ACT Math: Integrating Essential Skills (Core Quantitative Tools)

Rates, Ratios, and Proportional Relationships

What ratios and rates are (and why you keep seeing them)

A ratio compares two quantities using division. You can write a ratio as a:b, \frac{a}{b}, or “a to b.” Ratios matter because they describe relative size—for example, “3 cups of water for every 1 cup of rice” tells you a recipe can scale up or down.

A rate is a ratio that compares quantities with different units (miles per hour, dollars per pound). Rates are how you describe speed, cost, density, and many real-world relationships. A unit rate is a rate with denominator 1 (like \$2.50 per 1 pound). Unit rates are useful because they make comparisons easy: the smaller unit price is the better deal.

A proportional relationship happens when two quantities grow (or shrink) at a constant multiplicative rate. If y is proportional to x, then the ratio \frac{y}{x} stays constant. That constant is called the constant of proportionality.

How proportional relationships work

If y is proportional to x, you can model it with:

y = kx

Here k is the constant of proportionality. You can find k from any pair (x,y) by:

k = \frac{y}{x}

This shows up constantly on the ACT because it connects multiple skills:

• Algebra (writing equations, solving for unknowns)
• Tables and graphs (recognizing straight lines through the origin)
• Word problems (scaling, similar figures, unit pricing)

A common confusion is mixing up additive change with multiplicative change. Proportional relationships are multiplicative: doubling x doubles y.

Solving proportions (the mechanism)

A proportion is an equation stating two ratios are equal:

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

A reliable method is cross-multiplication, based on the idea that both sides represent the same number:

ad = bc

This is especially helpful when you have one unknown.

Worked examples

Example 1: Unit rate and comparison
A 12-ounce bag costs \$3.00 and a 20-ounce bag costs \$4.60. Which is the better deal?

Compute unit price:

\text{Price per ounce for 12 oz} = \frac{3.00}{12} = 0.25

\text{Price per ounce for 20 oz} = \frac{4.60}{20} = 0.23

Since 0.23 < 0.25, the 20-ounce bag is cheaper per ounce.

Example 2: Proportion from a scaling situation
A map uses a scale where 1 inch represents 30 miles. Two cities are 4.5 inches apart on the map. How far apart are they?

Set up a proportion:

\frac{1}{30} = \frac{4.5}{x}

Cross-multiply:

1\cdot x = 30\cdot 4.5

x = 135

They are 135 miles apart.

Exam Focus

• Typical question patterns:
  • “If a is to b as c is to d, find x” (a direct proportion).
  • Unit-rate comparisons (best buy, fastest speed, cheapest per unit).
  • Tables/graphs asking whether a relationship is proportional and to find k.
• Common mistakes:
  • Swapping numerator/denominator inconsistently when setting up a proportion.
  • Treating proportional change as additive (assuming “plus 5 each time” implies proportionality).
  • Forgetting units and comparing rates with different unit bases.

Percentages and Percent Change

What a percent really means

A percent is a ratio out of 100. Saying “35\%” means:

35\% = \frac{35}{100} = 0.35

Percents matter because many real contexts are communicated this way: discounts, tax, interest, markups, and population changes. ACT questions often test whether you can translate flexible language (“increased by,” “is what percent of,” “percent of change”) into math.

Percent of a number

“p\% of N” means multiply N by \frac{p}{100}:

p\%\text{ of }N = \frac{p}{100}N

You’re converting a comparison (out of 100) into a scaling factor.

Percent change: increase vs decrease

Percent change compares how much something changed relative to its original amount.

\text{Percent change} = \frac{\text{new} - \text{original}}{\text{original}}\times 100\%

• If the result is positive, it is a percent increase.
• If the result is negative, it is a percent decrease.

Another extremely useful view is the multiplier method:

• Increase by r\% means multiply by 1 + \frac{r}{100}.
• Decrease by r\% means multiply by 1 - \frac{r}{100}.

This matters because multi-step percent problems (discount then tax, or repeated growth) are easiest with multipliers.

Worked examples

Example 1: Percent of a number
A jacket costs \$80. A coupon gives 15\% off. What is the discount amount and the sale price?

