Showing posts with label SAT Prep. Show all posts

Friday, July 17, 2026

Guide to Percentages for Digital SAT Math, GCSE, IGCSE, ACT, and High School Algebra

Master Percentages with Step by Step Explanations, Worked Examples

Introduction

Percentages are one of the most useful mathematical concepts you will ever learn. Whether you are calculating a discount while shopping, comparing examination scores, interpreting statistical reports, reading graphs, analysing scientific data, or solving algebra problems, percentages appear almost everywhere.

In mathematics examinations, percentage questions often look simple, but they frequently test several concepts at the same time. A single problem may combine percentages with fractions, decimals, ratios, equations, graphs, probability, data analysis, or financial mathematics. Learning to recognise these connections is an important step towards becoming a confident problem solver.

A solid understanding of percentages is valuable for students preparing for the Digital SAT Math, PSAT, ACT Math, GCSE Mathematics, IGCSE Mathematics, Cambridge IGCSE Mathematics, Edexcel GCSE Mathematics, AQA GCSE Mathematics, OCR GCSE Mathematics, Scottish National 5 Mathematics, and many other secondary school mathematics courses around the world. Although examination styles may differ, the mathematical principles remain exactly the same.

This guide has been written from first principles. Every method is explained carefully, every algebraic step is shown, and every worked example follows a logical sequence so that you understand why each step works instead of simply memorising a formula.

After studying this chapter, you will be able to

• Understand the meaning of a percentage.

• Convert between percentages, fractions and decimals.

• Find the percentage of any quantity.

• Determine what percentage one number is of another.

• Solve percentage increase and percentage decrease problems.

• Apply percentage concepts to algebra and word problems.

• Develop the mathematical reasoning required for college entrance examinations and secondary school mathematics.

What Does the Word Percentage Mean?

The word percentage comes from the Latin phrase meaning per hundred.

Therefore,

1% means 1 out of every 100 equal parts.

Similarly,

10% means 10 parts out of 100.

25% means 25 parts out of 100.

75% means 75 parts out of 100.

100% means the entire quantity.

Understanding this simple idea makes every percentage calculation much easier.

Writing Percentages as Fractions

Every percentage can be written as a fraction whose denominator is 100.

Examples

25%

= 25/100

= 1/4

50%

= 50/100

= 1/2

75%

= 75/100

= 3/4

80%

= 80/100

= 4/5

125%

= 125/100

= 5/4

Notice that percentages greater than 100% are perfectly possible. They simply represent quantities larger than the original amount.

Writing Percentages as Decimals

Many Digital SAT, ACT, GCSE and IGCSE questions require changing percentages into decimals.

The rule is simple.

Divide the percentage by 100.

Examples

45%

= 45 ÷ 100

= 0.45

= 8 ÷ 100

= 0.08

150%

= 150 ÷ 100

= 1.5

0.5%

= 0.5 ÷ 100

= 0.005

Moving the decimal point two places to the left produces exactly the same result.

Converting Decimals into Percentages

To change a decimal into a percentage,

multiply by 100.

Examples

0.6

= 0.6 × 100

= 60%

0.08

= 0.08 × 100

= 8%

1.25

= 1.25 × 100

= 125%

Always remember to write the percentage symbol after multiplying by 100.

Converting Fractions into Percentages

There are two common methods.

Method 1

Convert the fraction into a decimal first.

Example

3/5

Divide.

3 ÷ 5

= 0.6

Multiply by 100.

0.6 × 100

= 60%

Method 2

Multiply the fraction directly by 100.

Example

3/5 × 100

= 300/5

= 60%

Both methods produce the same answer.

Choose whichever method you find easier.

Finding the Percentage of a Number

One of the most common examination questions asks you to calculate a certain percentage of a quantity.

The general rule is

Percentage of a number = Percentage × Number ÷ 100

Example 1

Find 25% of 80.

Step 1

Write the formula.

