prediction.

Example: If 5 workers complete a task in 8 hours, then 10 workers (inverse proportion) complete it in 4 hours, since $5 \times 8 = 10 \times 4 = 40$ .

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✖️ 4. Translating real-world word problems into formal algebraic proportion equations

📝 Word Problems to Proportions

Identify the two ratios being compared in the problem.
Assign a variable to the unknown quantity.
Write the proportion with matching units in the same position (top-to-top, bottom-to-bottom).
Use cross-multiplication to solve for the variable.
Always check that units cancel correctly.

Example: If 5 apples cost 10 dollars, how much do 8 apples cost? Set up $\frac{5}{10} = \frac{8}{x}$ . Cross-multiply: $5x = 80$ , so $x = 16$ dollars.

💡 Match units vertically: apples over apples, dollars over dollars.

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4. Translating real-world word problems into formal algebraic proportion equations

Translating Word Problems into Proportion Equations

Translation requires identifying the two equivalent ratios from problem context and expressing them as $\frac{a}{b} = \frac{c}{d}$ where one term is unknown.

Intuition: Match corresponding quantities (same units or roles) in numerators and denominators to preserve the proportional relationship.

Core Rules:

Identify the constant relationship: Determine what remains proportional (e.g., cost per item, speed)
Assign variables systematically: Let $x$ represent the unknown quantity
Maintain unit consistency: Numerators must share units; denominators must share units
Verify directionality: Ensure the proportion reflects whether quantities increase or decrease together

Consequence: Proper setup guarantees that cross-multiplication yields the correct equation for solving.

Example: "If 3 pencils cost 12 dollars, how much do 7 pencils cost?" becomes $\frac{3 \text{ pencils}}{12 \text{ dollars}} = \frac{7 \text{ pencils}}{x \text{ dollars}}$ , giving $3x = 84$ , so $x = 28$ dollars.

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✖️ 5. Applications in stoichiometry, Hooke's Law, and currency conversion

🔬 Real-World Proportion Applications

Stoichiometry: Use mole ratios from balanced equations ( $\frac{\text{moles A}}{\text{moles B}} = \frac{\text{coefficient A}}{\text{coefficient B}}$ ).
Hooke's Law: Force is directly proportional to extension ( $F = kx$ , so $\frac{F_1}{x_1} = \frac{F_2}{x_2}$ ).
Currency conversion: Exchange rates create direct proportions ( $\frac{\text{dollars}}{\text{euros}} = \text{constant rate}$ ).
Set up the proportion with known values and solve for the unknown.

Example: If 2 moles of H₂ produce 2 moles of H₂O, then 5 moles of H₂ produce $x$ moles: $\frac{2}{2} = \frac{5}{x}$ , so $x = 5$ moles.

💡 Proportions are the universal translator between different measurement systems.

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5. Applications in stoichiometry, Hooke's Law, and currency conversion

Applications Across Scientific and Practical Domains

Proportions model diverse phenomena: stoichiometry (mole ratios in reactions), Hooke's Law ( $F = kx$ for spring force), and currency conversion (exchange rates).

Intuition: Each application exploits a fixed multiplicative relationship between measurable quantities.

Core Rules:

Stoichiometry: $\frac{\text{moles A}}{\text{coefficient A}} = \frac{\text{moles B}}{\text{coefficient B}}$ from balanced equations
Hooke's Law: Force $F$ is directly proportional to displacement $x$ ; $\frac{F_1}{x_1} = \frac{F_2}{x_2}$ when spring constant $k$ is fixed
Currency conversion: $\frac{\text{amount}_1}{\text{currency}_1} = \frac{\text{amount}_2}{\text{currency}_2}$ using exchange rate
Always verify units match within each ratio

Consequence: Mastery of proportions enables quantitative reasoning across chemistry, physics, and economics.

Example: If 2 moles of $\text{H}_2$ produce 2 moles of $\text{H}_2\text{O}$ , then 5 moles of $\text{H}_2$ produce $\frac{2}{2} = \frac{x}{5}$ , yielding $x = 5$ moles of $\text{H}_2\text{O}$ .

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An exchange rate dictates that 2 euros are equivalent to 3 dollars. Using the currency conversion proportion, how many dollars are equivalent to 10 euros?

DEEP_COMPUTE

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Fundamental property of proportions and solving proportion equations

✖️ 1. Defining a proportion as an equality of two ratios

⚖️ What Is a Proportion?

1. Defining a proportion as an equality of two ratios

Defining a Proportion as an Equality of Two Ratios

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✖️ 2. Deriving the cross-multiplication rule directly from fraction equivalence

✖️ Cross-Multiplication Rule

2. Deriving the cross-multiplication rule directly from fraction equivalence

Deriving the Cross-Multiplication Rule from Fraction Equivalence

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✖️ 3. Direct and inverse proportional relationships and their identification

🔄 Direct vs Inverse Proportions

3. Direct and inverse proportional relationships and their identification

Direct and Inverse Proportional Relationships

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✖️ 4. Translating real-world word problems into formal algebraic proportion equations

📝 Word Problems to Proportions

4. Translating real-world word problems into formal algebraic proportion equations

Translating Word Problems into Proportion Equations

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✖️ 5. Applications in stoichiometry, Hooke's Law, and currency conversion

🔬 Real-World Proportion Applications

5. Applications in stoichiometry, Hooke's Law, and currency conversion

Applications Across Scientific and Practical Domains

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