SATMath800 • FORMULA SHEETS

SAT Math Formula Sheets

Essential formulas, patterns, and shortcuts for the Digital SAT — organized from fundamental ideas to the models that build on them.

Build Your SAT Math Toolkit

Don’t just memorize formulas. Learn the ideas behind them, recognize when to use them, and connect each formula to the types of problems you will see on the SAT.

①

Build Your Foundation

Start with the mathematical building blocks.

Fractions & Ratios

Equivalent fractions, ratios, proportions, rates, and the relationships behind many SAT problems.

Fundamentals →

Percentages & Percent Change

Percent increase, decrease, percent change, discounts, markups, and real-world applications.

Fundamentals →

△

Geometry

Essential formulas for triangles, circles, area, perimeter, angles, and coordinate geometry.

Fundamentals →

②

Master the Algebra Toolkit

Build the tools needed for more advanced SAT models.

xⁿ

Powers & Exponents

Exponent rules, zero and negative exponents, multiplication, division, and power rules.

Algebra →

√

Radicals & Rational Exponents

Roots, simplifying radicals, rational exponents, and the connection between powers and roots.

Algebra →

y=mx+b

Linear Equations & Functions

Slope, equations, functions, tables, graphs, and recognizing linear relationships.

Algebra →

③

Master SAT Models

Use the foundations to solve recurring SAT problem types.

⌁

Quadratics & Parabolas

Standard, factored, and vertex forms, intercepts, transformations, and SAT strategies.

SAT Models →

↗

Exponential Growth & Decay

Growth, decay, compound interest, time units, tables, target values, and exponential models.

SAT Models →

The SATMath800 Learning Path

Fractions & Ratios → Percentages → Algebra Toolkit → SAT Models

Know the Formula. Recognize the Pattern.

The goal is not to memorize more formulas. It is to recognize which mathematical idea the SAT is testing — and use it efficiently.

SATMath800 • FOUNDATION 01

Fractions & Ratios

Build the numerical foundation behind proportions, percentages, rates, and many of the relationships you will encounter on the SAT.

1. Fraction Basics

A fraction represents a division relationship: the numerator is divided by the denominator.

\frac{a}{b}=a\div b

$b\neq0$

Numerator

The number on top tells you how many parts you have.

Denominator

The number on the bottom tells you how many equal parts make one whole.

Example

The fraction $\frac{3}{4}$ means 3 parts out of 4 equal parts.

\frac{3}{4}

↑ Back to Formula Sheets

2. Equivalent Fractions

Multiplying or dividing both numerator and denominator by the same nonzero number does not change the value.

\frac{a}{b} = \frac{ak}{bk}

Example

Multiply the numerator and denominator of $\frac{3}{4}$ by 5:

\frac{3}{4} = \frac{3(5)}{4(5)} = \frac{15}{20}

SAT Rule: Never change only the numerator or only the denominator when creating an equivalent fraction.

↑ Back to Formula Sheets

3. Ratios

A ratio compares two quantities using division.

a:b=\frac{a}{b}

Example

A class has 12 students who play soccer and 8 students who play basketball.

\[ 12:8=3:2 \]

The ratio of soccer players to basketball players is $3:2$.

Important: The order matters. The ratio of $a$ to $b$ is not the same as the ratio of $b$ to $a$.

↑ Back to Formula Sheets

4. Proportions

A proportion states that two ratios are equal.

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

Example

Solve:

\frac{3}{5} = \frac{x}{20}

Cross multiply:

\[ 3(20)=5x \]

\[ x=12 \]

Shortcut: For a proportion, cross multiplication turns the equation into one simple multiplication equation.

↑ Back to Formula Sheets

5. Rates

A rate compares quantities measured in different units.

\text{Rate} = \frac{\text{Quantity}}{\text{Time}}

Example

A car travels 180 miles in 3 hours.

\frac{180\text{ miles}} {3\text{ hours}} = 60\text{ miles/hour}

SAT Habit: Always check the units. A correct numerical answer with incorrect units can represent the wrong quantity.

\boxed{ \text{Fraction} \rightarrow \text{Ratio} \rightarrow \text{Proportion} \rightarrow \text{Rate} }

These ideas are closely connected through division and comparison.

↑ Back to Formula Sheets

SATMath800 • FORMULA SHEETS

SAT Percentages

Master the language of percentages, find a percentage of a quantity, handle discounts and percent change, and avoid the traps the SAT loves.

1. Percentage Basics

A percent simply means “out of 100.”

\[ p\%=\frac{p}{100} \]

Percent is another way to write a number.