Percentage of a number

= Percentage × Number ÷ 100

Step 2

Substitute the values.

25 × 80 ÷ 100

Step 3

Multiply.

25 × 80

= 2000

Step 4

Divide by 100.

2000 ÷ 100

= 20

Therefore,

25% of 80 is 20.

Example 2

Find 18% of 250.

Step 1

Write the formula.

Percentage × Number ÷ 100

Step 2

Substitute.

18 × 250 ÷ 100

Step 3

Multiply.

18 × 250

= 4500

Step 4

Divide.

4500 ÷ 100

= 45

Therefore,

18% of 250 equals 45.

Example 3

Find 12.5% of 96.

Step 1

Write the formula.

Percentage × Number ÷ 100

Step 2

Substitute.

12.5 × 96 ÷ 100

Step 3

Multiply.

12.5 × 96

= 1200

Step 4

Divide.

1200 ÷ 100

= 12

Therefore,

12.5% of 96 is 12.

Using Fractions Instead of Percentages

Sometimes converting the percentage into a fraction makes the calculation much faster.

Example

Find 50% of 240.

50%

= 1/2

Half of 240

= 120

No multiplication is necessary.

Find 25% of 64.

25%

= 1/4

One quarter of 64

= 16

Find 75% of 80.

75%

= 3/4

First find one quarter.

80 ÷ 4

= 20

Now multiply by 3.

20 × 3

= 60

This approach is often quicker during timed examinations.

What Percentage Is One Number of Another?

Another common examination question asks

"What percentage is one quantity of another?"

The formula is

Percentage

= (Part ÷ Whole) × 100

Example 4

A class contains 40 students.

Twenty-eight students passed an examination.

What percentage passed?

Step 1

Identify the part.

Step 2

Identify the whole.

Step 3

Use the formula.

(28 ÷ 40) × 100

Step 4

Divide.

28 ÷ 40

= 0.7

Step 5

Multiply.

0.7 × 100

= 70%

Therefore,

70% of the students passed the examination.

Example 5

A football team won 18 matches out of 24.

What percentage of matches did they win?

Step 1

Write the formula.

(Part ÷ Whole) × 100

Step 2

Substitute.

(18 ÷ 24) × 100

Step 3

Simplify.

18 ÷ 24

= 0.75

Step 4

Multiply.

0.75 × 100

= 75%

Therefore,

The team won 75% of its matches.

SAT Strategy

Many Digital SAT, ACT, GCSE and IGCSE questions disguise percentage problems inside word problems, graphs, tables or algebraic expressions. Before beginning any calculation, identify whether the question is asking you to find a percentage of a quantity, what percentage one quantity is of another, or how much a quantity changes by a given percentage. Recognising the type of problem before performing any arithmetic often saves valuable time during an examination.

Practice Questions

Find 35% of 240.
Find 12% of 350.
Find 62.5% of 160.
Express 7/20 as a percentage.
Express 0.84 as a percentage.
What percentage is 45 out of 60?
What percentage is 18 out of 48?
Find 5% of 640.
Find 125% of 48.
A school has 600 students. If 456 students attend on a particular day, what percentage attended?

Answers

84
42
100
35%
84%
75%
37.5%
32
60
76%

Friday, July 10, 2026

Solving Quadratic Equations by Completing the Square The Digital SAT Math Guide to Quadratic Equations (Part 2)

The Digital SAT Math Guide to Quadratic Equations (Part 2)

Solving Quadratic Equations by Completing the Square

In the previous chapter, you learned how to solve quadratic equations by factoring. Factoring is often the quickest method, but many quadratic equations on the Digital SAT cannot be factored easily. Some have large coefficients, some produce fractional values, and others have no integer factors at all.

For these equations, completing the square provides a systematic method that always works. Unlike factoring, you do not have to guess factor pairs or recognize patterns. Instead, you follow the same sequence of algebraic steps every time.

Before learning the procedure, remember one important rule.