Discount:

\frac{15}{100}\cdot 80 = 0.15\cdot 80 = 12

Sale price:

80 - 12 = 68

So the discount is \$12 and the sale price is \$68.

Example 2: Percent change
A town’s population went from 24,000 to 27,600. What is the percent increase?

Change:

27{,}600 - 24{,}000 = 3{,}600

Percent increase:

\frac{3{,}600}{24{,}000}\times 100\% = 0.15\times 100\% = 15\%

Example 3: Successive percent changes are not reversible
A price increases by 20\% and then decreases by 20\%. Is it back to the original?

Let the original be 100.

Increase by 20\%:

100\cdot 1.20 = 120

Decrease by 20\%:

120\cdot 0.80 = 96

It ends at 96, not 100. A common trap is assuming “up 20\% then down 20\% cancels.” Percent changes act on different bases.

Exam Focus

• Typical question patterns:
  • “What percent of A is B?” which translates to \frac{B}{A}\times 100\%.
  • Percent increase/decrease from an original to a new value.
  • Multi-step pricing (discounts, tax, tip) using multipliers.
• Common mistakes:
  • Using the new value as the denominator instead of the original in percent change.
  • Confusing “percent of” with “percent change.”
  • Adding/subtracting percents directly instead of multiplying successive changes.

Area, Surface Area, and Volume

Why geometry measurement is a core ACT skill

Area, surface area, and volume are ways of measuring “how much space” in 2D and 3D. ACT problems often combine these formulas with unit conversions, proportional reasoning, and multi-step setups (for example, finding a radius from a circumference before computing area).

A key habit: identify the shape, identify the measurement type (area vs volume), and keep track of units (square units for area, cubic units for volume).

Area (2D)

Area measures the amount of 2D region inside a boundary.

Common ACT formulas:

A_{\text{rectangle}} = lw

A_{\text{triangle}} = \frac{1}{2}bh

A_{\text{circle}} = \pi r^2

Variables: l length, w width, b base, h height (perpendicular to base), r radius.

A frequent mistake is using a slanted side as the triangle’s height. The height must be perpendicular to the chosen base.

Surface area (3D “outer area”)

Surface area is the total area of all faces on the outside of a 3D object.

Two common solids:

SA_{\text{rectangular prism}} = 2(lw + lh + wh)

SA_{\text{cylinder}} = 2\pi r^2 + 2\pi rh

For a cylinder, 2\pi r^2 is the area of the two circular bases, and 2\pi rh is the “wrap-around” lateral area (circumference times height).

Volume (3D “space inside”)

Volume measures how much 3D space a solid contains.

Common formulas:

V_{\text{rectangular prism}} = lwh

V_{\text{cylinder}} = \pi r^2h

V_{\text{sphere}} = \frac{4}{3}\pi r^3

A major ACT theme is that volume scales faster than length: changing dimensions by a factor changes volume by the cube of that factor.

Scaling and similarity (how size changes affect area and volume)

If every linear dimension is multiplied by a scale factor k:

• Perimeter scales by k
• Area scales by k^2
• Volume scales by k^3

This matters in problems about similar figures, resized photos, and models.

Worked examples

Example 1: Triangle area with given information
A triangle has base 14 cm and height 9 cm. Find its area.

A = \frac{1}{2}bh = \frac{1}{2}\cdot 14\cdot 9 = 63

Area is 63 square centimeters.

Example 2: Cylinder volume from diameter
A cylinder has diameter 10 inches and height 12 inches. Find its volume.

First find radius:

r = \frac{10}{2} = 5

Then volume:

V = \pi r^2h = \pi\cdot 5^2\cdot 12 = 300\pi

Units are cubic inches.

Example 3: Scaling volume
A cube’s side length doubles. How does its volume change?

If side length scales by k = 2, volume scales by k^3 = 8. The volume becomes 8 times as large.

