Model Comparison: Integrating Essential Skills

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Gemini 3 Pro

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What You Need to Know

• It is the largest category: Integrating Essential Skills (IES) typically accounts for 40–43% of the ACT Math section. Mastery here is non-negotiable for a high score.

• It tests fluency, not just memory: These questions often use middle-school concepts (ratios, percentages, averages) in complex, multi-step high-school contexts.

• Context is king: Most IES questions are word problems requiring you to translate real-world scenarios into mathematical models.

• Speed matters: Since the math is conceptually simpler than trigonometry or logs, these questions are where you should bank time—but simple errors are common due to rushing.

Rates, Ratios, and Proportional Relationships

At the heart of the ACT is the ability to compare quantities. A ratio expresses the relationship between two numbers, while a proportion states that two ratios are equal.

Solving Proportions

When two ratios are equivalent, you can set them up as a fraction equation:

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

To solve for an unknown variable, use the cross-multiplication method:

$a \cdot d = b \cdot c$

Direct and Inverse Variation

• Direct Variation: As one variable increases, the other increases at a constant rate. The formula is $y = kx$, where $k$ is the constant of proportionality.

• Inverse Variation: As one variable increases, the other decreases. The formula is $y = \frac{k}{x}$ or $xy = k$.

Exam Focus

• Why it matters: Proportions are the quickest way to solve scaling problems, such as recipe adjustments or map distances.

• Typical question patterns:

  • "If 3 widgets cost $x$ dollars, how much do $y$ widgets cost?"

  • "$y$ varies inversely as $x$. If $y=4$ when $x=10$, what is $y$ when $x=2$?"

• Common mistakes: Confusing direct and inverse variation. Always write out the base formula ($y=kx$ vs $y=k/x$) before plugging in numbers.

Percentages and Percent Change

Percentage questions on the ACT often involve multiple steps, such as applying a tax after a discount.

The Basic Percent Equation

Translate the sentence "$Part$ is $Percent$ of $Whole$" directly into math:

$Part = \frac{Percent}{100} \times Whole$

Percent Change

To find the percentage increase or decrease between two values:

$\text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100$

If the result is negative, it is a percent decrease.

Growth and Decay Factors

Instead of finding $15\%$ of a number and adding it, multiply by the growth factor directly to save time:

• Increase by $p\%$: Multiply by $(1 + \frac{p}{100})$. (e.g., $+20\%$ means multiply by $1.20$).

• Decrease by $p\%$: Multiply by $(1 - \frac{p}{100})$. (e.g., $-20\%$ means multiply by $0.80$).

Exam Focus

• Why it matters: These are guaranteed points if you know the shortcuts.

• Typical question patterns:

  • Reverse percentages: "A shirt costs $40$ dollars after a $20\%$ discount. What was the original price?" (Solve $0.80x = 40$, do NOT calculate $20\%$ of $40$).

  • Compound interest or population growth scenarios.

• Common mistakes: Calculating the percent of the new number rather than the original. Always divide by the "Old Value" (the starting number).

Mean, Median, and Measures of Center

Statistics in the Essential Skills section focus on analyzing datasets and missing data problems.

Definitions

• Mean (Average): The sum of data points divided by the count.
  $\text{Mean} = \frac{\text{Sum of items}}{\text{Number of items}}$

• Median: The middle value when data is ordered from least to greatest. If there is an even number of items, average the two middle numbers.

• Mode: The most frequently occurring value.

• Range: The difference between the maximum and minimum values.

The Missing Value Problem

This is a classic ACT setup. If you know the average of a set and need to find a new value to reach a target average, use the Sum concept:

$\text{Sum} = \text{Mean} \times \text{Count}$

Example: If a student averages $80$ on 4 tests, their total points are $80 \times 4 = 320$. To average $82$ over 5 tests, they need $82 \times 5 = 410$ total points. The 5th test score must be $410 - 320 = 90$.

Exam Focus

• Why it matters: Missing value problems appear on nearly every ACT Math test.

• Typical question patterns:

  • "What must x be so that the mean of 2, 4, 8, and x is 6?"