The coefficient of x² should be 1 before you begin completing the square.

A quadratic equation whose coefficient of x² is 1 is called a monic quadratic equation.

If the equation is not monic, divide every term on both sides of the equation by the coefficient of x². This makes the remaining steps much easier and reduces mistakes.

Universal Method for Completing the Square

Always follow these steps.

Write the equation in standard form.
If the coefficient of x² is not 1, divide every term on both sides by that coefficient.
Move the constant term to the opposite side.
Take half of the coefficient of x.
Square that number.
Add the squared value to both sides.
Rewrite the left side as the square of a binomial.
Take the square root of both sides.
Remember both the positive and negative square roots.
Solve the resulting linear equations.
Check every solution in the original equation.

Example 1

Solve

2x² + 12x + 4 = 0

Step 1

The coefficient of x² is 2.

Divide every term on both sides by 2.

2x² ÷ 2 + 12x ÷ 2 + 4 ÷ 2 = 0 ÷ 2

Simplify.

x² + 6x + 2 = 0

The equation is now monic.

Step 2

Subtract 2 from both sides.

x² + 6x + 2 − 2 = 0 − 2

Simplify.

x² + 6x = −2

Step 3

Take half of 6.

6 ÷ 2 = 3

Square it.

3² = 9

Step 4

Add 9 to both sides.

x² + 6x + 9 = −2 + 9

Simplify.

x² + 6x + 9 = 7

Step 5

Rewrite the left side.

(x + 3)² = 7

Step 6

Take square roots.

√((x + 3)²) = ±√7

Simplify.

x + 3 = ±√7

Step 7

Subtract 3 from both sides.

Positive solution:

x = −3 + √7

Negative solution:

x = −3 − √7

Example 2

Now solve a question that produces fractions immediately after making the quadratic monic.

4x² + 10x − 3 = 0

Step 1

The coefficient of x² is 4.

Divide every term by 4.

4x² ÷ 4 + 10x ÷ 4 − 3 ÷ 4 = 0 ÷ 4

Simplify.

x² + ⁵⁄₂x − ³⁄₄ = 0

Notice that fractions are perfectly acceptable. Do not convert them to decimals because exact fractions make later calculations more accurate.

Step 2

Move the constant term.

Add ³⁄₄ to both sides.

x² + ⁵⁄₂x = ³⁄₄

Step 3

Take half of the coefficient of x.

The coefficient is ⁵⁄₂.

Half of ⁵⁄₂ is

⁵⁄₂ ÷ 2 = ⁵⁄₄

Now square the result.

(⁵⁄₄)² = ²⁵⁄₁₆

Step 4

Add ²⁵⁄₁₆ to both sides.

x² + ⁵⁄₂x + ²⁵⁄₁₆ = ³⁄₄ + ²⁵⁄₁₆

Convert ³⁄₄ to sixteenths.

³⁄₄ = ¹²⁄₁₆

Now add.

¹²⁄₁₆ + ²⁵⁄₁₆ = ³⁷⁄₁₆

The equation becomes

x² + ⁵⁄₂x + ²⁵⁄₁₆ = ³⁷⁄₁₆

Step 5

Rewrite the left side.

(x + ⁵⁄₄)² = ³⁷⁄₁₆

Step 6

Take square roots.

√((x + ⁵⁄₄)²) = ±√(³⁷⁄₁₆)

Simplify.

x + ⁵⁄₄ = ±√37⁄4

Step 7

Subtract ⁵⁄₄ from both sides.

x = −⁵⁄₄ ± √37⁄4

These are the exact solutions.

Notice that completing the square works just as well with fractions as it does with whole numbers. On the Digital SAT, leaving answers in exact fractional or radical form is often the correct approach unless the question specifically asks for a decimal approximation.

