Study guide
2.1 Ratios, Unit Rates, & Dimensional Analysis
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
2.1 Ratios, Unit Rates, & Dimensional Analysis
Key Takeaways Explain This Topic AI
A ratio compares two quantities by division (written a:b or a/b); it can describe part-to-part or part-to-whole relationships.
To convert a part-to-part ratio a:b into part-to-whole fractions, sum the terms (a + b) to form the denominator: a/(a+b) and b/(a+b).
A unit rate simplifies a comparison between two different measurement units so that the denominator equals exactly 1 unit (e.g., miles per gallon or cost per ounce).
Economic 'best-buy' comparisons require computing the unit price (total cost divided by quantity) across competing product sizes to determine the lowest cost per unit.
Dimensional analysis chains unit conversion factors equal to 1, algebraically canceling unwanted units across numerators and denominators.
Last updated: August 2026
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 a a to quantity b b b (where b ≠ 0 b \ne 0 b = 0 ) can be written in three equivalent mathematical formats:
- Word Form : a to b a\text{ to }b a to b
- Colon Notation : a : b a : b a : b
- Fraction Form : a b \frac{a}{b} b a
- 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 k k . a b = a ÷ GCD ( a , b ) b ÷ GCD ( a , b ) = a ⋅ k b ⋅ k \frac{a}{b} = \frac{a \div \text{GCD}(a,b)}{b \div \text{GCD}(a,b)} = \frac{a \cdot k}{b \cdot k} b a = b ÷ GCD ( a , b ) a ÷ GCD ( a , b ) = b ⋅ k a ⋅ k
Example : A chemistry solution contains 18 mL 18\text{ mL} 18 mL of acid and 42 mL 42\text{ mL} 42 mL of distilled water. The ratio of acid to water is 18 42 \frac{18}{42} 42 18 . Dividing numerator and denominator by GCD ( 18 , 42 ) = 6 \text{GCD}(18, 42) = 6 GCD ( 18 , 42 ) = 6 gives the simplified ratio 3 7 \frac{3}{7} 7 3 , or 3 : 7 3:7 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.
Relationship Type Definition Fraction Representation Example Scenario
Part-to-Part Compares one subgroup to another subgroup within a whole Part 1 Part 2 = a b \frac{\text{Part}_1}{\text{Part}_2} = \frac{a}{b} Part 2 Part 1 = b a Ratio of enrolled boys to girls is 4 : 5 4:5 4 : 5
Part-to-Whole Compares one subgroup to the entire population Part 1 Total = a a + b \frac{\text{Part}_1}{\text{Total}} = \frac{a}{a + b} Total Part 1 = a + b a Fraction of class that is boys is 4 4 + 5 = 4 9 \frac{4}{4 + 5} = \frac{4}{9} 4 + 5 4 = 9 4
The "Total Parts" Algorithm for Multi-Part Distribution
When dividing a total quantity T T T according to a ratio a : b : c a : b : c a : b : c :
Compute the total number of parts : N parts = a + b + c N_{\text{parts}} = a + b + c N parts = a + b + c .
Determine the value of one single part : Value of 1 part = T N parts \text{Value of } 1 \text{ part} = \frac{T}{N_{\text{parts}}} Value of 1 part = N parts T .
Multiply each ratio term by the single-part value: Quantity A = a ⋅ ( T N parts ) , Quantity B = b ⋅ ( T N parts ) , Quantity C = c ⋅ ( T N parts ) \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) Quantity A = a ⋅ ( N parts T ) , Quantity B = b ⋅ ( N parts T ) , Quantity C = c ⋅ ( N parts T )
Worked Example 1: Three-Way Ratio Allocation
A concrete mixture is prepared by mixing cement, sand, and gravel in the ratio 2 : 3 : 5 2 : 3 : 5 2 : 3 : 5 by weight. If a construction project requires a total of 3 , 600 pounds 3,600\text{ pounds} 3 , 600 pounds of the mixed concrete, how many pounds of sand are required?