Percent → Decimal

Divide by 100.

40\%=0.40

Decimal → Percent

Multiply by 100.

0.08=8\%

Fraction → Percent

Convert the fraction into a decimal or equivalent fraction out of 100.

\frac15=20\%

Percent → Fraction

Write the percent over 100 and simplify.

25\%=\frac{25}{100}=\frac14

First Thought:

Whenever you see a percentage, ask: Would this be easier as a fraction or a decimal?

50% One-half

25% One-fourth

75% Three-fourths

20% One-fifth

10% One-tenth

5% Half of 10%

2. Finding a Percentage of a Quantity

Nearly every percentage problem begins with one question: What is the whole?

\[ \boxed{\text{Part}=\text{Percent}\times\text{Whole}} \]

Think of this as a relationship, not a formula you must memorize.

Whole

The total amount.

Percent

The portion, written as a decimal or fraction.

Part

The amount represented by that percentage.

Guided Example 1

What is 25% of 80?

25\%=\frac14

\frac14(80)=20

Answer: 20

Notice that we did not need to calculate $0.25\times80$.

Guided Example 2

Find 15% of 200.

10\%\text{ of }200=20

5\%\text{ of }200=10

15\%=10\%+5\%

\[ 20+10=30 \]

Answer: 30

Guided Example 3

Find 35% of 80.

Method A:

35\%=0.35 \] \[ 0.35(80)=28

Method B:

35\%=30\%+5\% \] \[ 24+4=28

Both methods work. Choose whichever requires less work.

3. Percentage Number Sense

Many percentages can be built from simpler, familiar percentages.

5% Half of 10%

15% 10% + 5%

20% Double 10%

25% One-fourth

50% One-half

75% Three-fourths

Efficiency Note:

The goal is not to memorize tricks. The goal is to recognize useful relationships.

Example

Find 25% of 96.

25\%=\frac14 \] \[ 96\div4=24

Answer: 24

4. Discounts & Sale Prices

A common SAT trap is finding the discount correctly but forgetting that the question asks for the sale price.

Trap Example

A shirt costs $80 and is 25% off.

25\%\text{ of }80=20

The $20 is the discount.

\[ 80-20=60 \]

Sale price: $60

⚠️ SAT Trap Alert

If the question asks for the sale price, don’t stop after finding the discount.

Always ask: “What is the question actually asking for?”

Challenge

A jacket is discounted by 20%. After the discount, the price is $72. What was the original price?

After a 20% discount, the customer pays 80% of the original price.

\[ 0.80P=72 \]

P=\frac{72}{0.80}=90

Answer: $90

5. Percent Increase, Decrease & Change

These problems have two important steps: find the amount of change, then determine what the question wants you to do with it.

\[ \boxed{ \text{Percent Change} = \frac{\text{Change}} {\text{Original}} \times100\% } \]

The original amount is the reference point.

First Thought:

Before calculating, ask: Am I finding the change, or the final amount?

Those are not always the same.

Guided Example 1 — Increase

A school’s enrollment was 400 students. This year it increased by 15%. How many students are enrolled now?

10\%\text{ of }400=40 \] \[ 5\%\text{ of }400=20 \] \[ 15\%=40+20=60 \] \[ 400+60=460

Answer: 460

Guided Example 2 — Percent Change

A price increases from $80 to $100. By what percent did it increase?

\text{Change}=100-80=20

Compare the change with the original amount:

\frac{20}{80} = \frac14 = 25\%

Answer: 25%

⚠️ Another SAT Trap

A price goes from $80 to $100.

The change is $20, but the percent increase is 25%.

\frac{20}{80}=25\%

Do not divide by the new amount. Compare the change with the original amount.

6. Recognize Percentage Clues

Certain words should immediately make you think about percentage relationships.

Increased ↑

Decreased ↓

Discounted Sale

Reduced ↓

Grew ↑

Dropped ↓

SAT Habit:

Recognizing these clues quickly can save valuable time because you know what mathematical relationship to look for before calculating.

7. SAT-Style Practice

Apply the ideas above. Try these before looking at the solutions.

Question 1

A school has 800 students. If 35% participate in at least one club, how many students participate?

Question 2

A concert hall sold 240 tickets. If 75% were purchased online, how many tickets were purchased online?

Question 3

A store advertises a 20% discount on a backpack that originally costs $65. What is the sale price?

Question 4

A science class contains 24 students. If 3 students are absent, what percent of the class is absent?

Question 5 — Student-Produced Response

A bicycle originally costs $240. It is discounted by 15%. What is the amount of the discount?