Thursday, July 9, 2026

The Digital SAT Math Guide to Quadratic Equations (Part 1)

Master Quadratic Equations for the Digital SAT with Step-by-Step Explanations

Quadratic equations are one of the most important algebra topics on the Digital SAT. They appear in many forms, from straightforward equation-solving questions to graph interpretation, mathematical modeling, and real-world word problems. A strong understanding of quadratics also makes it much easier to learn functions, parabolas, coordinate geometry, and polynomial expressions.

Unlike linear equations, which produce straight lines when graphed, quadratic equations create curved graphs called parabolas. Learning how these equations behave will help you answer a wide variety of SAT Math questions quickly and accurately.

This guide is written for students who want to build a solid understanding of quadratics from the ground up. Every solution is explained one step at a time, with no skipped steps or unexplained shortcuts. By the time you finish this chapter, you'll understand what quadratic equations are, how to recognize them, and how to solve many of them by factoring.

Learning Goals

In this chapter, you will learn how to:

Recognize a quadratic equation.
Understand why quadratic equations are different from linear equations.
Identify the standard form of a quadratic equation.
Understand quadratic expressions and quadratic functions.
Solve simple quadratic equations by factoring.
Apply the Zero Product Property.
Check your answers correctly.
Avoid common mistakes made on the Digital SAT.

These concepts form the foundation for more advanced methods such as completing the square and using the quadratic formula, which will be covered in later chapters.

What Is a Quadratic Equation?

A quadratic equation is an equation in which the highest exponent of the variable is 2.

Examples include:

x² = 49

x² + 5x + 6 = 0

2x² − 7x + 3 = 0

4x² = 100

Notice that each equation contains x².

That squared variable is what makes the equation quadratic.

Compare these two equations.

Linear equation:

2x + 7 = 13

Highest exponent = 1

Quadratic equation:

x² + 2x − 15 = 0

Highest exponent = 2

The difference may seem small, but it changes how the equation behaves. A linear equation usually has one solution, while a quadratic equation can have two solutions, one solution, or no real solutions.

The Standard Form of a Quadratic Equation

Most quadratic equations on the SAT are written in standard form:

ax² + bx + c = 0

Each letter has a meaning.

a is the coefficient of x².

b is the coefficient of x.

c is the constant term.

For example,

3x² + 8x − 11 = 0

Here,

a = 3

b = 8

c = −11

Learning to identify these three values is important because later methods, especially the quadratic formula, use them directly.

Understanding the Parts of a Quadratic Equation

Consider

2x² + 9x − 18 = 0

This equation has three terms.

First term:

2x²

This is called the quadratic term because it contains x².

Second term:

This is called the linear term because it contains x.

Third term:

−18

This is the constant term because it contains no variable.

Recognizing these parts helps you identify which solving method to use.

What Does It Mean to Solve a Quadratic Equation?

Solving a quadratic equation means finding every value of the variable that makes the equation true.

For example,

x² = 25

Which numbers produce 25 when squared?

5² = 25

(−5)² = 25

Therefore,

x = 5

and

x = −5

Unlike linear equations, quadratic equations often have more than one correct answer.

Why Are There Two Answers?

Many students are surprised to discover that one equation can have two solutions.

The reason is simple.

Squaring removes the negative sign.

Positive example:

5 × 5 = 25

Negative example:

−5 × −5 = 25

Both calculations produce the same answer.

Whenever you solve an equation involving x², always ask yourself whether both a positive and a negative solution are possible.

Introduction to Factoring

Factoring x² + bx + c type when a = 1

Factoring is one of the fastest methods for solving many quadratic equations on the Digital SAT.

Factoring means rewriting an expression as the product of two smaller expressions.

Example:

x² + 5x + 6

can be written as

(x + 2)(x + 3)

When multiplied together,

(x + 2)(x + 3)

= x² + 3x + 2x + 6

= x² + 5x + 6

The original expression and its factored form are mathematically identical.

The Zero Product Property

Factoring works because of an important algebra rule.

A × B = 0

then

A = 0

B = 0

or both.