Exam Focus

• Typical question patterns:
  • Compute area/volume after finding missing dimensions (radius from diameter, height from a diagram, etc.).
  • Composite figures: add/subtract areas or volumes of parts.
  • Similarity/scaling: “If the scale factor is k, what happens to area/volume?”
• Common mistakes:
  • Mixing up area and perimeter units (square vs linear) or volume units (cubic).
  • Using diameter where radius is required in circle-based formulas.
  • Forgetting to double counts in surface area (two bases on a cylinder, three distinct face pairs on a rectangular prism).

Mean, Median, and Other Measures of Center

What “center” means in data

When you have a list of numbers (a data set), a measure of center describes a typical or representative value. ACT questions use these ideas because they test quantitative reasoning with real contexts: salaries, test scores, temperatures, and survey results.

Different measures of center answer slightly different questions. Choosing the right one depends on whether the data have outliers (extreme values) or are skewed.

Mean (average)

The mean is the sum of the values divided by the number of values.

If the data are x1, x2, \dots, x_n, then:

\text{mean} = \frac{x1 + x2 + \cdots + x_n}{n}

The mean uses every value, so it is sensitive to outliers.

Median (middle value)

The median is the middle value when the data are ordered.

• If n is odd, the median is the single middle number.
• If n is even, the median is the average of the two middle numbers.

The median is more resistant to outliers than the mean.

Mode and midrange (sometimes tested)

The mode is the most frequent value. A set can have more than one mode or no mode.

The midrange is the average of the minimum and maximum:

\text{midrange} = \frac{\min + \max}{2}

Midrange is easy to compute but very sensitive to extremes, so it’s mainly used in simpler questions.

Weighted mean (average with different “weights”)

A weighted mean happens when some values count more than others (like course grades where tests are worth more than homework).

If values x1, x2, \dots, xn have weights w1, w2, \dots, wn, then:

\text{weighted mean} = \frac{w1x1 + w2x2 + \cdots + wnxn}{w1 + w2 + \cdots + w_n}

A common error is dividing by the number of categories instead of by the sum of weights.

Worked examples

Example 1: Mean and median comparison with an outlier
Data: 4, 5, 5, 6, 30

Mean:

\frac{4+5+5+6+30}{5} = \frac{50}{5} = 10

Median: the middle value is 5.

The mean (10) is pulled upward by the outlier 30, while the median (5) better reflects the typical value.

Example 2: Weighted mean (grade calculation)
A course grade is 40% exams and 60% homework. A student has 70 on exams and 90 on homework. Find the final average.

Use weights 0.40 and 0.60:

0.40\cdot 70 + 0.60\cdot 90 = 28 + 54 = 82

Final average is 82.

Exam Focus

• Typical question patterns:
  • Compute mean/median after adding or removing a data point.
  • Interpret which measure of center is more appropriate given an outlier.
  • Weighted average contexts (grades, mixture costs, combined rates in some setups).
• Common mistakes:
  • Forgetting to sort the data before finding the median.
  • Averaging the two middle values incorrectly when n is even.
  • Using an unweighted mean when the problem clearly implies different weights.

Unit Conversions

Why unit conversions are really about multiplication by 1

A unit conversion changes a measurement into different units without changing the actual quantity. The cleanest way to understand conversions is that you multiply by a conversion factor equal to 1.

For example, since 1 foot equals 12 inches, you can write:

\frac{12\text{ in}}{1\text{ ft}} = 1

and also:

\frac{1\text{ ft}}{12\text{ in}} = 1

You choose the factor that cancels the unit you don’t want.

Dimensional analysis (unit-canceling method)

In dimensional analysis, you treat units like algebraic symbols. Multiply by conversion factors so unwanted units cancel.

This reduces mistakes because you can “see” whether your setup makes sense.

Worked examples

Example 1: Converting speed
Convert 90 miles per hour to miles per minute.

Since 1 hour is 60 minutes:

90\frac{\text{mi}}{\text{hr}}\cdot \frac{1\text{ hr}}{60\text{ min}} = \frac{90}{60}\frac{\text{mi}}{\text{min}} = 1.5\frac{\text{mi}}{\text{min}}

Example 2: Converting area units (a common trap)
A square has side length 2 feet. Find its area in square inches.

