Study guide
13.1 Ratios, Rates, Proportions, and Scaling Relationships
California Subject Examinations for Teachers CSET
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1 Chapter 1: The CSET Program and the Multiple Subjects Examination
1.1 The CSET Program: Which Tests a California Teacher Actually Needs 1.2 CSET Multiple Subjects: Test Structure, Scoring, and Test-Day Mechanics
2 Chapter 2: Language Structure, Linguistics, and Non-Written Communication
2.1 Linguistics and Language Structure: Phonology, Morphology, Syntax, Semantics, and Pragmatics 2.2 First and Second Language Acquisition and ELD Instruction 2.3 Non-Written Communication, Listening, Speaking, and Media Analysis
3 Chapter 3: Literacy Development, Reading Comprehension, and Text Complexity
3.1 Foundations of Early Literacy: Print Concepts, Phonemic Awareness, and Phonics 3.2 Reading Fluency and Vocabulary Acquisition 3.3 Reading Comprehension of Literary and Informational Texts 3.4 Text Complexity and Matching Readers to Texts and Tasks
4 Chapter 4: Writing, Conventions, Research, and Literacy Assessment
4.1 The Writing Process, Discourse Modes, and Writing Applications 4.2 Conventions of Language: Grammar, Usage, Mechanics, and Spelling 4.3 Research Practices, Source Credibility, and Presenting Knowledge 4.4 Diagnostic Literacy Assessment, Running Records, and Miscue Analysis
5 Chapter 5: Ancient Civilizations and Medieval World History
5.1 Early River Valley Civilizations and the Foundations of Society 5.2 Classical Mediterranean Civilizations: Ancient Greece and Rome 5.3 Medieval and Early Modern Eras: Byzantium, Islam, Africa, the Americas, and the Renaissance
6 Chapter 6: United States History and Government
6.1 Indigenous North America, Colonial Settlement, and the American Revolution 6.2 Constitutional Foundations, Federalism, and the American System of Government 6.3 Westward Expansion, Sectional Conflict, the Civil War, and Reconstruction 6.4 Industrialization, the Progressive Era, World Wars, and the Civil Rights Movement
7 Chapter 7: California History and Geography
7.1 California Geography, Indigenous Nations, and the Spanish Mission Era 7.2 The Mexican Rancho Period, the Bear Flag Revolt, and the 1849 Gold Rush 7.3 California Statehood, Governance, Water Infrastructure, and Modern Economic Growth
8 Chapter 8: Social Science Skills: Geography, Civics, Economics, and Source Analysis
8.1 Chronological and Spatial Thinking: Timelines, Maps, and Geographic Tools 8.2 Core Concepts in Civics, Economics, and the Social Sciences 8.3 Analyzing Primary and Secondary Sources and Writing Historical Arguments
9 Chapter 9: Scientific Practices and Physical Science
9.1 Scientific Investigation, Engineering Design, and Crosscutting Concepts 9.2 Structure and Properties of Matter: Atoms, Elements, Compounds, and Chemical Reactions 9.3 Forces, Motion, Energy Transformations, and Waves
10 Chapter 10: Life Science and Ecology
10.1 Cell Biology, Cellular Organization, and Metabolic Processes 10.2 Genetics, Heredity, Molecular Biology, and Evolution 10.3 Ecosystems, Ecological Interactions, Energy Flow, and Biogeochemical Cycles
11 Chapter 11: Earth, Water, and Space Science
11.1 Earth's Interior, Plate Tectonics, and Geological Processes 11.2 Atmospheric Structure, Hydrology, Weather Systems, and Climate 11.3 Earth's Water: Oceans, Estuaries, Freshwater Systems, and Tidal Dynamics 11.4 Earth-Sun-Moon Dynamics, the Solar System, and the Universe
12 Chapter 12: Number Sense, Operations, and Fractions
12.1 Number Classifications, Place Value, Properties of Operations, and Integers 12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents 12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving
13 Chapter 13: Algebra, Functions, and Geometry
13.1 Ratios, Rates, Proportions, and Scaling Relationships 13.2 Algebraic Expressions, Linear Equations, Systems, and Functions 13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
14 Chapter 14: Coordinate Geometry, Measurement, Statistics, and Mathematical Practices
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations 14.2 Measurement Systems, Dimensional Analysis, and Precision 14.3 Descriptive Statistics, Data Displays, and Probability Theory 14.4 Mathematical Practices: Reasoning, Modeling, Estimation, and Justification