This rule is called the Zero Product Property.

Example:

(x + 4)(x − 7) = 0

Either

x + 4 = 0

x − 7 = 0

Solve each equation separately.

First equation:

x + 4 = 0

Subtract 4 from both sides.

x + 4 − 4 = 0 − 4

Simplify.

x = −4

Second equation:

x − 7 = 0

Add 7 to both sides.

x − 7 + 7 = 0 + 7

Simplify.

x = 7

Therefore,

the two solutions are

x = −4

and

x = 7

Example 1

Solve

x² + 7x + 12 = 0

Step 1

Write the equation.

x² + 7x + 12 = 0

Step 2

Find two numbers whose product is 12 and whose sum is 7.

Possible factor pairs of 12 are

1 and 12

2 and 6

3 and 4

Only

3 and 4

add to 7.

Step 3

Write the factors.

(x + 3)(x + 4) = 0

Step 4

Apply the Zero Product Property.

Either

x + 3 = 0

x + 4 = 0

Step 5

Solve the first equation.

Subtract 3 from both sides.

x + 3 − 3 = 0 − 3

Simplify.

x = −3

Step 6

Solve the second equation.

Subtract 4 from both sides.

x + 4 − 4 = 0 − 4

Simplify.

x = −4

Final Answer

x = −3

x = −4

Example 2

Solve

x² − 9x + 20 = 0

Step 1

Find two numbers whose product is 20.

1 and 20

2 and 10

4 and 5

Step 2

Which pair adds to −9?

Since the product is positive and the sum is negative,

both numbers must be negative.

−4 and −5

Step 3

Write the factors.

(x − 4)(x − 5) = 0

Step 4

Set each factor equal to zero.

x − 4 = 0

x − 5 = 0

Step 5

Solve.

Add 4 to both sides.

x = 4

Add 5 to both sides.

x = 5

Check

4² − 9(4) + 20

16 − 36 + 20

Correct.

Now check 5.

25 − 45 + 20

Correct.

Both answers satisfy the equation.

Example 3

Solve

x² + x − 12 = 0

Step 1

Find two numbers whose product is −12.

Possible pairs include

1 and −12

2 and −6

3 and −4

Step 2

Find the pair whose sum equals 1.

4 and −3

Step 3

Write the factors.

(x + 4)(x − 3) = 0

Step 4

Set each factor equal to zero.

x + 4 = 0

x − 3 = 0

Step 5

Solve.

Subtract 4 from both sides.

x = −4

Add 3 to both sides.

x = 3

Final Answer

x = −4

x = 3

A Quick Factoring Strategy

Whenever you see

x² + bx + c

ask yourself two questions.

Question 1

Which two numbers multiply to give c?

Question 2

Do those same numbers add to give b?

If the answer is yes,

you have found the correct factors.

With practice, this process becomes much faster.

Forgetting to Check

Substitute every solution back into the original equation.

If the equation balances,

your solution is correct.

Practice Questions

Solve by factoring.

x² + 5x + 6 = 0
x² − 8x + 15 = 0
x² + 9x + 20 = 0
x² − 7x + 10 = 0
x² + 2x − 15 = 0
x² − x − 12 = 0
x² + 11x + 24 = 0
x² − 10x + 24 = 0

Answers

x = −2, −3
x = 3, 5
x = −4, −5
x = 2, 5
x = 3, −5
x = 4, −3
x = −3, −8
x = 4, 6

A quadratic equation is an equation whose highest exponent is two. Before attempting to solve it, identify whether it is already in standard form and determine the values of a, b, and c. When the equation can be factored, rewriting it as the product of two binomials often provides the quickest solution. The Zero Product Property then allows each factor to be solved separately, producing all possible solutions. As you continue practicing, you'll begin to recognize common factor patterns quickly, an essential skill for success on the Digital SAT Math section.

