Number & Quantity

Master numbers, operations, and quantities to boost your ACT Math score!

0%

Number & Quantity Fundamentals

Number Systems

Understanding different number systems is essential for the ACT Math section.

Key ACT Concept: Types of Numbers

  • Natural Numbers: Positive integers (1, 2, 3, ...)
  • Whole Numbers: Natural numbers and zero (0, 1, 2, 3, ...)
  • Integers: Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
  • Rational Numbers: Numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0
  • Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., π, √2)
  • Real Numbers: All rational and irrational numbers
  • Complex Numbers: Numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)

Example:

Classify the following numbers: -7, 0, 3.14, √9, 2/3, √2

-7: Integer, Rational, Real

0: Whole Number, Integer, Rational, Real

3.14: Rational (can be written as 314/100), Real

√9 = 3: Natural, Whole, Integer, Rational, Real

2/3: Rational, Real

√2: Irrational, Real

Properties of Real Numbers

Understanding the properties of real numbers helps solve equations and simplify expressions.

Key ACT Concept: Number Properties

  • Commutative Property:
    • Addition: a + b = b + a
    • Multiplication: a × b = b × a
  • Associative Property:
    • Addition: (a + b) + c = a + (b + c)
    • Multiplication: (a × b) × c = a × (b × c)
  • Distributive Property: a × (b + c) = a × b + a × c
  • Identity Property:
    • Addition: a + 0 = a
    • Multiplication: a × 1 = a
  • Inverse Property:
    • Addition: a + (-a) = 0
    • Multiplication: a × (1/a) = 1 (for a ≠ 0)

Example:

Simplify the expression: 3(x + 2) - 2(x - 1)

Step 1: Use the distributive property.

\begin{align} 3(x + 2) - 2(x - 1) &= 3x + 6 - 2x + 2\\ &= 3x - 2x + 6 + 2\\ &= x + 8 \end{align}

Exponents and Radicals

Exponents and radicals are common in the ACT Math section.

Key ACT Concept: Exponent Rules

  • Product Rule: x^a × x^b = x^(a+b)
  • Quotient Rule: x^a ÷ x^b = x^(a-b)
  • Power Rule: (x^a)^b = x^(a×b)
  • Negative Exponents: x^(-a) = 1/(x^a)
  • Zero Exponent: x^0 = 1 (for x ≠ 0)
  • Fractional Exponents: x^(a/b) = (x^a)^(1/b) = (x^(1/b))^a = b√(x^a)

Example:

Simplify the expression: (2^3 × 2^2) ÷ 2^4

Step 1: Use the product rule for the numerator.

\begin{align} 2^3 \times 2^2 &= 2^{3+2}\\ &= 2^5 \end{align}

Step 2: Use the quotient rule.

\begin{align} \frac{2^5}{2^4} &= 2^{5-4}\\ &= 2^1\\ &= 2 \end{align}

Key ACT Concept: Radical Rules

  • Product Rule: √(a × b) = √a × √b
  • Quotient Rule: √(a ÷ b) = √a ÷ √b
  • Power Rule: √(a^n) = (√a)^n
  • Rationalizing the Denominator: Multiply by a form of 1 to eliminate radicals in the denominator

Example:

Simplify the expression: √12 + √27

Step 1: Express each radical in simplest form.

\begin{align} \sqrt{12} &= \sqrt{4 \times 3}\\ &= \sqrt{4} \times \sqrt{3}\\ &= 2\sqrt{3} \end{align}
\begin{align} \sqrt{27} &= \sqrt{9 \times 3}\\ &= \sqrt{9} \times \sqrt{3}\\ &= 3\sqrt{3} \end{align}

Step 2: Add the like terms.

\begin{align} \sqrt{12} + \sqrt{27} &= 2\sqrt{3} + 3\sqrt{3}\\ &= 5\sqrt{3} \end{align}

Complex Numbers

Complex numbers may appear in the ACT Math section, especially in questions involving quadratic equations.

Key ACT Concept: Complex Number Operations

  • Definition: a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1)
  • Addition/Subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
  • Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
  • Division: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$
  • Complex Conjugate: The complex conjugate of a + bi is a - bi

Example:

Simplify the expression: (3 + 2i) + (4 - 5i)

Step 1: Add the real parts and the imaginary parts separately.

\begin{align} (3 + 2i) + (4 - 5i) &= (3 + 4) + (2 - 5)i\\ &= 7 - 3i \end{align}

Ratios, Proportions, and Percentages

Ratios, proportions, and percentages are frequently tested on the ACT.

