6.3 Ratio, Proportional Relationships, and Unit Conversions

Ratio, Proportional Relationships & Unit Conversions

Core Competency: Proportional reasoning forms the computational backbone of real-world applied mathematics. The ACCUPLACER QAS exam tests your ability to translate verbal relationships into ratios, compute unit rates and compound velocities, execute multi-step dimensional conversions, interpret geometric scale factors, and distinguish direct from inverse variation models.

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Ratios, Rates, and Unit Rates

A ratio is a mathematical comparison of two quantities with respect to magnitude. Ratios can be expressed in three standard formats: with a colon ($a:b$), as a fraction ($\frac{a}{b}$), or with the word "to" ($a\text{ to }b$).

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

• Part-to-Part: Compares one subset of a group to another subset. In a chemistry cohort with 12 biology majors and 18 chemistry majors, the ratio of biology to chemistry majors is $12:18 = 2:3$.
• Part-to-Whole: Compares a subset to the entire aggregate. The ratio of biology majors to total students is $\frac{12}{12 + 18} = \frac{12}{30} = \frac{2}{5}$ (or 40%).

The Ratio Partitioning Method

When given a ratio $a:b:c$ and an aggregate sum $T$, solve for individual quantities by calculating the value of a single base share:

$$\text{Base Share Multiplier } (x) = \frac{\text{Total Quantity } (T)}{a + b + c}$$

Worked Example: Ratio Partitioning

A manufacturing plant produces brass alloy using copper, zinc, and lead in the ratio $7:4:1$ by weight. How many pounds of zinc are needed to produce 2,160 pounds of the alloy?

• Step 1 (Sum of ratio parts): $7 + 4 + 1 = 12\text{ total parts}$.
• Step 2 (Calculate base share): $x = \frac{2,160\text{ lbs}}{12} = 180\text{ lbs per share}$.
• Step 3 (Determine zinc quantity): $\text{Zinc weight} = 4 \times 180\text{ lbs} = 720\text{ lbs}$.
• Verification: Copper = $7(180) = 1,260\text{ lbs}$; Lead = $1(180) = 180\text{ lbs}$. Total = $1,260 + 720 + 180 = 2,160\text{ lbs}$.

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Rates, Unit Rates & Multi-Leg Average Speed

A rate is a ratio comparing two measurements with different units (e.g., 280 miles per 8 gallons, $15.60 for 24 ounces). A unit rate is simplified so that its denominator is exactly 1 unit.

$$\text{Unit Rate} = \frac{\text{Quantity } A}{\text{Quantity } B} \implies \frac{\$15.60}{24\text{ oz}} = \$0.65\text{ per ounce}$$

Motion and Speed Relationships ($d = r \cdot t$)

The relationship between distance ($d$), constant rate/speed ($r$), and elapsed time ($t$) is governed by:

$$d = r \cdot t \quad \iff \quad r = \frac{d}{t} \quad \iff \quad t = \frac{d}{r}$$

The Multi-Leg Average Speed Trap

A frequent distractor trap on the ACCUPLACER occurs when computing average speed over a journey with varying velocities. You cannot compute the simple arithmetic mean of the two speeds. You must compute total distance divided by total elapsed time.

$$v_{\text{avg}} = \frac{\text{Total Distance } (d_1 + d_2)}{\text{Total Time } (t_1 + t_2)}$$

Worked Example: Harmonic Average Speed

A delivery truck travels 120 miles from Warehouse A to Distribution Hub B at an average speed of 60 mph, and returns along the exact same 120-mile route at an average speed of 40 mph due to heavy traffic. What is the average speed of the truck for the entire round trip?

• Step 1 (Total distance): $d_{\text{total}} = 120 + 120 = 240\text{ miles}$.
• Step 2 (Outbound time): $t_1 = \frac{120\text{ miles}}{60\text{ mph}} = 2.0\text{ hours}$.
• Step 3 (Return time): $t_2 = \frac{120\text{ miles}}{40\text{ mph}} = 3.0\text{ hours}$.
• Step 4 (Total time): $t_{\text{total}} = 2.0 + 3.0 = 5.0\text{ hours}$.
• Step 5 (Compute average velocity): $v_{\text{avg}} = \frac{240\text{ miles}}{5.0\text{ hours}} = 48\text{ mph}$ (Note: The arithmetic average $\frac{60 + 40}{2} = 50\text{ mph}$ is incorrect!).