First find area in square feet:

A = 2\cdot 2 = 4\text{ ft}^2

Now convert square feet to square inches. Since 1 ft is 12 in, then:

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

So:

4\text{ ft}^2 = 4\cdot 144\text{ in}^2 = 576\text{ in}^2

The mistake to avoid: converting by 12 instead of 144 for square units.

Exam Focus

• Typical question patterns:
  • One-step conversions (minutes to hours, inches to feet, dollars to cents).
  • Multi-step conversions (miles to feet to inches, or mixed time units).
  • Area/volume conversions that require squaring or cubing the conversion factor.
• Common mistakes:
  • Flipping the conversion factor so units don’t cancel correctly.
  • Forgetting that squared and cubed units scale by the square or cube of the linear conversion.
  • Mixing units inside a formula (using feet for one dimension and inches for another).

Expressing Numbers in Different Ways

Why “multiple representations” matters

ACT math often tests whether you recognize that the same quantity can be written in different but equivalent forms. Being fluent with these forms helps you simplify, compare values quickly, and choose efficient solving methods.

Fractions, decimals, and percents

These are three common representations of the same idea (a ratio).

To convert:

• Fraction to decimal: divide numerator by denominator.
• Decimal to percent: multiply by 100 and add the percent sign.
• Percent to decimal: divide by 100.

Example equivalence:

\frac{3}{8} = 0.375 = 37.5\%

A common mistake is moving the decimal the wrong direction when converting between decimal and percent.

Scientific notation

Scientific notation expresses very large or very small numbers as:

a\times 10^n

where 1 \le a < 10 and n is an integer.

Why it matters: it makes calculations and comparisons manageable, and it appears in some ACT contexts involving measurement, population, or computing.

Rules you’ll use:

(a\times 10^m)(b\times 10^n) = (ab)\times 10^{m+n}

\frac{a\times 10^m}{b\times 10^n} = \frac{a}{b}\times 10^{m-n}

Prime factorization and simplifying radicals

Prime factorization writes an integer as a product of primes. It helps with reducing fractions and simplifying square roots.

For radicals, a key idea is pulling out perfect squares:

\sqrt{ab} = \sqrt{a}\sqrt{b}

If a is a perfect square, \sqrt{a} becomes an integer.

Worked examples

Example 1: Fraction to percent
Convert \frac{7}{20} to a percent.

First to decimal:

\frac{7}{20} = 0.35

Then to percent:

0.35\times 100\% = 35\%

Example 2: Scientific notation multiplication
Compute (3\times 10^4)(2\times 10^3).

Multiply coefficients and add exponents:

(3\cdot 2)\times 10^{4+3} = 6\times 10^7

Example 3: Simplifying a radical
Simplify \sqrt{72}.

Factor 72 into a perfect square times something else:

72 = 36\cdot 2

Then:

\sqrt{72} = \sqrt{36\cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}

Exam Focus

• Typical question patterns:
  • Convert among fraction/decimal/percent to compare quantities.
  • Scientific notation operations and interpretation of exponent size.
  • Simplifying radicals and rational expressions as part of a larger algebra problem.
• Common mistakes:
  • Leaving scientific notation with a\ge 10 (not in proper form).
  • Treating \sqrt{a+b} as \sqrt{a}+\sqrt{b} (this is not valid in general).
  • Converting percents by dividing/multiplying by 100 in the wrong direction.

Multi-Step Problem Solving

What makes a problem “multi-step”

A multi-step problem is one where you cannot answer in a single direct calculation. You have to connect several ideas—often translating words to equations, solving for an intermediate value, then using that result in another computation.

On the ACT, multi-step problems are less about advanced math and more about discipline:

• Define variables clearly.
• Keep units consistent.
• Track what the question is actually asking (final target).

A reliable process you can use

1. Restate the goal: What quantity must you find?
2. List givens: Numbers, relationships, constraints.
3. Choose representations: Equation, table, diagram, or proportion.
4. Solve for intermediate values carefully.
5. Check reasonableness: sign, size, units.

This approach prevents common ACT pitfalls like answering an intermediate value instead of the requested final value.

