3.1 Ratios, Rates, Unit Rates & Proportions

Understanding Ratios: Structure, Notation, and Simplification

A ratio is a mathematical comparison of two quantities by division. Ratios express how many times one number contains another or how two values scale relative to each other. On the HiSET Mathematics subtest, ratio questions appear in pure numeric forms, geometric scaling scenarios, and contextual word problems.

Three Notations for Ratios

A ratio comparing quantity $a$ to quantity $b$ (where $b \neq 0$) can be written in three equivalent mathematical formats:

1. Word notation: $a \text{ to } b$
2. Colon notation: $a : b$
3. Fraction notation: $\frac{a}{b}$

In all formats, the order of terms is strictly governed by the problem phrasing. The ratio of apples to oranges is $\frac{\text{apples}}{\text{oranges}}$, which is the reciprocal of the ratio of oranges to apples.

   Ratio Notations for 8 Defective Parts to 100 Total Parts
   ┌────────────────────────────────────────────────────────┐
   │  Fraction:  8/100  =  2/25                             │
   │  Colon:     8 : 100  =  2 : 25                         │
   │  Word:      8 to 100  =  2 to 25                       │
   └────────────────────────────────────────────────────────┘

Simplifying Ratios

Ratios should always be simplified to their lowest terms by dividing both quantities by their Greatest Common Divisor (GCD), exactly like simplifying fractions. Ratios can also involve decimals or fractions, which must be converted to whole-number ratios:

• Decimal Ratios: To simplify $1.25 : 3.75$, multiply both terms by 100 to clear decimals: $\frac{125}{375} = \frac{1}{3} \implies 1:3$.
• Fractional Ratios: To simplify $\frac{2}{3} : \frac{5}{6}$, multiply both terms by the least common denominator (6): $\left(\frac{2}{3} \times 6\right) : \left(\frac{5}{6} \times 6\right) = 4 : 5$.

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Part-to-Part vs. Part-to-Whole Relationships

One of the most frequent traps on the HiSET is confusing part-to-part ratios with part-to-whole ratios. Standardized test writers intentionally place part-to-part options in questions that ask for fractions or percentages of the entire group.

Extended Ratios ($a : b : c$)

An extended ratio compares three or more quantities simultaneously. To solve problems involving extended ratios, assign a common scaling factor variable $x$ to each part:

$\text{Total Quantity} = a x + b x + c x = (a + b + c)x$

Worked Example: Concrete Aggregate Mix

A construction contractor mixes cement, sand, and gravel in an extended ratio of $1 : 2 : 4$ by volume to produce structural concrete. If a foundation pour requires $280\text{ cubic feet}$ of concrete, how many cubic feet of sand are needed?

1. Define the terms with variable $x$: $\text{Cement} = 1x, \quad \text{Sand} = 2x, \quad \text{Gravel} = 4x$
2. Set up the total sum equation: $1x + 2x + 4x = 280 \implies 7x = 280$
3. Solve for the unit multiplier $x$: $x = \frac{280}{7} = 40\text{ cu ft}$
4. Calculate the requested volume (Sand): $\text{Sand} = 2x = 2(40) = 80\text{ cubic feet}$
5. Verify the total volume: $\text{Cement} = 40, \quad \text{Sand} = 80, \quad \text{Gravel} = 160 \implies 40 + 80 + 160 = 280\text{ cu ft}$

Rates and Calculating Unit Rates

While a ratio often compares quantities with the same units (making the ratio unitless), a rate is a specific type of ratio that compares two quantities measured in different physical units (e.g., miles per hour, dollars per pound, heartbeats per minute).

A unit rate is a rate in which the denominator is simplified to exactly 1 unit. Unit rates allow direct, standardized comparisons between competing options or changing speeds.

$\text{Unit Rate} = \frac{\text{Numerator Quantity}}{\text{Denominator Quantity}} = \frac{a}{b} \text{ units of } a \text{ per } 1 \text{ unit of } b$

High-Frequency HiSET Unit Rate Applications

• Unit Pricing: $\text{Price per Unit} = \frac{\text{Total Cost}}{\text{Total Quantity (oz, lbs, items)}}$
• Speed (Velocity): $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{t}$
• Fuel Efficiency: $\text{Fuel Economy} = \frac{\text{Distance Traveled}}{\text{Gallons Consumed}} = \frac{\text{miles}}{\text{gal}}$
• Labor Productivity: $\text{Work Rate} = \frac{\text{Units Produced}}{\text{Hours Worked}}$

The "Better Buy" Unit Price Comparison

Consumer math questions on the HiSET frequently ask you to identify the most economical product packaging by computing unit prices. When packaging sizes use different measurement units (such as pounds versus ounces), you must convert them to a common unit before calculating rates ($1\text{ lb} = 16\text{ oz}$).

In this comparison, Brand B offers the lowest unit cost at $$0.210$ per ounce, making it the most cost-effective choice despite Brand C having a larger total package size.