  • Weighted averages (e.g., GPA calculations).

• Common mistakes: Forgetting to re-order the list of numbers before finding the median.

Area, Surface Area, and Volume

Unlike the SAT, the ACT does not provide a reference sheet with formulas. You must memorize them.

Essential Formulas

• Rectangle: $A = lw$, $P = 2l + 2w$

• Triangle: $A = \frac{1}{2}bh$

• Circle: $A = \pi r^2$, $C = 2\pi r$

• Rectangular Prism (Box) Volume: $V = lwh$

• Cylinder Volume: $V = \pi r^2 h$

Composite Shapes

Many questions involve "composite" figures—complex shapes made of simple ones (e.g., a rectangle with a semi-circle on top). Calculate the area of each part separately and add (or subtract) them.

Exam Focus

• Why it matters: Geometry is a significant chunk of the test, and basic area/volume questions are the easiest type of geometry problem offered.

• Typical question patterns:

  • "A room is $10$ ft by $12$ ft. Carpet costs $5$ dollars per square foot. How much to carpet the room?"

  • Finding the area of a shaded region (usually Area of Large Shape minus Area of Small Shape).

• Common mistakes: Mixing up Diameter ($d$) and Radius ($r$). Always check which one is given. If given $d$, immediately write down $r = d/2$.

Unit Conversions (Dimensional Analysis)

Unit conversion requires multiplying by conversion factors arranged so that unwanted units cancel out.

The Chain Method

To convert units, multiply your starting value by fractions where the numerator equals the denominator (e.g., $\frac{12 \text{ inches}}{1 \text{ foot}}$):

$\text{Given Value} \times \frac{\text{Target Unit}}{\text{Current Unit}} = \text{Target Value}$

Example: Convert $60$ miles per hour to feet per second ($1$ mile = $5280$ ft):

$60 \frac{\text{miles}}{\text{hr}} \times \frac{5280 \text{ft}}{1 \text{mile}} \times \frac{1 \text{hr}}{60 \text{min}} \times \frac{1 \text{min}}{60 \text{sec}}$

Notice how "miles", "hr", and "min" cancel out diagonally.

Exam Focus

• Why it matters: Real-world modeling problems often use mixed units (e.g., room dimensions in feet, tile costs in inches).

• Typical question patterns: Speed conversions or currency exchanges.

• Common mistakes: Flipping the conversion factor (multiplying when you should divide). Always write the units to ensure they cancel.

Expressing Numbers and Modeling

These questions test your ability to translate English sentences into algebraic expressions.

Numerical Representations

• Scientific Notation: $a \times 10^n$, where 1 \le a < 10. Positive $n$ for large numbers, negative $n$ for decimals.

• Rational vs. Irrational: Rational numbers can be written as fractions (e.g., $0.333…$, $\frac{1}{2}$). Irrational numbers cannot (e.g., $\pi$, $\sqrt{2}$).

Mathematical Modeling

You will often be asked to identify which equation represents a word problem.

1. Identify the variables. (Let $x$ = number of adult tickets).

2. Identify the constants. (Fixed fees, starting values).

3. Identify the operators.

   • "per", "each", "rate" $\rightarrow$ Multiplication

   • "initial fee", "starting at", "plus" $\rightarrow$ Addition

   • "less than", "discount" $\rightarrow$ Subtraction

Exam Focus

• Why it matters: Modeling connects basic arithmetic to algebra.

• Typical question patterns: "A taxi charges $3.00$ dollars plus $0.50$ dollars per mile. Which expression represents the cost for $m$ miles?" (Answer: $0.50m + 3.00$).

• Common mistakes: Swapping the variable and the constant (e.g., writing $3.00m + 0.50$).

Multi-Step Problem Solving

Integrated Essential Skills questions rarely ask for a single calculation. They require a "logic chain."

Strategy: Break It Down

1. Step 1: Calculate the intermediate value (e.g., the area of a floor).

2. Step 2: Apply the constraint or rate (e.g., multiply area by cost per sq ft).

3. Step 3: Apply final adjustments (e.g., add sales tax).

Exam Focus

• Why it matters: This separates high scorers from average scorers. You must keep track of your goal.