Find the total number of parts: Total Parts = 2 + 3 + 5 = 10 parts \text{Total Parts} = 2 + 3 + 5 = 10\text{ parts} Total Parts = 2 + 3 + 5 = 10 parts
Find the weight represented by each single part: Weight per part = 3600 lbs 10 = 360 lbs/part \text{Weight per part} = \frac{3600\text{ lbs}}{10} = 360\text{ lbs/part} Weight per part = 10 3600 lbs = 360 lbs/part
Calculate the required weight of sand (3 parts): Sand Weight = 3 × 360 lbs = 1 , 080 pounds \text{Sand Weight} = 3 \times 360\text{ lbs} = 1,080\text{ pounds} Sand Weight = 3 × 360 lbs = 1 , 080 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 1 1 unit of the reference measurement. Unit Rate = Quantity in Numerator Quantity in Denominator = A B units of A per 1 unit of B \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 Unit Rate = Quantity in Denominator Quantity in Numerator = B A units of A per 1 unit of B
Common Unit Rate Domains
Speed / Velocity : Miles Hour = mph \frac{\text{Miles}}{\text{Hour}} = \text{mph} Hour Miles = mph , Meters Second = m/s \frac{\text{Meters}}{\text{Second}} = \text{m/s} Second Meters = m/s
Worker Productivity : Parts Produced Hours Worked = parts/hour \frac{\text{Parts Produced}}{\text{Hours Worked}} = \text{parts/hour} Hours Worked Parts Produced = parts/hour
Fuel Efficiency : Miles Traveled Gallons Consumed = mpg \frac{\text{Miles Traveled}}{\text{Gallons Consumed}} = \text{mpg} Gallons Consumed Miles Traveled = mpg
Unit Pricing : Total Cost ($) Quantity (oz, lbs, units) = dollars/unit \frac{\text{Total Cost (\$)}}{\text{Quantity (oz, lbs, units)}} = \text{dollars/unit} Quantity (oz, lbs, units) Total Cost ($) = 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. Unit Price = Total Price Total Number of Units \text{Unit Price} = \frac{\text{Total Price}}{\text{Total Number of Units}} Unit Price = Total Number of Units Total Price
Worked Example 2: Best Buy Comparison
A supermarket offers three purchasing options for laundry detergent:
- Package A : 32 fluid ounces 32\text{ fluid ounces} 32 fluid ounces for $ 6.72 \$6.72 $6.72
- Package B : 48 fluid ounces 48\text{ fluid ounces} 48 fluid ounces for $ 9.60 \$9.60 $9.60
- Package C : 80 fluid ounces 80\text{ fluid ounces} 80 fluid ounces for $ 15.20 \$15.20 $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.
Calculate unit price for each option: Unit Price A = $ 6.72 32 oz = $ 0.210 per oz \text{Unit Price } A = \frac{\$6.72}{32\text{ oz}} = \$0.210\text{ per oz} Unit Price A = 32 oz $6.72 = $0.210 per oz Unit Price B = $ 9.60 48 oz = $ 0.200 per oz \text{Unit Price } B = \frac{\$9.60}{48\text{ oz}} = \$0.200\text{ per oz} Unit Price B = 48 oz $9.60 = $0.200 per oz Unit Price C = $ 15.20 80 oz = $ 0.190 per oz \text{Unit Price } C = \frac{\$15.20}{80\text{ oz}} = \$0.190\text{ per oz} Unit Price C = 80 oz $15.20 = $0.190 per oz
Compare unit prices:
Package C is the most economical at $ 0.190 / oz \$0.190/\text{oz} $0.190/ oz . Package A is the most expensive at $ 0.210 / oz \$0.210/\text{oz} $0.210/ oz .
Calculate savings per ounce: Savings = $ 0.210 − $ 0.190 = $ 0.020 (or 2.0 cents per ounce) \text{Savings} = \$0.210 - \$0.190 = \$0.020\text{ (or } 2.0\text{ cents per ounce)} Savings = $0.210 − $0.190 = $0.020 (or 2.0 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
Multiplication by Unity : A conversion factor is a fraction whose numerator and denominator represent equal quantities in different units (e.g., 1 foot 12 inches = 1 \frac{1\text{ foot}}{12\text{ inches}} = 1 12 inches 1 foot = 1 and 12 inches 1 foot = 1 \frac{12\text{ inches}}{1\text{ foot}} = 1 1 foot 12 inches = 1 ). Multiplying any quantity by 1 preserves its identity.
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: Given Unit × ( Desired Unit Given Unit ) = Desired Unit \text{Given Unit} \times \left(\frac{\text{Desired Unit}}{\text{Given Unit}}\right) = \text{Desired Unit} Given Unit × ( Given Unit Desired Unit ) = 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 miles per hour 135\text{ miles per hour} 135 miles per hour . What is the train's speed expressed in feet per second ?