Worked examples

Example 1: Combined geometry and percent
A circular pizza has diameter 14 inches. A coupon takes 25\% off the price of \$16. Find (1) the area of the pizza and (2) the sale price.

Part 1: area. Radius is half the diameter:

r = 7

Area:

A = \pi r^2 = \pi\cdot 7^2 = 49\pi

Part 2: sale price. A 25\% discount means multiply by 0.75:

16\cdot 0.75 = 12

Area is 49\pi square inches and sale price is \$12.

Example 2: Average after adding a value
The mean of 5 numbers is 12. A sixth number is added, and the new mean is 11. What is the sixth number?

Mean 12 for 5 numbers means total sum is:

5\cdot 12 = 60

New total sum with 6 numbers is:

6\cdot 11 = 66

So the sixth number is:

66 - 60 = 6

A common mistake is trying to “average the averages” instead of using total sums.

Exam Focus

• Typical question patterns:
  • Word problems requiring an intermediate step (find radius then area, find total then mean).
  • “After a change, what is the new value?” often mixing percents and basic operations.
  • Problems where setting up an equation from a scenario is the main challenge.
• Common mistakes:
  • Stopping after an intermediate result instead of answering the asked quantity.
  • Mixing units mid-solution (minutes with hours, inches with feet).
  • Rounding too early and accumulating error (keep exact values when possible, like \pi or fractions).

Mathematical Modeling in Real-World Contexts

What modeling means on the ACT

Mathematical modeling is translating a real situation into math you can compute with—typically an expression, equation, inequality, or function. On the ACT, modeling is often “hidden” inside word problems: you’re expected to decide what quantities matter, how they relate, and what assumptions are reasonable.

Good modeling connects several essential skills:

• Rates and proportions (constant speed, unit pricing)
• Percents (tax, growth, depreciation)
• Geometry (measurement and scaling)
• Statistics (interpreting averages)

Choosing a model: linear, proportional, or something else

Many ACT modeling situations are linear: the change is constant per unit.

A general linear model is:

y = mx + b

• m is the slope (rate of change).
• b is the starting value (when x = 0).

A proportional model is a special case where b = 0:

y = kx

A common mistake is assuming every straight-line situation is proportional. If there is a fixed starting fee, it is linear but not proportional.

Interpreting parameters (what the numbers mean)

Modeling is not just plugging numbers into formulas. You need to interpret:

• What does m represent in context? (dollars per mile, gallons per minute)
• What does b represent? (base fee, starting amount)
• What is a reasonable domain for x? (time cannot be negative; number of items is often an integer)

Worked examples

Example 1: Linear cost model with a base fee
A taxi charges a base fee of \$3 plus \$2.50 per mile. Write a model for the cost C after m miles, then find the cost for 8 miles.

Model:

C = 2.50m + 3

For 8 miles:

C = 2.50\cdot 8 + 3 = 20 + 3 = 23

Cost is \$23.

Example 2: Proportional model from constant speed
A cyclist travels at a constant speed of 15 miles per hour. Write a model for distance d after t hours.

Constant speed implies proportionality (starts at 0 miles when time is 0):

d = 15t

If t = 2.5, then:

d = 15\cdot 2.5 = 37.5

Example 3: Modeling with percent decrease (depreciation-like)
A phone costs \$500 and loses 30\% of its value after 1 year. What is its value after 1 year?

A 30\% decrease means multiply by 0.70:

500\cdot 0.70 = 350

Value is \$350.

What can go wrong in modeling

Modeling errors usually come from translation, not arithmetic:

• Misreading “per” (which indicates division and a rate).
• Confusing a one-time fee with a repeating fee.
• Using the wrong base for percent change (original vs new).
• Ignoring constraints (negative distance, fractional people, etc.).

Exam Focus

• Typical question patterns:
  • Write an equation from a word description and evaluate it for a given input.
  • Identify slope and intercept meaning in a context (rate vs starting value).
  • Decide whether a situation is proportional y = kx or general linear y = mx + b.
• Common mistakes:
  • Forcing proportionality when there is a nonzero starting amount.
  • Mixing up which variable is the input (independent) vs output (dependent).
  • Interpreting slope as total change rather than change per unit.