15 Chapter 15: Physical Education: Movement, Fitness, Social Development, and Inclusion
15.1 Motor Development, Fundamental Movement Skills, and Movement Concepts 15.2 Fitness Principles, Exercise Physiology, Nutrition, and Safety 15.3 Self-Image, Social Development, and Personal and Social Responsibility 15.4 Movement Forms, Cultural Traditions, and Inclusive Adapted Physical Education
16 Chapter 16: Human Development: Cognition, Intelligence, Social and Physical Growth
16.1 Cognitive and Linguistic Development: Piaget, Vygotsky, and Information Processing 16.2 Perspectives on Intelligence, Play, and Individual Differences 16.3 Social-Emotional and Moral Development: Erikson, Kohlberg, and Attachment Theory 16.4 Physical Development, Ecological Systems, and Mandated Reporting
17 Chapter 17: Visual and Performing Arts: Dance, Music, and Theatre
17.1 Fundamental Elements of Dance, Choreographic Principles, and Cultural Styles 17.2 Core Elements of Music, Staff Notation, Rhythm, and Auditory Analysis 17.3 Music History, World Traditions, and Elementary Music Pedagogy 17.4 Elements of Theatre, Dramatic Structures, Acting Techniques, and Stagecraft
18 Chapter 18: Visual Art, Arts Integration, and Constructed-Response Mastery
18.1 Elements of Art, Principles of Design, and Visual Analysis 18.2 Art Media, Techniques, World Art History, and Art Criticism 18.3 California Arts Standards and Cross-Curricular Arts Integration 18.4 CSET Constructed-Response Strategy, Scoring Scale, and Domain-Specific Exemplars
13.1 Ratios, Rates, Proportions, and Scaling Relationships
Key Takeaways Explain This Topic AI
A ratio is a multiplicative comparison of two quantities; part-to-part ratios compare disjoint subsets ( a : b a:b a : b ), whereas part-to-whole ratios compare a subset to the total ( a : ( a + b ) a:(a+b) a : ( a + b ) ), converting directly into rational fractions.
Rates compare quantities measured in distinct units; a unit rate expresses the quantity per single unit of the reference denominator ( r = y / x r = y/x r = y / x ), serving as the constant of proportionality k k k in direct variation relationships ( y = k x y = kx y = k x ).
Concrete pedagogical models for proportional reasoning—including ratio tables, tape diagrams (bar models), double number lines, and Cartesian coordinate graphs—bridge arithmetic operations to linear relationships (a straight line ray passing through the origin ( 0 , 0 ) (0,0) ( 0 , 0 ) ).
Proportions ( a b = c d \frac{a}{b} = \frac{c}{d} b a = d c ) are formally solved via cross-multiplication ( a d = b c ad = bc a d = b c ) and multiplicative scale factor scaling; indirect measurement applies proportional similarity (e.g., shadow reckoning) to calculate inaccessible physical heights.
Multi-dimensional geometric scaling obeys power laws: if the linear 1D scale factor between similar figures is k k k , surface area scales by k 2 k^2 k 2 , and volume scales by k 3 k^3 k 3 .
Last updated: August 2026
13.1 Ratios, Rates, Proportions, and Scaling Relationships
CSET Focus: Proportional reasoning forms the mathematical bridge between elementary arithmetic and secondary algebra. On CSET Multiple Subjects Subtest II, you must demonstrate a deep conceptual and procedural mastery of ratios (part-to-part vs. part-to-whole), rates and unit rates, constant of proportionality ( k = y / x k = y/x k = y / x ), visual representations (tape diagrams, double number lines, and ratio tables), direct variation graphs passing through the origin ( 0 , 0 ) (0,0) ( 0 , 0 ) , indirect measurement via similar figures, and dimensional scaling laws across 1D length, 2D area, and 3D volume.
1. Foundations of Ratios and Multiplicative Comparisons
A ratio is a mathematical expression that compares two quantities multiplicatively. Unlike additive comparisons (which ask how much more or how much less one quantity is than another), multiplicative comparisons determine how many times greater one quantity is relative to another, or what fractional part one quantity represents of another.