Tuesday, July 7, 2026

SAT Math Formula Cheat Sheet for Quick Reference

SAT Math Formulas

The Digital SAT includes a reference sheet with some geometry formulas, but it does not include everything you'll need. Knowing the most common formulas before test day helps you solve problems more quickly and reduces the chance of making simple mistakes.

Exponent Rules

For any nonzero number a:

a⁰ = 1

a¹ = a

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

(aᵐ)ⁿ = aᵐⁿ

(ab)ⁿ = aⁿbⁿ

(a/b)ⁿ = aⁿ/bⁿ

a⁻ⁿ = 1/aⁿ

Remember

Multiply → add exponents.
Divide → subtract exponents.
A negative exponent means take the reciprocal.

Radicals

√a × √b = √(ab)

√a ÷ √b = √(a/b)

Examples

√49 = 7

√81 = 9

∛125 = 5

Perfect squares worth memorizing:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

121, 144, 169, 196, 225, 256, 289, 324, 361, 400

Linear Equations

Slope

m = (y₂ − y₁)/(x₂ − x₁)

Slope-intercept form

y = mx + b

Point-slope form

y − y₁ = m(x − x₁)

Standard form

Ax + By = C

Parallel lines have the same slope.

Perpendicular lines have negative reciprocal slopes.

Coordinate Geometry

Distance Formula

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint Formula

((x₁ + x₂)/2, (y₁ + y₂)/2)

Quadratic Equations

Standard form

ax² + bx + c = 0

Quadratic Formula

x = (−b ± √(b² − 4ac))/2a

Discriminant

b² − 4ac

Positive → two real solutions

Zero → one real solution

Negative → no real solutions

Vertex

x = −b/(2a)

Factoring Identities

Difference of Squares

a² − b² = (a + b)(a − b)

Perfect Square Trinomials

a² + 2ab + b² = (a + b)²

a² − 2ab + b² = (a − b)²

Circle Formulas

Circumference

C = 2πr

Area

A = πr²

Arc Length

(θ/360) × 2πr

Sector Area

(θ/360) × πr²

Diameter = 2r

Rectangles and Squares

Rectangle

Area = length × width

Perimeter = 2(length + width)

Square

Area = side²

Perimeter = 4 × side

Diagonal = side√2

Triangles

Area

½ × base × height

Pythagorean Theorem

a² + b² = c²

The angles inside every triangle add up to 180°.

An exterior angle equals the sum of the two opposite interior angles.

Special Right Triangles

45°–45°–90°

1 : 1 : √2

30°–60°–90°

1 : √3 : 2

These ratios are tested regularly.

Trigonometry

SOH CAH TOA

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

tan θ = sin θ/cos θ

Polygons

Interior Angle Sum

(n − 2) × 180°

Each Interior Angle of a Regular Polygon of n vertics

[(n − 2) × 180°]/n

Volume

Cube

s³

Rectangular Prism

lwh

Cylinder

πr²h

Cone

⅓πr²h

Sphere

⁴⁄₃πr³

Surface Area

Cube

6s²

Cylinder

2πr² + 2πrh

Sphere

4πr²

Mean

Sum of all values ÷ Number of values

Weighted Mean

Σ(value × weight) ÷ Σ(weights)

Probability

P(Event)

Number of Favorable Outcomes ÷ Total number of Outcomes

Complement Rule

P(complement event ) =1 − P(Event)

A probability is always between 0 and 1.

Percents

Increase

Original × (1 + rate)

Decrease

Original × (1 − rate)

Percent Change

(New − Original)/Original × 100%

Simple Interest

I = Prt

P = Principal

r = Interest Rate

t = Time

Exponential Growth and Decay

Growth

A = P(1 + r)ᵗ

Decay

A = P(1 − r)ᵗ

Functions

Example

f(x) = 2x + 3

f(5) = 13

Replace x with the given value.

Useful Constants

π ≈ 3.14

√2 ≈ 1.414

√3 ≈ 1.732