Solving Proportions via Cross-Multiplication

A proportion is an equation stating that two ratios or rates are mathematically equal:

$\frac{a}{b} = \frac{c}{d} \quad (b \neq 0, \, d \neq 0)$

In a proportion, $a$ and $d$ are called the extremes, while $b$ and $c$ are called the means.

The Fundamental Property of Proportions (Means-Extremes Theorem)

For any valid proportion, the product of the extremes equals the product of the means:

$\text{If } \frac{a}{b} = \frac{c}{d}, \quad \text{then } a \cdot d = b \cdot c$

This cross-multiplication principle transforms rational equations into straightforward linear equations.

      Cross-Multiplication Visual Layout
            a          c
             \        /
              \      /
               \    /
                 \/
                 /\
                /  \
               /    \
              b      d
         ───►  a · d  =  b · c  ◄───

Solving Linear and Binomial Proportions

When a variable appears in a numerator or denominator, cross-multiply and apply inverse algebraic operations.

Worked Example 1: Direct Proportion

If a vehicle travels $165\text{ miles}$ on $5.5\text{ gallons}$ of fuel, how many gallons $g$ are required to drive $420\text{ miles}$ at the same rate of consumption?

1. Set up the proportion with consistent unit placement: $\frac{165\text{ miles}}{5.5\text{ gallons}} = \frac{420\text{ miles}}{g\text{ gallons}}$
2. Cross-multiply: $165 \cdot g = 5.5 \cdot 420$ $165g = 2,310$
3. Divide by the coefficient of $g$: $g = \frac{2,310}{165} = 14\text{ gallons}$

Worked Example 2: Binomial Expression in Proportion

Solve for $x$ in the algebraic proportion:

$\frac{3x - 2}{4} = \frac{2x + 5}{3}$

1. Cross-multiply (ensuring binomial terms are enclosed in parentheses): $3(3x - 2) = 4(2x + 5)$
2. Apply the distributive property: $9x - 6 = 8x + 20$
3. Isolate variable terms by subtracting $8x$ from both sides: $x - 6 = 20$
4. Add 6 to both sides: $x = 26$
5. Check solution in original proportion: $\frac{3(26) - 2}{4} = \frac{78 - 2}{4} = \frac{76}{4} = 19 \quad \text{and} \quad \frac{2(26) + 5}{3} = \frac{52 + 5}{3} = \frac{57}{3} = 19 \quad (\text{True})$

Scale Drawings, Blueprints, and Dimensional Models

A scale drawing or scale model is an exact proportional representation of an object enlarged or reduced by a specific scale factor $k$. Architectural floor plans, road maps, and mechanical blueprints rely entirely on proportional scaling.

$\text{Scale Factor } (k) = \frac{\text{Measurement on Drawing or Model}}{\text{Actual Measurement in Reality}}$

Setting Up Scale Proportions

When converting blueprint dimensions to real-world measurements, maintain identical unit orientations in both ratios:

$\frac{\text{Scale Drawing Length}}{\text{Actual Real-World Length}} = \frac{\text{Blueprint Measurement}}{\text{Actual Measurement}}$

Blueprint Area vs. Linear Scaling Warning

Critical HiSET Rule: While linear dimensions scale by the factor $k$, areas scale by $k^2$ and volumes scale by $k^3$.

If a blueprint has a scale of $1\text{ inch} = 5\text{ feet}$, a room drawn as $2\text{ in} \times 3\text{ in}$ has:

• Actual width: $2\text{ in} \times 5\text{ ft/in} = 10\text{ ft}$
• Actual length: $3\text{ in} \times 5\text{ ft/in} = 15\text{ ft}$
• Actual Area: $10\text{ ft} \times 15\text{ ft} = 150\text{ sq ft}$
• Area check via scale factor squared: $\text{Drawing Area} = 2 \times 3 = 6\text{ sq in}$. Real area $= 6\text{ sq in} \times (5\text{ ft/in})^2 = 6 \times 25 = 150\text{ sq ft}$.

      Linear Scaling vs. Area Scaling
      
      Drawing (1 in = 5 ft)          Actual Room (10 ft x 15 ft)
      ┌───────────┐                  ┌──────────────────────────────┐
      │  2 in     │                  │                              │
      │           │                  │ 10 ft                        │
      │  3 in     │                  │                              │
      └───────────┘                  │           15 ft              │
      Area = 6 sq in                 └──────────────────────────────┘
                                     Area = 150 sq ft (= 6 x 5²)

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Summary of Proportional Reasoning Traps

1. Inverted Units Trap: Writing $\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}$. Both ratios must have the same unit in the numerator and the same unit in the denominator.
2. Part-to-Part Misinterpreted as Probability or Fraction: If the ratio of passed to failed students is $7 : 3$, the fraction of students who passed is $\frac{7}{7 + 3} = \frac{7}{10} = 70%$, not $\frac{7}{3}$.
3. Premature Rounding: Never round intermediate unit rates before multiplying by the target quantity. Keep exact fractions or memory values in your calculator.