Set up the chain of unit conversion factors so that miles cancel to feet and hours cancel to seconds: Speed = 135 mi 1 hr × ( 5 , 280 ft 1 mi ) × ( 1 hr 60 min ) × ( 1 min 60 sec ) \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) Speed = 1 hr 135 mi × ( 1 mi 5 , 280 ft ) × ( 60 min 1 hr ) × ( 60 sec 1 min )
Cancel matching units across numerators and denominators: Speed = 135 × 5 , 280 × 1 × 1 1 × 1 × 60 × 60 ft/sec = 712 , 800 3 , 600 ft/sec \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} Speed = 1 × 1 × 60 × 60 135 × 5 , 280 × 1 × 1 ft/sec = 3 , 600 712 , 800 ft/sec
Simplify the resulting fraction: Speed = 198 feet per second \text{Speed} = 198\text{ feet per second} Speed = 198 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 2 2 for area, 3 3 3 for volume): Area Conversion: 1 yd 2 = ( 1 yd ) 2 = ( 3 ft ) 2 = 9 ft 2 \text{Area Conversion: } 1\text{ yd}^2 = (1\text{ yd})^2 = (3\text{ ft})^2 = 9\text{ ft}^2 Area Conversion: 1 yd 2 = ( 1 yd ) 2 = ( 3 ft ) 2 = 9 ft 2 Volume Conversion: 1 yd 3 = ( 1 yd ) 3 = ( 3 ft ) 3 = 27 ft 3 \text{Volume Conversion: } 1\text{ yd}^3 = (1\text{ yd})^3 = (3\text{ ft})^3 = 27\text{ ft}^3 Volume Conversion: 1 yd 3 = ( 1 yd ) 3 = ( 3 ft ) 3 = 27 ft 3 Area Conversion: 1 ft 2 = ( 12 in ) 2 = 144 in 2 \text{Area Conversion: } 1\text{ ft}^2 = (12\text{ in})^2 = 144\text{ in}^2 Area Conversion: 1 ft 2 = ( 12 in ) 2 = 144 in 2
Worked Example 4: Area Conversion
A homeowner plans to install hardwood flooring in a rectangular living room measuring 18 feet 18\text{ feet} 18 feet by 24 feet 24\text{ feet} 24 feet . Flooring materials are priced and sold exclusively by the square yard . How many square yards of flooring are required?
Calculate area in square feet: Area = 18 ft × 24 ft = 432 ft 2 \text{Area} = 18\text{ ft} \times 24\text{ ft} = 432\text{ ft}^2 Area = 18 ft × 24 ft = 432 ft 2
Convert square feet to square yards using the squared conversion factor ( 1 yd 3 ft ) 2 = 1 yd 2 9 ft 2 \left(\frac{1\text{ yd}}{3\text{ ft}}\right)^2 = \frac{1\text{ yd}^2}{9\text{ ft}^2} ( 3 ft 1 yd ) 2 = 9 ft 2 1 yd 2 : Area = 432 ft 2 × ( 1 yd 2 9 ft 2 ) = 432 9 yd 2 = 48 square yards \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} Area = 432 ft 2 × ( 9 ft 2 1 yd 2 ) = 9 432 yd 2 = 48 square yards
(Exam Trap Alert: Dividing 432 432 432 by 3 3 3 gives 144 144 144 , which is an incorrect distractor created by using a 1D length conversion for a 2D area!)
Common Pitfalls & ACCUPLACER Exam Traps
Confusing Part-to-Part with Part-to-Whole : If the ratio of red marbles to blue marbles is 3 : 7 3:7 3 : 7 , the probability or fraction of selecting a red marble is 3 3 + 7 = 3 10 = 30 % \frac{3}{3 + 7} = \frac{3}{10} = 30\% 3 + 7 3 = 10 3 = 30% , NOT 3 7 \frac{3}{7} 7 3 .
Inverting the Unit Price Ratio : Always divide total currency cost by physical quantity ( Dollars Ounces \frac{\text{Dollars}}{\text{Ounces}} Ounces Dollars ), not quantity by cost ( Ounces Dollars \frac{\text{Ounces}}{\text{Dollars}} Dollars Ounces ), unless specifically asked for 'ounces per dollar'.
Failing to Square or Cube Conversion Factors for Multi-Dimensional Units : Always remember that 1 square foot = 144 square inches 1\text{ square foot} = 144\text{ square inches} 1 square foot = 144 square inches (since 12 × 12 = 144 12 \times 12 = 144 12 × 12 = 144 ) and 1 cubic foot = 1 , 728 cubic inches 1\text{ cubic foot} = 1,728\text{ cubic inches} 1 cubic foot = 1 , 728 cubic inches ( 12 3 = 1 , 728 12^3 = 1,728 1 2 3 = 1 , 728 ).
Mismatched Time Units in Rate Problems : When multiplying rate by time, verify that time units match (e.g., convert 45 minutes 45\text{ minutes} 45 minutes to 0.75 hours 0.75\text{ hours} 0.75 hours before multiplying by speed in miles per hour \text{miles per hour} miles per hour ).