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Proportions & Cross-Multiplication Mechanics

A proportion is a statement equating two rational fractions:

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

By the Fundamental Property of Proportions (Cross-Products Property):

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

Solving Multi-Step Algebraic Proportions

When variables appear in binomial expressions within fractions, apply parentheses before cross-multiplying to ensure accurate distribution.

$$\frac{2x - 3}{5} = \frac{x + 4}{3} \implies 3(2x - 3) = 5(x + 4) \implies 6x - 9 = 5x + 20 \implies x = 29$$

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Dimensional Analysis & Multi-Step Unit Conversions

Dimensional analysis (factor-label method) converts a measurement from one unit system to another by multiplying by a sequence of conversion factors (fractions equal to 1). Units are arranged diagonally across numerators and denominators so that unwanted units cancel algebraically.

Essential Unit Conversion Equalities

                               Dimensional Analysis Cancellation Chain
  [Given Unit A]   [Desired Unit B]   [Desired Unit C]
  ────────────── × ──────────────── × ──────────────── = [Calculated Unit C]
      [1]          [Given Unit A]     [Desired Unit B]
                        ▲                  ▲
                   (Unit A Cancels)   (Unit B Cancels)

Worked Example: Chained Compound Conversion

Convert a fluid flow rate of $45\text{ gallons per minute}$ into quarts per second.

$$\frac{45\text{ gal}}{1\text{ min}} \times \frac{4\text{ qt}}{1\text{ gal}} \times \frac{1\text{ min}}{60\text{ s}} = \frac{45 \times 4 \times 1}{1 \times 1 \times 60}\text{ qt/s} = \frac{180}{60}\text{ qt/s} = 3.0\text{ quarts per second}$$

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Scale Drawings and Geometric Scale Factors

A scale factor ($k$) defines the ratio of a linear dimension on a model/map to the corresponding linear dimension on the actual object:

$$k = \frac{\text{Scale Model Length}}{\text{Actual Real-World Length}}$$

The Linear vs. Area vs. Volume Scaling Rules

When scaling geometric figures by a linear scale factor $k$:

1. Linear Dimensions (Perimeter, Length, Radius): Scale by $k$.
2. Surface Area & Cross-Sectional Area: Scale by $k^2$.
3. Volume & Capacity: Scale by $k^3$.

Example: If a blueprint uses a scale factor of $1\text{ inch} = 5\text{ feet}$ ($k = \frac{1}{5}$ linear scale), an actual room with a floor area of $300\text{ sq ft}$ will occupy $300 \times (\frac{1}{5})^2 = 300 \times \frac{1}{25} = 12\text{ square inches}$ on the blueprint.

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Direct Variation vs. Inverse Variation

Variation equations model how changes in one independent variable dictate changes in a dependent variable.

+-----------------------------------------------------------------------------------------+
|                        VARIATION MATHEMATICAL MODELS COMPARISON                         |
+-----------------------+---------------------------------+-------------------------------+
| Dimension             | Direct Variation                | Inverse Variation             |
+-----------------------+---------------------------------+-------------------------------+
| Algebraic Formula     | y = k · x                       | y = k / x   (or x · y = k)    |
| Constant Formula (k)  | k = y / x                       | k = x · y                     |
| Functional Behavior   | When x doubles, y doubles       | When x doubles, y halves      |
| Graphical Shape       | Straight line through origin    | Rectangular hyperbola         |
| Coordinate Signature  | Constant ratio between points   | Constant product between pts  |
| Proportion Identity   | y₁ / x₁ = y₂ / x₂               | x₁ · y₁ = x₂ · y₂             |
+-----------------------+---------------------------------+-------------------------------+

Step-by-Step Protocol for Variation Problems

1. Identify the Variation Type: Search for verbal cues ("varies directly as", "is proportional to" vs. "varies inversely as", "inversely proportional to").
2. Set Up the Base Equation: Write $y = kx$ or $y = \frac{k}{x}$.
3. Solve for the Constant of Variation ($k$): Substitute the initial pair of given values $(x_1, y_1)$.
4. Evaluate the Target Unknown: Substitute $k$ and the second given value into the equation to solve for the missing variable.