• Typical question patterns: Best-value comparisons (e.g., "Is it cheaper to buy two 12-inch pizzas or one 18-inch pizza?").

• Common mistakes: Stopping after Step 1. Always re-read the last sentence: "What is the question asking for?"

Quick Review Checklist

• Can you find a new average when a number is added or removed from a set?

• Do you know the difference between direct ($y=kx$) and inverse ($xy=k$) variation?

• Can you calculate percent change using $\frac{\text{New}-\text{Old}}{\text{Old}}$?

• Can you convert "$30\%$ off" into a multiplication by $0.70$?

• Do you have the formulas for Area (triangle, circle, trapezoid) and Volume (cylinder, box) memorized?

• Can you set up a unit conversion chain to go from seconds to hours?

• Can you translate "5 less than 3 times a number" into $3x - 5$?

Final Exam Pitfalls

1. The "Reverse Percent" Trap:

   • Mistake: Thinking that if a price goes up $20\%$ then down $20\%$, it returns to the original price.

   • Correction: It does not. $(1.20) \times (0.80) = 0.96$. The price is now $96\%$ of the original.

2. Units Mismatch:

   • Mistake: Multiplying feet by inches without converting.

   • Correction: Always convert to the same unit before calculating area or volume.

3. Average Rates:

   • Mistake: Averaging speeds directly. If you go $60$ mph there and $40$ mph back, the average speed is NOT $50$ mph.

   • Correction: Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$.

4. Radius vs. Diameter:

   • Mistake: Plugging the diameter into $A = \pi r^2$.

   • Correction: If the problem says "diameter," circle it, divide by 2, and write $r=…$ immediately.

5. Stopping Early:

   • Mistake: Finding the value of $x$ when the question asks for $2x + 5$.

   • Correction: Circle the goal in the question stem. When you finish, check if your answer matches the goal.

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GPT 5.2 Pro

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What You Need to Know

• Integrating Essential Skills on ACT Math emphasizes practical, cross-topic skills—rates/ratios, percent, measurement/geometry, basic statistics, and unit/number representations.

• Many questions are word problems—your job is to translate text into equations, track units, and choose the most efficient strategy.

• Accuracy comes from fundamentals: proportions, percent equations, geometry formulas, and clean arithmetic (including conversions).

Curriculum alignment note: ACT publicly describes reporting categories for ACT Math, including Integrating Essential Skills and a cross-cutting Modeling category. These notes follow those official category descriptions (skills-based rather than a fixed “syllabus” of named lessons). Where exact percentages/weights vary by test form and official updates, focus on the repeatable skill patterns below.

Rates, Ratios, and Proportional Relationships

Core ideas

• Ratio compares quantities by division, e.g. $\text{ratio} = \frac{a}{b}$.

• Rate is a ratio with units, e.g. $\frac{\text{miles}}{\text{hour}}$.

• Proportion sets two ratios equal: $\frac{a}{b} = \frac{c}{d}$.

• Unit rate simplifies to “per $1$”: $\text{unit rate} = \frac{\text{quantity}}{1}$.

Key setups

• Distance–rate–time: $d = rt$ (and $r = \frac{d}{t}$, $t = \frac{d}{r}$).

• Scale/mixture problems often reduce to a proportion with consistent units.

Worked example

A car travels $150$ miles in $3$ hours. At the same rate, how far in $5$ hours?

1. Find rate: $r = \frac{150}{3} = 50$ miles/hour.

2. Use $d = rt$: $d = 50 \cdot 5 = 250$ miles.

Exam Focus

• Why it matters: ACT frequently tests whether you can set up and solve a proportion correctly from a real context.

• Typical question patterns:

  • “At this constant rate, how long/how far…?” using $d = rt$.

  • Best-buy/unit-price comparisons (unit rates).

  • Ratio sharing: split a total in the ratio $a:b$.

• Common mistakes:

  • Mixing units (minutes vs hours) before applying $d = rt$.