Ratio Notational Conventions
A ratio comparing quantity a a a to quantity b b b (where b ≠ 0 b \ne 0 b = 0 ) can be formally expressed in three equivalent notations:
- Odds / Colon Notation: a : b a : b a : b
- Fraction / Quotient Notation: a b \frac{a}{b} b a
- Verbal Notation: " a a a to b b b "
- Part-to-Part vs. Part-to-Whole Comparisons
A critical conceptual distinction tested heavily on the CSET is the difference between part-to-part and part-to-whole ratios: CLASSROOM OF STUDENTS (TOTAL = 28)
- ┌─────────────────────────┴─────────────────────────┐
- BOYS (12) GIRLS (16)
- Part-to-Part Ratio: Boys : Girls = 12 : 16 = 3 : 4
- Part-to-Whole Ratio: Boys : Total = 12 : 28 = 3 : 7 (Fraction: 3/7 of class)
- Part-to-Whole Ratio: Girls : Total = 16 : 28 = 4 : 7 (Fraction: 4/7 of class)
Part-to-Part Ratio: Compares one distinct subgroup to another disjoint subgroup within the same universal set. In the example above, the ratio of boys to girls is 12 : 16 = 3 : 4 12:16 = 3:4 12 : 16 = 3 : 4 . Caution: A part-to-part ratio does not directly represent a fraction of the total group.
Part-to-Whole Ratio: Compares a single subgroup to the entire universal set (the sum of all parts). In the example above, the ratio of boys to total students is 12 : 28 = 3 : 7 12:28 = 3:7 12 : 28 = 3 : 7 . Every part-to-whole ratio directly corresponds to a rational fraction (e.g., 3 7 \frac{3}{7} 7 3 of the students are boys).
Converting Part-to-Part to Part-to-Whole: If the ratio of quantity A A A to quantity B B B is a : b a : b a : b , then: Fraction of A = a a + b , Fraction of B = b a + b \text{Fraction of } A = \frac{a}{a + b}, \qquad \text{Fraction of } B = \frac{b}{a + b} Fraction of A = a + b a , Fraction of B = a + b b
Equivalent Ratios
Two ratios a b \frac{a}{b} b a and c d \frac{c}{d} d c are equivalent if they represent the same multiplicative relationship. Equivalent ratios are generated by multiplying or dividing both terms by the same non-zero real number m m m : a b = a ⋅ m b ⋅ m ( m ≠ 0 ) \frac{a}{b} = \frac{a \cdot m}{b \cdot m} \quad (m \ne 0) b a = b ⋅ m a ⋅ m ( m = 0 )
2. Pedagogical Visual Models for Proportional Reasoning
The California Mathematics Framework emphasizes multiple visual and structural representations to develop students' proportional reasoning from concrete arithmetic to abstract algebra.
1. Tape Diagrams (Bar Models / Strip Diagrams)
Tape diagrams represent quantities as adjacent rectangular bars partitioned into equal unit segments. They are exceptionally powerful for solving ratio word problems without immediately invoking multi-step algebraic equations.
Example: The ratio of flour to sugar in a recipe is 5 : 2 5 : 2 5 : 2 . If a baker uses 350 grams 350\text{ grams} 350 grams of flour, how much sugar is required? Flour: [ 70g ] [ 70g ] [ 70g ] [ 70g ] [ 70g ] <-- 5 units = 350g (1 unit = 70g)
- Sugar: [ 70g ] [ 70g ] <-- 2 units = 2 × 70g = 140g
- 2. Double Number Lines
A double number line displays two parallel coordinate axes with aligned zero points, where corresponding tick marks indicate equivalent ratios across different units of measurement. It is ideal for visualizing continuous rates, scale conversions, and percentages. Distance (miles): 0 15 30 45 60 75
- |--------|--------|--------|--------|--------|
- Time (minutes): 0 20 40 60 80 100
- 3. Ratio Tables
A ratio table organizes equivalent ratios in structured rows and columns. Students use both additive build-up strategies (repeated addition of baseline ratio values) and multiplicative scaling strategies (multiplying/dividing rows or columns by scale factors).
