4.1 Ratios, Rates, Unit Rates, & Proportional Relationships

4.1 Ratios, Rates, Unit Rates, & Proportional Relationships

Proportional reasoning is one of the most heavily tested conceptual domains on the TABE 13&14 Mathematics test across Levels M, D, and A. Workplace tasks in healthcare, construction, manufacturing, and logistics depend constantly on comparing quantities, scaling measurements, calculating unit costs, and interpreting linear rate models. This section provides a comprehensive guide to ratios, unit rates, proportion equations, direct variation ($y = kx$), and scale drawings.

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1. Ratio Concepts & Notation

A ratio is a mathematical comparison of two quantities by division. When comparing quantity $a$ to quantity $b$ (where $b \neq 0$), ratios can be expressed in three equivalent notations:

1. Colon Notation: $a : b$ (read as "$a$ to $b$")
2. Word Notation: $a\text{ to }b$
3. Fraction Notation: $\frac{a}{b}$

Part-to-Part vs. Part-to-Whole Ratios

A crucial distinction on the TABE exam is whether a ratio compares subparts to one another or a subpart to the combined entire total:

• Part-to-Part Ratio: Compares one distinct category directly to another category within a group. If a concrete mix contains $3$ bags of portland cement, $5$ bags of sand, and $2$ bags of gravel, the ratio of cement to sand is $3:5$.
• Part-to-Whole Ratio: Compares one category to the total aggregate of all categories combined. In the same concrete mix, the total number of bags is $3 + 5 + 2 = 10$. The ratio of cement to total concrete mix is $3:10$ (or $\frac{3}{10}$).

+---------------------------------------------------------------+
|                   CONCRETE MIXTURE (10 BAGS TOTAL)            |
+-----------------------+-------------------------------+-------+
|   Cement (3 bags)     |         Sand (5 bags)         |Gravel |
|        [ 30% ]        |            [ 50% ]            |(2) 20%|
+-----------------------+-------------------------------+-------+
  <--- Cement : Sand = 3:5 --->   <--- Cement : Total = 3:10 --->

Simplifying Ratios

Like fractions, ratios must be expressed in simplest integer form by dividing all terms by their Greatest Common Factor (GCF). In standard ratio notation, decimals and mixed numbers should be converted into integer values:

$$\text{Ratio } 2.5 : 7.5 \implies \frac{2.5 \times 10}{7.5 \times 10} = \frac{25}{75} = \frac{25 \div 25}{75 \div 25} = \frac{1}{3} \implies 1:3$$

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2. Rates, Unit Rates, & Unit Price

A rate is a specialized ratio that compares two quantities measured in different units of measurement (such as miles per hour, dollars per gallon, or beats per minute).

The Unit Rate

A unit rate is a rate simplified so that the denominator equals exactly $1$ unit of the second quantity. To calculate a unit rate, divide the numerator by the denominator:

$$\text{Unit Rate} = \frac{\text{Quantity}_1}{\text{Quantity}_2} = \frac{\text{Quantity}_1 \div \text{Quantity}_2}{1}$$

Unit Price & Comparative Shopping ("Best Buy")

A unit price expresses the monetary cost per standard unit of item (e.g., price per ounce, price per foot). To find the best economic value among competing products, calculate and compare their unit prices:

$$\text{Unit Price} = \frac{\text{Total Dollar Cost}}{\text{Total Quantity / Units}}$$

Unit Rates with Complex Fractions

On TABE Levels D and A, rates frequently involve fractional quantities in both numerator and denominator:

$$\text{Example: A runner covers } \frac{3}{4}\text{ mile in } \frac{1}{10}\text{ hour. What is their speed in mph?}$$ $$\text{Unit Rate} = \frac{\frac{3}{4}\text{ mile}}{\frac{1}{10}\text{ hour}} = \frac{3}{4} \div \frac{1}{10} = \frac{3}{4} \times \frac{10}{1} = \frac{30}{4} = 7.5\text{ miles per hour}$$

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3. Proportions & Solving Unknown Quantities