  • Flipping a ratio inconsistently (e.g. $\frac{\text{miles}}{\text{hours}}$ vs $\frac{\text{hours}}{\text{miles}}$).

  • Treating non-proportional situations as proportional (watch for fixed starting fees).

Percentages and Percent Change

Core ideas

• Percent means “per $100$”: $p\% = \frac{p}{100}$.

• Percent of: $\text{part} = \left(\frac{p}{100}\right)!(\text{whole})$.

• Percent change:
  $\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$

• Multiplier method (fast on ACT):

  • Increase by $k\%$: multiply by $\left(1 + \frac{k}{100}\right)$.

  • Decrease by $k\%$: multiply by $\left(1 - \frac{k}{100}\right)$.

Worked example

A jacket costs $80$ dollars and is discounted $25\%$. What is the sale price?

1. Discount multiplier: $1 - \frac{25}{100} = 0.75$.

2. Sale price: $80 \cdot 0.75 = 60$ dollars.

Exam Focus

• Why it matters: Percent shows up constantly in practical contexts (discounts, tax, markups, growth/decay).

• Typical question patterns:

  • “After a $x\%$ increase/decrease, what is the new amount?”

  • “What percent of $A$ is $B$?” using $\frac{B}{A} \times 100\%$.

  • Successive changes (e.g. decrease then increase) using multipliers.

• Common mistakes:

  • Using the wrong base in percent change (must divide by original).

  • Adding/subtracting percents directly instead of multiplying successive factors.

  • Confusing “$p\%$ of” with “$p\%$ more than.”

Area, Surface Area, and Volume

Core ideas (high-frequency formulas)

• Rectangle area: $A = lw$.

• Triangle area: $A = \frac{1}{2}bh$.

• Circle area: $A = \pi r^2$; circumference: $C = 2\pi r$.

• Rectangular prism volume: $V = lwh$.

• Cylinder volume: $V = \pi r^2 h$.

ACT often combines geometry with unit conversion (e.g. $\text{cm}^2$ to $\text{m}^2$) or with word-problem interpretation.

Worked example

A cylindrical can has radius $3$ cm and height $10$ cm. Find volume.

1. Use $V = \pi r^2 h$.

2. Substitute: $V = \pi(3^2)(10) = 90\pi$ cubic cm.

Exam Focus

• Why it matters: Geometry/measurement questions reward formula fluency and careful attention to units.

• Typical question patterns:

  • Choose the correct formula, then substitute values (often with $\pi$ left exact).

  • Composite shapes: add/subtract areas or volumes.

  • Unit changes involving squared/cubed units.

• Common mistakes:

  • Forgetting to square the radius in $\pi r^2$.

  • Converting linear units but not area/volume units (see Unit Conversions section).

  • Using diameter as radius (if diameter is $d$, then $r = \frac{d}{2}$).

Mean, Median, and Other Measures of Center

Core ideas

• Mean (average): $\text{mean} = \frac{\text{sum of values}}{\text{number of values}}$.

• Median: middle value when ordered (or average of the two middle values for an even count).

• Mode: most frequent value.

• Weighted mean: $\bar{x} = \frac{\sum w<em>ix</em>i}{\sum w_i}$.

Worked example

Test scores: $70, 80, 80, 90, 100$.

• Mean: $\frac{70+80+80+90+100}{5} = \frac{420}{5} = 84$.

• Median: middle of ordered list $= 80$.

• Mode: $80$.

Exam Focus

• Why it matters: ACT tests whether you can compute and interpret basic statistics quickly, sometimes from tables.

• Typical question patterns:

  • Find mean/median from a list (may include an unknown like $x$).

  • “If one value changes, how does the mean change?” using sums.

  • Weighted average contexts (grades, prices).

• Common mistakes:

  • Finding the median without sorting.

  • Dividing by the wrong count (especially after removing/adding a value).

  • Treating a weighted average as a simple average.

Unit Conversions

Core ideas

• Convert by multiplying by a conversion factor equal to $1$ (a ratio of equal quantities).

• Track units like algebra—units should cancel.

