2.1 Ratios, Unit Rates, & Dimensional Analysis

Foundations of Ratios

A ratio is a mathematical comparison of two numbers or quantities by division. Ratios describe relative size, frequency, or concentration between quantities. On the ACCUPLACER Quantitative Reasoning, Algebra, and Statistics (QAS) test, ratios appear across arithmetic word problems, geometry similarity problems, and algebraic modeling.

Standard Notations for Ratios

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

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

Simplifying and Scaling Ratios

Like standard fractions, ratios should be simplified to lowest terms by dividing all terms by their Greatest Common Divisor (GCD). Conversely, ratios can be scaled up by multiplying both terms by a common positive multiplier $k$.

$\frac{a}{b} = \frac{a \div \text{GCD}(a,b)}{b \div \text{GCD}(a,b)} = \frac{a \cdot k}{b \cdot k}$

Example: A chemistry solution contains $18\text{ mL}$ of acid and $42\text{ mL}$ of distilled water. The ratio of acid to water is $\frac{18}{42}$. Dividing numerator and denominator by $\text{GCD}(18, 42) = 6$ gives the simplified ratio $\frac{3}{7}$, or $3:7$.

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

A critical distinction on the ACCUPLACER test is recognizing whether a given ratio represents a part-to-part comparison or a part-to-whole comparison.

The "Total Parts" Algorithm for Multi-Part Distribution

When dividing a total quantity $T$ according to a ratio $a : b : c$:

1. Compute the total number of parts: $N_{\text{parts}} = a + b + c$.
2. Determine the value of one single part: $\text{Value of } 1 \text{ part} = \frac{T}{N_{\text{parts}}}$.
3. Multiply each ratio term by the single-part value: $\text{Quantity } A = a \cdot \left(\frac{T}{N_{\text{parts}}}\right), \quad \text{Quantity } B = b \cdot \left(\frac{T}{N_{\text{parts}}}\right), \quad \text{Quantity } C = c \cdot \left(\frac{T}{N_{\text{parts}}}\right)$

Worked Example 1: Three-Way Ratio Allocation

A concrete mixture is prepared by mixing cement, sand, and gravel in the ratio $2 : 3 : 5$ by weight. If a construction project requires a total of $3,600\text{ pounds}$ of the mixed concrete, how many pounds of sand are required?

1. Find the total number of parts: $\text{Total Parts} = 2 + 3 + 5 = 10\text{ parts}$
2. Find the weight represented by each single part: $\text{Weight per part} = \frac{3600\text{ lbs}}{10} = 360\text{ lbs/part}$
3. Calculate the required weight of sand (3 parts): $\text{Sand Weight} = 3 \times 360\text{ lbs} = 1,080\text{ pounds}$

Rates and Unit Rates

While a ratio often compares two quantities with identical units (such as inches to inches), a rate compares two quantities measured in different units (such as miles to hours, dollars to pounds, or rotations to minutes).

Unit Rate Definition

A unit rate is a specialized rate simplified so that the denominator is scaled to exactly $1$ unit of the reference measurement.

$\text{Unit Rate} = \frac{\text{Quantity in Numerator}}{\text{Quantity in Denominator}} = \frac{A}{B}\text{ units of } A \text{ per } 1\text{ unit of } B$

Common Unit Rate Domains

• Speed / Velocity: $\frac{\text{Miles}}{\text{Hour}} = \text{mph}$, $\frac{\text{Meters}}{\text{Second}} = \text{m/s}$
• Worker Productivity: $\frac{\text{Parts Produced}}{\text{Hours Worked}} = \text{parts/hour}$
• Fuel Efficiency: $\frac{\text{Miles Traveled}}{\text{Gallons Consumed}} = \text{mpg}$
• Unit Pricing: $\frac{\text{Total Cost (\$)}}{\text{Quantity (oz, lbs, units)}} = \text{dollars/unit}$

Unit Price and "Best Buy" Comparative Analysis

Consumer arithmetic questions frequently ask test-takers to identify the most economical packaging option by comparing unit prices across different volume or weight sizes.

$\text{Unit Price} = \frac{\text{Total Price}}{\text{Total Number of Units}}$

Worked Example 2: Best Buy Comparison

A supermarket offers three purchasing options for laundry detergent:

• Package A: $32\text{ fluid ounces}$ for $\$6.72$
• Package B: $48\text{ fluid ounces}$ for $\$9.60$
• Package C: $80\text{ fluid ounces}$ for $\$15.20$

Determine which package provides the lowest unit cost per fluid ounce and calculate the exact savings per ounce compared to the most expensive option.

1. Calculate unit price for each option: $\text{Unit Price } A = \frac{\$6.72}{32\text{ oz}} = \$0.210\text{ per oz}$ $\text{Unit Price } B = \frac{\$9.60}{48\text{ oz}} = \$0.200\text{ per oz}$ $\text{Unit Price } C = \frac{\$15.20}{80\text{ oz}} = \$0.190\text{ per oz}$
2. Compare unit prices: Package C is the most economical at $\$0.190/\text{oz}$. Package A is the most expensive at $\$0.210/\text{oz}$.
3. Calculate savings per ounce: $\text{Savings} = \$0.210 - \$0.190 = \$0.020\text{ (or } 2.0\text{ cents per ounce)}$

Dimensional Analysis (The Unit Factor Method)

Dimensional analysis (also known as the factor-label method or unit conversion method) is an algebraic technique that uses conversion factors to transform a quantity from one unit of measurement to another without altering its actual magnitude.