Proportional Visual Models Comparison Matrix
Pedagogical Model Visual Structure Primary Conceptual Focus Best Classroom Problem Scenarios
Tape Diagram (Bar Model) Segmented adjacent rectangular strips Visualizing discrete unit values and part-part-whole relationships Word problems with known totals, differences between quantities, or multi-step ratio mixtures
Double Number Line Two parallel coordinate axes anchored at 0 0 0 Continuous proportional scaling across two distinct measurement units Speed/time conversions, unit pricing, percentage benchmarks ( 0 % 0\% 0% to 100 % 100\% 100% )
Ratio Table Tabular rows/columns of equivalent pairs Additive iteration and multiplicative factor combinations Scaling recipes, financial budgeting, identifying patterns and missing values
Coordinate Graph 2D Cartesian plane ray passing through ( 0 , 0 ) (0,0) ( 0 , 0 ) Geometric representation of continuous linear direct variation ( y = k x y=kx y = k x ) Analyzing slope as unit rate, comparing multiple rates of change visually
- 3. Rates, Unit Rates, and the Constant of Proportionality
- Rates and Unit Rates Defined
Rate: A specific type of ratio that compares two quantities measured in different units (e.g., 180 miles 180\text{ miles} 180 miles per 3 hours 3\text{ hours} 3 hours , $ 4.50 \text{\textdollar}4.50 $ 4.50 for 12 ounces 12\text{ ounces} 12 ounces , 480 words 480\text{ words} 480 words in 8 minutes 8\text{ minutes} 8 minutes ).
Unit Rate: A rate simplified so that the quantity in the denominator is exactly 1 unit of the reference measurement: Unit Rate r = Numerator Quantity Denominator Quantity \text{Unit Rate } r = \frac{\text{Numerator Quantity}}{\text{Denominator Quantity}} Unit Rate r = Denominator Quantity Numerator Quantity
Example 1 (Speed): 180 miles 3 hours = 60 miles/hour = 60 mph \frac{180\text{ miles}}{3\text{ hours}} = 60\text{ miles/hour} = 60\text{ mph} 3 hours 180 miles = 60 miles/hour = 60 mph .
Example 2 (Unit Price): $ 4.50 12 oz = $ 0.375 per ounce ≈ $ 0.38 /oz \frac{\text{\textdollar}4.50}{12\text{ oz}} = \text{\textdollar}0.375\text{ per ounce} \approx \text{\textdollar}0.38\text{/oz} 12 oz $ 4.50 = $ 0.375 per ounce ≈ $ 0.38 /oz . Unit pricing enables consumers to compare product values across varying package volumes.
The Constant of Proportionality ( k k k )
When two variables x x x and y y y are in a proportional relationship (direct variation), the ratio of the dependent variable y y y to the independent variable x x x is constant for all non-zero pairs ( x , y ) (x, y) ( x , y ) . This constant value is designated as the constant of proportionality , denoted by k k k : k = y x ⟺ y = k x k = \frac{y}{x} \iff y = kx k = x y ⟺ y = k x
In this formulation, k k k is identically equal to the unit rate of the relationship.
Coordinate Plane Graphical Characteristics of Proportional Relationships
On a Cartesian coordinate plane, a relationship between x x x and y y y is directly proportional if and only if its graph satisfies two mandatory geometric criteria :
Linearity: The graph must be a perfectly straight line.
- Origin Intersection: The line must pass directly through the origin ( 0 , 0 ) (0, 0) ( 0 , 0 ) . y (Cost in $)
- ^
- | / Graph of y = 15x (Proportional)
- 45 | / - Straight line
- 30 | / * - Passes through origin (0, 0)
- 15 | / * - Point (1, k) = (1, 15) gives unit rate k = 15
- 0 └────────*─────────> x (Hours)
- 0 1 2 3
CSET Key Distinction: The linear equation y = 3 x + 5 y = 3x + 5 y = 3 x + 5 represents a linear relationship, but it is NOT a proportional relationship because its graph has a non-zero y y y -intercept ( b = 5 b = 5 b = 5 ) and does not pass through ( 0 , 0 ) (0, 0) ( 0 , 0 ) . The ratio y x = 3 x + 5 x = 3 + 5 x \frac{y}{x} = \frac{3x+5}{x} = 3 + \frac{5}{x} x y = x 3 x + 5 = 3 + x 5 is not constant!