Power rule for area/volume conversions

If $1\ \text{ft} = 12\ \text{in}$, then

• $1\ \text{ft}^2 = (12\ \text{in})^2 = 144\ \text{in}^2$

• $1\ \text{ft}^3 = (12\ \text{in})^3 = 1728\ \text{in}^3$

Worked example

Convert $3$ hours to minutes.

1. Use $1\ \text{hour} = 60\ \text{minutes}$.

2. $3\ \text{hours} \cdot \frac{60\ \text{minutes}}{1\ \text{hour}} = 180\ \text{minutes}$.

Exam Focus

• Why it matters: Conversion errors are a common “easy-point loss,” and ACT loves mixed-unit word problems.

• Typical question patterns:

  • Time, speed, and distance with minutes/hours.

  • Geometry with $\text{in}^2$, $\text{ft}^2$, $\text{cm}^3$, etc.

  • Rates like $\frac{\text{dollars}}{\text{pound}}$ or $\frac{\text{miles}}{\text{gallon}}$.

• Common mistakes:

  • Forgetting to square/cube the conversion for area/volume.

  • Using the reciprocal conversion factor (units don’t cancel).

  • Rounding too early—keep exact values until the end when possible.

Expressing Numbers in Different Ways

Core ideas

ACT often tests flexibility moving between fractions, decimals, percents, and scientific notation.

High-yield conversions

• Fraction to decimal: $\frac{a}{b} = a \div b$.

• Decimal to percent: multiply by $100\%$.

• Percent to decimal: divide by $100$.

• Scientific notation: $a \times 10^n$ where 1 \le a < 10 and $n$ is an integer.

Worked example

Write $0.00056$ in scientific notation.

1. Move decimal $4$ places right to get $5.6$.

2. Therefore $0.00056 = 5.6 \times 10^{-4}$.

Exam Focus

• Why it matters: Efficient number representation speeds up computation and helps you estimate/check reasonableness.

• Typical question patterns:

  • Convert among fraction/decimal/percent.

  • Operations with scientific notation (match exponents).

  • Ordering numbers written in different forms.

• Common mistakes:

  • Misplacing the decimal in scientific notation (sign of exponent wrong).

  • Confusing percent points vs percent change.

  • Simplifying fractions incorrectly (not dividing numerator and denominator by the same factor).

Multi-Step Problem Solving

Core ideas

• Many ACT questions require a plan: translate → set up equation(s) → solve → interpret.

• Common multi-step tools:

  • Distribute/factor, combine like terms.

  • Solve linear equations: isolate the variable.

  • Use systems when two conditions are given.

Worked example

A gym charges a $20$ dollar signup fee plus $15$ dollars per month. You paid $95$ dollars total. How many months?

1. Model total cost: $95 = 20 + 15m$.

2. Subtract $20$: $75 = 15m$.

3. Solve: $m = \frac{75}{15} = 5$ months.

Exam Focus

• Why it matters: “Integrating” means blending skills—algebra with units, percents with equations, geometry with rates.

• Typical question patterns:

  • Translate a scenario into $y = mx + b$ style structure.

  • Work backward from an outcome to an initial value.

  • Combine two relationships (e.g. percent discount plus tax).

• Common mistakes:

  • Solving the equation correctly but answering the wrong question (months vs total cost).

  • Skipping steps and losing sign/operation accuracy.

  • Not checking whether the answer is reasonable in context (negative time, impossible counts).

Mathematical Modeling in Real-World Contexts

Core ideas

• Mathematical modeling means representing a situation with math (equations, inequalities, functions, tables) and using it to make a decision.

• Common model types on ACT:

  • Linear model: $y = mx + b$ (constant rate + initial value).

  • Proportional model: $y = kx$ (passes through the origin).

  • Constraints with inequalities: e.g. $ax + by \le C$.

Worked example

A taxi charges $3$ dollars plus $2.5$ dollars per mile. Write a cost model for $m$ miles and find cost for $8$ miles.

1. Model: $C = 3 + 2.5m$.

2. Evaluate: $C = 3 + 2.5(8) = 3 + 20 = 23$ dollars.

Exam Focus

• Why it matters: ACT often embeds modeling inside Integrating Essential Skills—interpreting slope, intercepts, or constraints from context.