Fundamental Principles of Dimensional Analysis

1. Multiplication by Unity: A conversion factor is a fraction whose numerator and denominator represent equal quantities in different units (e.g., $\frac{1\text{ foot}}{12\text{ inches}} = 1$ and $\frac{12\text{ inches}}{1\text{ foot}} = 1$). Multiplying any quantity by 1 preserves its identity.
2. Diagonal Unit Cancellation: Units obey the same algebraic rules as numerical variables. A unit appearing in the numerator cancels with the identical unit appearing in the denominator: $\text{Given Unit} \times \left(\frac{\text{Desired Unit}}{\text{Given Unit}}\right) = \text{Desired Unit}$

Essential Reference Conversion Factors

U.S. Customary Length & Weight:
  1 foot (ft) = 12 inches (in)         1 pound (lb) = 16 ounces (oz)
  1 yard (yd) = 3 feet (ft)            1 ton (T) = 2,000 pounds (lbs)
  1 mile (mi) = 5,280 feet = 1,760 yd

U.S. Customary Liquid Capacity:
  1 cup = 8 fluid ounces (fl oz)       1 quart (qt) = 2 pints = 4 cups
  1 pint (pt) = 2 cups = 16 fl oz      1 gallon (gal) = 4 quarts = 128 fl oz

Time & Metric System:
  1 hour = 60 minutes = 3,600 seconds  1 kilogram (kg) = 1,000 grams (g)
  1 day = 24 hours                     1 meter (m) = 100 centimeters (cm) = 1,000 mm
  1 kilometer (km) = 1,000 meters      1 liter (L) = 1,000 milliliters (mL)

Worked Example 3: Multi-Step Rate Conversion

A high-speed commuter train travels at a constant velocity of $135\text{ miles per hour}$. What is the train's speed expressed in feet per second?

1. Set up the chain of unit conversion factors so that miles cancel to feet and hours cancel to seconds: $\text{Speed} = \frac{135\text{ mi}}{1\text{ hr}} \times \left(\frac{5,280\text{ ft}}{1\text{ mi}}\right) \times \left(\frac{1\text{ hr}}{60\text{ min}}\right) \times \left(\frac{1\text{ min}}{60\text{ sec}}\right)$
2. Cancel matching units across numerators and denominators: $\text{Speed} = \frac{135 \times 5,280 \times 1 \times 1}{1 \times 1 \times 60 \times 60}\text{ ft/sec} = \frac{712,800}{3,600}\text{ ft/sec}$
3. Simplify the resulting fraction: $\text{Speed} = 198\text{ feet per second}$

Higher-Dimensional Unit Conversions (Area & Volume Trap)

A frequent trap on standardized mathematics exams occurs when converting units of area (squared units) or volume (cubic units).

The Dimension Rule

When converting squared or cubic units, the linear conversion factor must be raised to the corresponding power ($2$ for area, $3$ for volume):

$\text{Area Conversion: } 1\text{ yd}^2 = (1\text{ yd})^2 = (3\text{ ft})^2 = 9\text{ ft}^2$ $\text{Volume Conversion: } 1\text{ yd}^3 = (1\text{ yd})^3 = (3\text{ ft})^3 = 27\text{ ft}^3$ $\text{Area Conversion: } 1\text{ ft}^2 = (12\text{ in})^2 = 144\text{ in}^2$

Worked Example 4: Area Conversion

A homeowner plans to install hardwood flooring in a rectangular living room measuring $18\text{ feet}$ by $24\text{ feet}$. Flooring materials are priced and sold exclusively by the square yard. How many square yards of flooring are required?

1. Calculate area in square feet: $\text{Area} = 18\text{ ft} \times 24\text{ ft} = 432\text{ ft}^2$
2. Convert square feet to square yards using the squared conversion factor $\left(\frac{1\text{ yd}}{3\text{ ft}}\right)^2 = \frac{1\text{ yd}^2}{9\text{ ft}^2}$: $\text{Area} = 432\text{ ft}^2 \times \left(\frac{1\text{ yd}^2}{9\text{ ft}^2}\right) = \frac{432}{9}\text{ yd}^2 = 48\text{ square yards}$

(Exam Trap Alert: Dividing $432$ by $3$ gives $144$, which is an incorrect distractor created by using a 1D length conversion for a 2D area!)

Common Pitfalls & ACCUPLACER Exam Traps

1. Confusing Part-to-Part with Part-to-Whole: If the ratio of red marbles to blue marbles is $3:7$, the probability or fraction of selecting a red marble is $\frac{3}{3 + 7} = \frac{3}{10} = 30\%$, NOT $\frac{3}{7}$.
2. Inverting the Unit Price Ratio: Always divide total currency cost by physical quantity ($\frac{\text{Dollars}}{\text{Ounces}}$), not quantity by cost ($\frac{\text{Ounces}}{\text{Dollars}}$), unless specifically asked for 'ounces per dollar'.
3. Failing to Square or Cube Conversion Factors for Multi-Dimensional Units: Always remember that $1\text{ square foot} = 144\text{ square inches}$ (since $12 \times 12 = 144$) and $1\text{ cubic foot} = 1,728\text{ cubic inches}$ ($12^3 = 1,728$).
4. Mismatched Time Units in Rate Problems: When multiplying rate by time, verify that time units match (e.g., convert $45\text{ minutes}$ to $0.75\text{ hours}$ before multiplying by speed in $\text{miles per hour}$).