Question 6 — Thinking Question

A store advertises “15% off” and another store advertises “$12 off.” A backpack costs $80. Which store gives the larger discount?

🔴 Challenge

A store offers 20% off, then an additional 20% off the sale price.

Is this the same as 40% off the original price? Explain your reasoning.

8. Answers & Solutions

Question 1: $35\%$ of 800 = 280.

Question 2: $75\%=\frac34$, so $\frac34(240)=180$.

Question 3: $20\%$ of 65 = 13. Sale price = $65-13=\$52$.

Question 4: \[ \frac{3}{24}=\frac18=12.5\% \] Answer: 12.5%.

Question 5: \[ 15\%\text{ of }240=36 \] Answer: $36.

Question 6: $15\%$ of $80 is $12. Both discounts are the same.

🔴 Challenge Solution

Let the original price be $100.

100\rightarrow80

The second 20% discount is taken from $80, not from the original $100.

20\%\text{ of }80=16 \] \[ 80-16=64

The final price is $64, so the total discount is $36, or 36%.

Therefore, two successive 20% discounts are not the same as 40% off.

🎯 SAT Connection

Percentage questions can appear in many different SAT contexts.

Discounts

Sales, prices, and promotions.

Surveys

Percent of a population or group.

Probability

Representing portions as percentages.

Data Tables

Interpreting percentages in data.

Scientific Experiments

Comparing quantities and changes.

Financial Situations

Prices, discounts, and real-world quantities.

Final SAT Habit:

Before calculating, identify the whole, the part, and exactly what the question is asking for.

SAT MATH ESSENTIALS

Geometry

A last-minute reference for the essential geometric relationships, formulas, and visual patterns you need to recognize quickly on the SAT.

Build Your Geometry Foundation

Geometry questions on the SAT often become much easier when you recognize the relationship being tested. Start by identifying whether angles are equal (congruent), add to 90°, or add to 180°. Then use the properties of the figure instead of trying to memorize every possible diagram.

Lines & Angles

The basic language of geometric figures.

Angles & Angle Measure

The basic language of geometric figures

An angle measures the amount of rotation between two rays that share a common endpoint.

Angle measure is expressed in degrees (°).

Complementary Angles

Two angles that add to 90°

Together, the two angles form a right angle.

$x+y=90^\circ$

Supplementary Angles

Two angles that add to 180°

Together, the two angles form a straight line.

$x+y=180^\circ$

Vertical Angles

Opposite angles formed by intersecting lines

When two lines intersect, the opposite angles are congruent.

Opposite angles are congruent.

Parallel Lines & A Transversal

When a third line crosses two parallel lines, it is called a transversal. The resulting angles create several important SAT relationships.

Corresponding Angles

Angles in the same relative position at the two intersections are congruent.

∠1 ≅ ∠5

∠2 ≅ ∠6

∠3 ≅ ∠7

∠4 ≅ ∠8

Alternate Interior Angles

Interior angles on opposite sides of the transversal are congruent.

∠3 ≅ ∠6

∠4 ≅ ∠5

Same-Side Interior Angles

Interior angles on the same side of the transversal are supplementary.

∠3 + ∠5 = 180°

∠4 + ∠6 = 180°

On the SAT, look for repeated angle measures first. Parallel lines create congruent relationships; when two angles form a straight line, use 180°.

SAT

Representative SAT-Style Questions

Original questions applying the angle relationships from this section.

Vertical Angles + Linear Pair

Two lines intersect as shown. One angle measures $3x+20^\circ$. Its vertical angle measures $y^\circ$, while a neighboring angle measures $(y+20)^\circ$.

What is the value of $x$?

A) 15

B) 18

C) 20

D) 24

Parallel Lines + Transversal

In the diagram below, the two horizontal lines are parallel. The labeled angles have measures $5x+12^\circ$ and $3x+8^\circ$. The angle adjacent to $5x+12^\circ$ is labeled $Q$.

What is the value of $Q$?

A) 62°

B) 68°

C) 74°

D) 80°

✓

Solutions

Translate the geometric relationships into equations.

Vertical Angles + Linear Pair

Step 1 — Use vertical angles

The angle $3x+20^\circ$ and the angle $y^\circ$ are vertical angles. Vertical angles are congruent.

$3x+20=y$

Step 2 — Use the linear pair

The bottom angle $y^\circ$ and a neighboring angle $(y+20)^\circ$ form a straight line. Therefore, they are supplementary.