4. Solving Proportions and Indirect Measurement Applications
A proportion is a formal mathematical statement declaring that two ratios are strictly equal: a b = c d ( b , d ≠ 0 ) \frac{a}{b} = \frac{c}{d} \quad (b, d \ne 0) b a = d c ( b , d = 0 )
Methods for Solving Proportions
Cross-Multiplication (Means-Extremes Property): In any valid proportion a b = c d \frac{a}{b} = \frac{c}{d} b a = d c , the product of the extremes ( a ⋅ d a \cdot d a ⋅ d ) equals the product of the means ( b ⋅ c b \cdot c b ⋅ c ): a b = c d ⟹ a ⋅ d = b ⋅ c ⟹ x = b ⋅ c a \frac{a}{b} = \frac{c}{d} \implies a \cdot d = b \cdot c \implies x = \frac{b \cdot c}{a} b a = d c ⟹ a ⋅ d = b ⋅ c ⟹ x = a b ⋅ c
Mathematical Proof: Multiply both sides of a b = c d \frac{a}{b} = \frac{c}{d} b a = d c by the common denominator b d bd b d : b d ( a b ) = b d ( c d ) ⟹ a d = b c bd\left(\frac{a}{b}\right) = bd\left(\frac{c}{d}\right) \implies ad = bc b d ( b a ) = b d ( d c ) ⟹ a d = b c .
Scale Factor (Horizontal / Vertical Multiplier) Method: Identify the scalar factor connecting corresponding terms: 4 7 = x 35 \frac{4}{7} = \frac{x}{35} 7 4 = 35 x . Since the denominator is multiplied by 5 5 5 ( 7 × 5 = 35 7 \times 5 = 35 7 × 5 = 35 ), the numerator must also be multiplied by 5 5 5 : x = 4 × 5 = 20 x = 4 \times 5 = 20 x = 4 × 5 = 20 .
Indirect Measurement and Shadow Reckoning
Indirect measurement uses proportional relationships between similar geometric figures to determine physical distances or heights that cannot be measured directly (e.g., the height of a flagpole, tree, or building).
The Principle of Shadow Reckoning: At the same time of day and in the same geographic location, incoming solar rays are parallel, forming similar right triangles between vertical objects and their cast horizontal shadows. Tree Person
- |\ |\
- | \ | \
- H | \ h | \
- | \ | \
- └───-─\ └───-─\
S (Shadow) s (Shadow) Height of Tree ( H ) Shadow of Tree ( S ) = Height of Person ( h ) Shadow of Person ( s ) ⟹ H = h ⋅ S s \frac{\text{Height of Tree } (H)}{\text{Shadow of Tree } (S)} = \frac{\text{Height of Person } (h)}{\text{Shadow of Person } (s)} \implies H = \frac{h \cdot S}{s} Shadow of Tree ( S ) Height of Tree ( H ) = Shadow of Person ( s ) Height of Person ( h ) ⟹ H = s h ⋅ S
Worked Example: A 6 -foot 6\text{-foot} 6 -foot tall person casts a 4 -foot 4\text{-foot} 4 -foot shadow. At the exact same moment, a nearby communications tower casts a 56 -foot 56\text{-foot} 56 -foot shadow. Calculate the height of the tower: H 56 = 6 4 ⟹ 4 H = 56 × 6 = 336 ⟹ H = 84 feet . \frac{H}{56} = \frac{6}{4} \implies 4H = 56 \times 6 = 336 \implies H = 84\text{ feet}. 56 H = 4 6 ⟹ 4 H = 56 × 6 = 336 ⟹ H = 84 feet .
Mixture and Scale Map Applications
Map Scale Ratios: A map scale of 1 inch : 25 miles 1\text{ inch} : 25\text{ miles} 1 inch : 25 miles means that a distance of 3.6 inches 3.6\text{ inches} 3.6 inches on the map represents 3.6 × 25 = 90 miles 3.6 \times 25 = 90\text{ miles} 3.6 × 25 = 90 miles in physical reality.
Mixture Concentration: If a chemist has 400 mL 400\text{ mL} 400 mL of a 15 % 15\% 15% saline solution, the pure salt content is 400 × 0.15 = 60 mL 400 \times 0.15 = 60\text{ mL} 400 × 0.15 = 60 mL . To dilute this to a 10 % 10\% 10% saline solution by adding pure water ( w w w ), establish the proportion: 60 400 + w = 10 100 = 0.10 ⟹ 60 = 0.10 ( 400 + w ) ⟹ 60 = 40 + 0.10 w ⟹ 0.10 w = 20 ⟹ w = 200 mL . \frac{60}{400 + w} = \frac{10}{100} = 0.10 \implies 60 = 0.10(400 + w) \implies 60 = 40 + 0.10w \implies 0.10w = 20 \implies w = 200\text{ mL}. 400 + w 60 = 100 10 = 0.10 ⟹ 60 = 0.10 ( 400 + w ) ⟹ 60 = 40 + 0.10 w ⟹ 0.10 w = 20 ⟹ w = 200 mL .
