ARITHMETIC

FRACTIONS

A fraction is a number that represents a part of a whole or a collection of objects.

Parts of a Fraction

A fraction has two main numbers separated by a fraction line.

Numerator

- The top number.

- Shows how many parts are being considered or taken.

Denominator

- The bottom number.

- Shows the total number of equal parts in the whole.

Example: 3/4

- 3 = numerator → 3 parts are being considered.

- 4 = denominator → the whole is divided into 4 equal parts.

- The fraction line can also mean division: 3 ÷ 4.

Common Types of Fractions

Proper Fraction

- Numerator is smaller than the denominator.

- Example: 3/5

Improper Fraction

- Numerator is equal to or larger than the denominator.

- Example: 5/3

Mixed Number

- A whole number combined with a proper fraction.

- Example: 1 2/3

PERCENT CONVERSION

A percentage represents a number as a part of 100. Percentages can be converted into decimals and fractions.

Common Conversions

Percent to Decimal

- Divide by 100.

- Move the decimal point 2 places to the left.

- Example: 50% = 0.50 = 0.5

Decimal to Percent

- Multiply by 100.

- Move the decimal point 2 places to the right.

- Example: 0.75 = 75%

Percent to Fraction

- Write the percentage number over 100.

- Simplify the fraction.

- Example: 25% = 25/100 = 1/4

Core Formulas

Find a Percentage of a Number

Part = (Percentage ÷ 100) × Whole

Example:

20% of 50 = (20 ÷ 100) × 50 = 10

Find What Percent One Number Is of Another

Percentage = (Part ÷ Whole) × 100

Example:

10 is what percent of 50?

(10 ÷ 50) × 100 = 20%

Percentage Increase

New Value = Original + (Original × Percentage ÷ 100)

Percentage Decrease

New Value = Original − (Original × Percentage ÷ 100)

Quick Review

Percent → Decimal → ÷ 100

Decimal → Percent → × 100

Percent → Fraction → over 100, then simplify

Percentage of a number → (P ÷ 100) × Whole

What percent? → (Part ÷ Whole) × 100

EXPONENTS

Exponents are a shorthand way to show repeated multiplication of a number by itself.

Parts of an Exponent

An exponential expression has two main parts:

Base

- The main number being multiplied.

Exponent

- The small raised number.

- Tells how many times the base is multiplied by itself.

Example: 3²

- 3 = base

- 2 = exponent

- 3² = 3 × 3 = 9

Example: 4³

- 4 = base

- 3 = exponent

- 4³ = 4 × 4 × 4 = 64

Core Exponent Rules

Product Rule

When multiplying powers with the same base, add the exponents.

aᵐ × aⁿ = aᵐ⁺ⁿ

Example:

2³ × 2² = 2⁵ = 32

Quotient Rule

When dividing powers with the same base, subtract the exponents.

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example:

2⁵ ÷ 2² = 2³ = 8

Zero Exponent

Any non-zero number raised to the power of 0 equals 1.

a⁰ = 1

Example:

7⁰ = 1

Negative Exponent

A negative exponent means the reciprocal of the base with a positive exponent.

a⁻ᵐ = 1/aᵐ

Example:

2⁻³ = 1/2³ = 1/8

Power of a Power

When a power is raised to another power, multiply the exponents.

(aᵐ)ⁿ = aᵐⁿ

Example:

(2³)² = 2⁶ = 64

Quick Review

Base → number being multiplied

Exponent → tells how many times the base is used

Multiply same bases → add exponents

Divide same bases → subtract exponents

Power of 0 → 1

Negative exponent → reciprocal

Power of a power → multiply exponents