$y+(y+20)=180$

$2y+20=180$
$2y=160$

$y=80$

Step 3 — Solve for $x$

Substitute $y=80$ into $3x+20=y$.

$3x+20=80$

$3x=60$

$x=20$

$x=20$ — Choice C

Alternate Exterior Angles + Linear Pair

Step 1 — Use alternate exterior angles

$Q$ and $3x+8^\circ$ are alternate exterior angles formed by two parallel lines. Therefore, they are congruent.

$Q=3x+8$

Step 2 — Use the linear pair

$Q$ and $5x+12^\circ$ form a straight line at the upper intersection. Therefore, they are supplementary.

$Q+(5x+12)=180$

Step 3 — Substitute

Replace $Q$ with $3x+8$.

$(3x+8)+(5x+12)=180$

$8x+20=180$

$8x=160$

Step 4 — Find $Q$

$x=20$. Therefore,

$Q=3(20)+8$

$Q=68^\circ$

$Q=68^\circ$ — Choice B

SAT takeaway: With parallel lines, identify the relationship first. Here, alternate exterior angles are congruent, while $Q$ and $5x+12^\circ$ form a linear pair.

Triangles

Essential triangle properties, formulas, and patterns for the SAT.

Triangles appear constantly on the SAT. Before calculating, identify what type of triangle you have and look for a known relationship. Remember the most important basics: the angles of every triangle add to $180^\circ$, a right triangle has a $90^\circ$ angle, and special triangles give you additional shortcuts.

Triangle Angle Sum

The three interior angles always total 180°

Add the three interior angles and set the sum equal to $180^\circ$.

$A+B+C=180^\circ$

Right Triangles

Hypotenuse, legs, and the Pythagorean theorem

The side opposite the $90^\circ$ angle is the hypotenuse and is always the longest side.

$a^2+b^2=c^2$

Equilateral Triangle

Three equal sides and three 60° angles

Every side has the same length, and every interior angle measures $60^\circ$.

$a=b=c\qquad\text{and}\qquad A=B=C=60^\circ$

Isosceles Triangle

Equal sides create equal base angles

The two equal sides face the two equal base angles. The height from the vertex also bisects the base.

$x+x+(180^\circ-2x)=180^\circ$

Area & Height

Base and perpendicular height

The height must be perpendicular to the chosen base. It does not have to lie inside the triangle for every possible triangle.

$A=\frac12bh$

Triangle Inequality

The third side must fit between two bounds

The sum of any two sides must be greater than the third side.

\(|a-b|

Similar Triangles

Same shape, possibly different size

AA Similarity

Two corresponding angles are congruent. The third angle must also match.

Angle–Angle

SAS Similarity

Two corresponding side ratios are equal and the included angles are congruent.

Side–Angle–Side

SSS Similarity

All three pairs of corresponding sides have the same ratio.

Side–Side–Side

Corresponding sides are proportional.

SAT strategy: Before calculating, identify the triangle type. A right triangle, equilateral triangle, isosceles triangle, or pair of similar triangles may give you the relationship you need immediately.

Latest Updates

Categories

SAT Math Formula Sheets

Build Your SAT Math Toolkit

Build Your Foundation

Fractions & Ratios

Percentages & Percent Change

Geometry

Master the Algebra Toolkit

Powers & Exponents

Radicals & Rational Exponents

Linear Equations & Functions

Master SAT Models

Quadratics & Parabolas

Exponential Growth & Decay

The SATMath800 Learning Path

Know the Formula. Recognize the Pattern.

Fractions & Ratios

1. Fraction Basics

Numerator

Denominator

2. Equivalent Fractions

3. Ratios

4. Proportions

5. Rates

SAT Percentages

1. Percentage Basics

Percent → Decimal

Decimal → Percent

Fraction → Percent

Percent → Fraction

2. Finding a Percentage of a Quantity

Whole

Percent

Part

3. Percentage Number Sense

4. Discounts & Sale Prices

5. Percent Increase, Decrease & Change

6. Recognize Percentage Clues

7. SAT-Style Practice

8. Answers & Solutions

🎯 SAT Connection

Discounts

Surveys

Probability

Data Tables

Scientific Experiments

Financial Situations

Build Your Geometry Foundation

Angles & Angle Measure

Complementary Angles

Supplementary Angles

Vertical Angles

Parallel Lines & A Transversal

Representative SAT-Style Questions

Solutions

Triangles

Triangle Angle Sum

Right Triangles

Equilateral Triangle

Isosceles Triangle

Area & Height

Triangle Inequality

Similar Triangles

AA Similarity

SAS Similarity

SSS Similarity

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