5. Multi-Dimensional Scaling Relationships (1D, 2D, and 3D Laws)
One of the most frequently tested concepts on the CSET exam is the non-linear relationship between linear scale factors and higher-dimensional measurements (area and volume).
The Fundamental Dimensional Scaling Laws
When all linear dimensions of a geometric figure or solid are multiplied by a constant linear scale factor k k k :
1D Linear Dimensions (Perimeter, Circumference, Side Length, Height, Radius): Scale directly by k 1 = k k^1 = k k 1 = k . Perimeter new = k ⋅ Perimeter original \text{Perimeter}_{\text{new}} = k \cdot \text{Perimeter}_{\text{original}} Perimeter new = k ⋅ Perimeter original
2D Surface Measurements (Area, Surface Area, Base Area, Lateral Area): Scale quadratically by k 2 k^2 k 2 . Area new = k 2 ⋅ Area original \text{Area}_{\text{new}} = k^2 \cdot \text{Area}_{\text{original}} Area new = k 2 ⋅ Area original
3D Volumetric Measurements (Volume, Capacity, Mass of homogeneous material): Scale cubically by k 3 k^3 k 3 . Volume new = k 3 ⋅ Volume original \text{Volume}_{\text{new}} = k^3 \cdot \text{Volume}_{\text{original}} Volume new = k 3 ⋅ Volume original
Multi-Dimensional Scaling Matrix
Dimension Geometric Attribute Original Measurement Scale Factor ( k k k ) New Scaled Formula Exemplar Calculation ( k = 3 k = 3 k = 3 )
1D (Linear) Length, Width, Perimeter ( P P P ) L , W , P L, W, P L , W , P k k k P new = k ⋅ P P_{\text{new}} = k \cdot P P new = k ⋅ P If P = 20 cm P = 20\text{ cm} P = 20 cm , P new = 3 × 20 = 60 cm P_{\text{new}} = 3 \times 20 = 60\text{ cm} P new = 3 × 20 = 60 cm ( 3 1 = 3 × 3^1 = 3\times 3 1 = 3 × )
2D (Area) Area ( A A A ), Surface Area ( S A SA S A ) A = l w A = lw A = lw k k k A new = ( k L ) ( k W ) = k 2 A A_{\text{new}} = (kL)(kW) = k^2 A A new = ( k L ) ( kW ) = k 2 A If A = 40 cm 2 A = 40\text{ cm}^2 A = 40 cm 2 , A new = 3 2 × 40 = 9 × 40 = 360 cm 2 A_{\text{new}} = 3^2 \times 40 = 9 \times 40 = 360\text{ cm}^2 A new = 3 2 × 40 = 9 × 40 = 360 cm 2 ( 9 × 9\times 9 × )
3D (Volume) Volume ( V V V ), Enclosed Space V = l w h V = lwh V = lw h k k k V new = ( k L ) ( k W ) ( k H ) = k 3 V V_{\text{new}} = (kL)(kW)(kH) = k^3 V V new = ( k L ) ( kW ) ( k H ) = k 3 V If V = 50 cm 3 V = 50\text{ cm}^3 V = 50 cm 3 , V new = 3 3 × 50 = 27 × 50 = 1 , 350 cm 3 V_{\text{new}} = 3^3 \times 50 = 27 \times 50 = 1,350\text{ cm}^3 V new = 3 3 × 50 = 27 × 50 = 1 , 350 cm 3 ( 27 × 27\times 27 × )
CSET Exam Warning: If a problem states that the dimensions of a rectangular box are doubled ( k = 2 k = 2 k = 2 ), its surface area increases by a factor of 2 2 = 4 2^2 = 4 2 2 = 4 , and its volume/weight capacity increases by a factor of 2 3 = 8 2^3 = 8 2 3 = 8 ! If only one or two dimensions are changed, apply the specific product formula directly rather than uniform scaling.