A proportion is a mathematical equation stating that two ratios or rates are strictly equal:

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

The Cross-Product Property (Means-Extremes Theorem)

In any true proportion, the product of the extremes ($a$ and $d$) equals the product of the means ($b$ and $c$):

$$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$$

Step-by-Step Algorithm for Solving Proportions

1. Set up matching units: Ensure the units in both numerators match, and the units in both denominators match ($\frac{\text{Unit A}}{\text{Unit B}} = \frac{\text{Unit A}}{\text{Unit B}}$).
2. Cross-multiply: Multiply diagonally across the equals sign ($a \cdot d = b \cdot c$).
3. Isolate the unknown variable: Divide both sides by the coefficient attached to the variable.
4. Verify: Substitute the solution back into the original proportion to check equivalence.

Worked Example: Solving an Unknown

Problem: An industrial metal stamping press produces $140$ parts in $35\text{ minutes}$. At this identical rate, how many minutes ($m$) will it take to produce $360$ parts?

$$\frac{140\text{ parts}}{35\text{ minutes}} = \frac{360\text{ parts}}{m\text{ minutes}}$$ $$140 \times m = 35 \times 360$$ $$140m = 12,600$$ $$m = \frac{12,600}{140} = 90\text{ minutes (1 hour 30 minutes)}$$

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4. Constant of Proportionality ($k$) & Direct Variation ($y = kx$)

Two variables $x$ (independent variable) and $y$ (dependent variable) are in a proportional relationship (or direct variation) if their quotient is constant for all non-zero pairs:

$$k = \frac{y}{x} \qquad \Longleftrightarrow \qquad y = kx$$

The constant $k$ is called the constant of proportionality, unit rate, or rate of change.

Testing Proportionality in Data Tables

To determine if a data table represents a proportional relationship, calculate the ratio $\frac{y}{x}$ for every single row. If all ratios equal the exact same constant $k$, the relationship is proportional.

Because $k = 22.50$ across all rows, the direct variation equation is $y = 22.50x$.

Graphical Representation of Proportional Relationships

A graph represents a proportional relationship if and only if it satisfies two mandatory geometric criteria:

1. The graph is a perfectly straight line (constant slope).
2. The line passes directly through the origin $(0, 0)$.

  y (Total Cost $)
  ^
40|                        * (4, 40)
30|                  * (3, 30)
20|            * (2, 20)
10|      * (1, 10)  <-- Unit rate point (1, k)
 0+------+-----+-----+-----+-----> x (Items)
 (0,0)   1     2     3     4

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5. Scale Drawings, Blueprint Calculations, & Map Scales

A scale drawing or blueprint is a proportional, scaled-down or scaled-up 2D representation of an actual object. The scale expresses the ratio of the drawing's dimensions to the actual object's dimensions:

$$\text{Scale Factor} = \frac{\text{Drawing Measurement}}{\text{Actual Measurement}}$$

Worked Example: Blueprint Scale Calculation

Problem: An architectural floor plan has a scale of $\frac{1}{4}\text{ inch} = 3\text{ feet}$. On the blueprint, a master conference room measures $4.5\text{ inches}$ wide by $6\text{ inches}$ long. What are the actual dimensions and the actual floor area of the conference room?

1. Calculate Actual Width: $$\frac{0.25\text{ in}}{3\text{ ft}} = \frac{4.5\text{ in}}{W\text{ ft}} \implies 0.25W = 3 \times 4.5 = 13.5 \implies W = \frac{13.5}{0.25} = 54\text{ feet}$$
2. Calculate Actual Length: $$\frac{0.25\text{ in}}{3\text{ ft}} = \frac{6\text{ in}}{L\text{ ft}} \implies 0.25L = 3 \times 6 = 18 \implies L = \frac{18}{0.25} = 72\text{ feet}$$
3. Calculate Actual Area: $$\text{Area} = W \times L = 54\text{ ft} \times 72\text{ ft} = 3,888\text{ sq ft}$$

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TABE Exam Traps & Workplace Applications