• Typical question patterns:

  • Identify which equation matches a word description.

  • Interpret $m$ (slope) as “per unit” and $b$ as starting amount.

  • Decide between options using a model (compare costs, break-even).

• Common mistakes:

  • Using a proportional model $y = kx$ when there is a fixed fee (needs $+b$).

  • Interpreting slope/intercept with wrong units.

  • Ignoring constraints (e.g. negative miles/months not allowed).

Quick Review Checklist

• Can you set up and solve a proportion $\frac{a}{b} = \frac{c}{d}$ from a word problem?

• Can you compute percent change using $\frac{\text{new}-\text{original}}{\text{original}} \times 100\%$?

• Do you know the key geometry formulas $A = lw$, $A = \frac{1}{2}bh$, $A = \pi r^2$, $V = lwh$, $V = \pi r^2 h$?

• Can you compute mean/median/mode and recognize when a weighted mean $\frac{\sum w<em>ix</em>i}{\sum w_i}$ is needed?

• Can you convert units using cancellation (e.g. multiply by $\frac{60\ \text{min}}{1\ \text{hr}}$)?

• Can you convert between decimals, fractions, percents, and scientific notation $a \times 10^n$?

• Can you translate a multi-step scenario into an equation like $95 = 20 + 15m$ and solve for the variable?

• Can you interpret a linear model $y = mx + b$ in context (meaning of $m$ and $b$)?

Final Exam Pitfalls

1. Using the wrong “base” for percent change — Always divide by the original value in $\frac{\text{new}-\text{original}}{\text{original}}$.

2. Conversion errors with squared/cubed units — If you convert length by a factor of $k$, then area scales by $k^2$ and volume by $k^3$.

3. Treating non-proportional situations as proportional — Fixed fees require $y = mx + b$, not $y = kx$.

4. Median without sorting / wrong count — Order the data and confirm how many values you have (especially after adding/removing a value).

5. Answering the wrong quantity — After solving, reread: are they asking for $m$, $C$, a rate, or a distance?

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Claude Opus 4.6

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What You Need to Know

• Integrating Essential Skills questions make up roughly 40–50% of the ACT Math section (approximately 24–28 of 60 questions). They test foundational math from pre-algebra through early geometry — concepts you've used since middle school but applied in trickier, multi-step contexts.

• These questions reward careful reading and methodical problem solving more than advanced math knowledge. Most errors come from rushing, misreading units, or skipping steps — not from lacking the underlying skill.

• You must be fluent with ratios, percentages, basic statistics, area/volume formulas, and unit conversions. The ACT doesn't provide a formula sheet, so commit key formulas to memory.

• Many Integrating Essential Skills problems are real-world word problems that require you to set up equations or expressions before solving — mathematical modeling is the core skill being tested.

Rates, Ratios, and Proportional Relationships

A rate compares two quantities with different units (e.g., miles per hour). A ratio compares two quantities with the same or related units (e.g., 3 boys to 5 girls). A proportion is an equation stating two ratios are equal:

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

To solve, cross-multiply: $a \cdot d = b \cdot c$.

Key Concepts

• Unit rate: a rate with denominator 1 (e.g., $\$3.50$ per gallon).

• Direct proportion: as one quantity increases, the other increases at a constant rate — $y = kx$.

• Setting up proportions: always match units in the same position (numerator-to-numerator, denominator-to-denominator).

Worked Example

A car travels 180 miles in 3 hours. At the same rate, how far will it travel in 5 hours?

$\frac{180 \text{ miles}}{3 \text{ hours}} = \frac{x \text{ miles}}{5 \text{ hours}}$

Cross-multiply: $180 \times 5 = 3x \implies x = 300$ miles.

Exam Focus

• Why it matters: Ratio and proportion questions appear frequently and are among the most straightforward points available on the ACT.

• Typical question patterns:

  • "If 4 widgets cost \$12, how much do 10 cost?"

  • Scale factor problems (maps, blueprints)

  • Speed/distance/time problems using $d = rt$

• Common mistakes:

  • Mixing up which quantity goes in the numerator vs. denominator — always label your units.

  • Forgetting that ratios can be part-to-part or part-to-whole; read the question carefully.

Percentages and Percent Change

Percent means "per hundred." To convert a percent to a decimal, divide by 100.

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$

Percent change measures increase or decrease relative to the original value:

$\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100$

Quick Multipliers

Using multipliers is faster and less error-prone than computing the percent separately and then adding or subtracting.

Exam Focus

• Why it matters: Percent problems appear in nearly every ACT, often embedded in word problems about sales tax, discounts, tips, or population change.

• Typical question patterns:

  • "A shirt originally costs \$40 and is 25% off. What is the sale price?"

  • Successive percent changes (e.g., increase then decrease)

  • Finding the original value given a percent and result

• Common mistakes:

  • Applying percent change to the new value instead of the original.

  • Assuming two successive 10% changes equal a 20% change — they don't. ($1.10 \times 0.90 = 0.99$, a net 1% decrease.)

Area, Surface Area, and Volume

Memorize these formulas — the ACT does not provide them.

Worked Example

Find the volume of a cylinder with radius 3 and height 10:

$V = \pi (3)^2 (10) = 90\pi \approx 282.7$

Exam Focus

• Why it matters: Geometry makes up a significant portion of ACT Math; area and volume questions are among the most common.

• Typical question patterns:

  • Finding the area of a composite shape (rectangle with a semicircle attached)

  • "How much paint is needed?" (surface area) or "How much water fits?" (volume)

  • Problems requiring you to solve for a missing dimension given area or volume

• Common mistakes:

  • Confusing radius and diameter — always check which one the problem gives you.

  • Forgetting to square the radius in area/volume formulas.

  • Using circumference when area is needed, or vice versa.

Mean, Median, and Other Measures of Center

• Mean (average): $\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

• Median: the middle value when data is ordered. For an even count, average the two middle values.

• Mode: the most frequently occurring value.

• Weighted average: when different values carry different weights — $\text{Weighted Mean} = \frac{\sum (w<em>i \cdot x</em>i)}{\sum w_i}$

Useful Rearrangement

If you know the mean and the number of values, you can find the total sum:

$\text{Sum} = \text{Mean} \times \text{Number of values}$

This is critical for problems like: "The average of 4 tests is 82. What score is needed on the 5th test to raise the average to 85?"

Sum of 4 tests: $82 \times 4 = 328$. Needed total for 5 tests: $85 \times 5 = 425$. Fifth score: $425 - 328 = 97$.

Exam Focus

• Why it matters: Statistics questions are reliable ACT staples, typically worth 2–4 questions per test.

• Typical question patterns:

  • "What score is needed to achieve a certain average?"

  • Finding the median from a list, table, or stem-and-leaf plot

  • How adding or removing a data point affects the mean or median

• Common mistakes:

  • Forgetting to order data before finding the median.

  • Dividing by the wrong count (e.g., number of categories vs. number of data points in a frequency table).

Unit Conversions

Dimensional analysis (factor-label method) is the most reliable approach. Multiply by conversion fractions so unwanted units cancel:

$60 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 88 \text{ ft/s}$

Common conversions to know: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5280 feet, 1 kg ≈ 2.2 lbs, 1 inch = 2.54 cm.

Exam Focus

• Why it matters: Unit conversion is frequently embedded inside larger word problems, making it easy to lose points if you're careless.

• Typical question patterns:

  • Converting units of area or volume (remember to square/cube the linear conversion factor)

  • Rate conversions (dollars per hour to dollars per minute)

• Common mistakes:

  • When converting area, forgetting to square the factor: $1 \text{ ft}^2 = 144 \text{ in}^2$, not 12.

  • Multiplying when you should divide, or vice versa — dimensional analysis prevents this.

Expressing Numbers in Different Ways

You should move fluently between fractions, decimals, and percents.

Also be comfortable with scientific notation: $a \times 10^n$ where 1 \leq a < 10.

Example: $0.00045 = 4.5 \times 10^{-4}$.

Exam Focus

• Why it matters: The ACT often presents answer choices in a different form than the one you naturally compute — you need to convert quickly.

• Typical question patterns:

  • Ordering fractions/decimals on a number line

  • Identifying equivalent expressions

• Common mistakes:

  • Rounding repeating decimals prematurely and picking a close-but-wrong answer.

  • Misplacing the decimal point when converting to/from scientific notation.

Multi-Step Problem Solving

Many ACT questions require two or more sequential operations. The math in each step may be simple, but chaining them correctly is the challenge.

Strategy

1. Read the entire problem before calculating.

2. Identify what's being asked — circle or underline the final question.

3. Work step-by-step, writing intermediate results clearly.

4. Check that your answer matches the question — a common trap is solving for an intermediate value and stopping too soon.

Exam Focus

• Why it matters: Multi-step problems are how the ACT increases difficulty without requiring advanced math.

• Typical question patterns:

  • Find a value, then use it in a second calculation

  • Apply a discount, then compute tax (or vice versa)

  • Compute area, then use it in a cost-per-square-foot calculation

• Common mistakes:

  • Answering with an intermediate result instead of the final answer.

  • Dropping negative signs or mishandling order of operations across steps.

Mathematical Modeling in Real-World Contexts

Mathematical modeling means translating a word problem into equations or expressions. The ACT loves scenarios about budgets, travel, mixtures, and measurements.

Approach

• Define variables clearly (e.g., let $x$ = number of hours).

• Write an equation that captures the relationship described.

• Solve and interpret — make sure your numerical answer makes sense in context (e.g., a negative number of hours doesn't make sense).

Example

A phone plan charges \$25/month plus \$0.10 per text. You can model the monthly cost as:

$C = 25 + 0.10t$

where $t$ is the number of texts. If your budget is \$40, solve $40 = 25 + 0.10t$ to get $t = 150$ texts.

Exam Focus

• Why it matters: Modeling questions test whether you can bridge the gap between a verbal description and mathematical reasoning — a skill the ACT values highly.

• Typical question patterns:

  • "Which equation represents…?"

  • Interpreting slope or y-intercept in context

  • Determining whether a linear or other model is appropriate

• Common mistakes:

  • Setting up the equation with the wrong operation (e.g., multiplying when the context calls for addition).

  • Ignoring constraints (e.g., $x$ must be a whole number if it represents people).

Quick Review Checklist

• Can you set up and solve a proportion using cross-multiplication?

• Can you compute percent change using $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$?

• Do you know the area, surface area, and volume formulas for rectangles, triangles, circles, rectangular prisms, and cylinders from memory?

• Can you find the mean, median, and mode of a data set — and rearrange the mean formula to find a missing value?

• Can you perform unit conversions using dimensional analysis, including squared and cubed units?

• Can you fluently convert between fractions, decimals, and percents?

• Can you read a multi-step word problem, identify the final question, and solve step-by-step without skipping ahead?

• Can you translate a real-world scenario into an algebraic equation and interpret the solution in context?

Final Exam Pitfalls

1. Answering the wrong question. Multi-step problems often have answer choices that match intermediate results. Always re-read the question before selecting your answer.

2. Confusing radius and diameter. Many geometry problems give the diameter. If a circle's diameter is 10, the radius is 5 — use $r = 5$ in $A = \pi r^2$.

3. Percent change denominator error. Percent change is always relative to the original value, not the new value. Mixing this up changes the answer significantly.

4. Forgetting to square or cube in unit conversions. Converting $\text{ft}^2$ to $\text{in}^2$ requires multiplying by $12^2 = 144$, not 12. For volume, multiply by $12^3 = 1728$.

5. Not ordering data before finding the median. The median requires a sorted list. If you pick the middle value from an unsorted list, you'll almost certainly get the wrong answer.

6. Successive percent changes aren't additive. A 20% increase followed by a 20% decrease does not return you to the original — it results in a 4% net decrease ($1.20 \times 0.80 = 0.96$). Watch for this trap.