Key ACT Concept: Ratio and Proportion

  • Ratio: A comparison of two quantities, written as a:b or a/b
  • Proportion: An equation stating that two ratios are equal, written as a/b = c/d
  • Cross Multiplication: If a/b = c/d, then ad = bc

Example:

If 3 gallons of paint cover 750 square feet, how many gallons are needed to cover 2000 square feet?

Step 1: Set up a proportion.

\frac{3 \text{ gallons}}{750 \text{ sq ft}} = \frac{x \text{ gallons}}{2000 \text{ sq ft}}

Step 2: Cross multiply and solve for x.

\begin{align} 3 \times 2000 &= 750 \times x\\ 6000 &= 750x\\ x &= \frac{6000}{750}\\ x &= 8 \end{align}

Therefore, 8 gallons of paint are needed to cover 2000 square feet.

Key ACT Concept: Percentages

  • Percentage: A ratio expressed as a fraction of 100
  • Converting to Decimal: Divide by 100 (move decimal point two places left)
  • Converting to Fraction: Express as a fraction with denominator 100, then simplify
  • Percentage Change: $\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$

Example:

A shirt originally priced at $40 is on sale for $30. What is the percentage discount?

Step 1: Calculate the amount of discount.

\text{Discount} = \$40 - \$30 = \$10

Step 2: Calculate the percentage discount.

\begin{align} \text{Percentage Discount} &= \frac{\text{Discount}}{\text{Original Price}} \times 100\%\\ &= \frac{\$10}{\$40} \times 100\%\\ &= 0.25 \times 100\%\\ &= 25\% \end{align}

Therefore, the percentage discount is 25%.

Units and Measurement

Understanding units and conversions is important for solving real-world problems on the ACT.

Key ACT Concept: Unit Conversions

  • Length:
    • 1 foot = 12 inches
    • 1 yard = 3 feet
    • 1 mile = 5280 feet
    • 1 meter = 100 centimeters
    • 1 kilometer = 1000 meters
  • Weight/Mass:
    • 1 pound = 16 ounces
    • 1 ton = 2000 pounds
    • 1 kilogram = 1000 grams
  • Volume:
    • 1 cup = 8 fluid ounces
    • 1 pint = 2 cups
    • 1 quart = 2 pints
    • 1 gallon = 4 quarts
    • 1 liter = 1000 milliliters
  • Time:
    • 1 minute = 60 seconds
    • 1 hour = 60 minutes
    • 1 day = 24 hours
    • 1 week = 7 days
    • 1 year = 365 days (or 366 in a leap year)

Example:

A car travels at a speed of 65 miles per hour. How many feet does it travel in 10 seconds?

Step 1: Convert miles per hour to feet per second.

\begin{align} 65 \text{ miles per hour} &= 65 \text{ miles} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}\\ &= \frac{65 \times 5280}{3600} \text{ feet per second}\\ &= \frac{343200}{3600} \text{ feet per second}\\ &= 95.33 \text{ feet per second} \end{align}

Step 2: Calculate the distance traveled in 10 seconds.

\begin{align} \text{Distance} &= \text{Speed} \times \text{Time}\\ &= 95.33 \text{ feet per second} \times 10 \text{ seconds}\\ &= 953.3 \text{ feet} \end{align}

Therefore, the car travels approximately 953 feet in 10 seconds.

Ready to Practice?

Now that you've learned the fundamentals of number and quantity, test your knowledge with practice problems!

Practice Problems

Test your understanding with these practice problems. Select a difficulty level to begin.

Select a difficulty level to load practice problems.

Number & Quantity Challenges

Put your number and quantity skills to the test with these timed challenges. Complete them to earn badges and level up your character!

Exponents & Radicals Challenge

100 XP

Solve 5 exponent and radical problems in 5 minutes.

5 minutes
5 problems

Ratios & Percentages Challenge

150 XP

Solve 5 ratio and percentage problems in 7 minutes.

7 minutes
5 problems

Complex Numbers Challenge

200 XP

Solve 5 complex number problems in 10 minutes.

10 minutes
5 problems

Number & Quantity ACT Test Simulation

Take a timed test that simulates the number and quantity questions you'll see on the ACT. This will help you identify your strengths and areas for improvement.

Test Details

  • Time Limit: 15 minutes
  • Questions: 10 number and quantity questions in ACT format
  • Topics Covered: Number systems, exponents, radicals, complex numbers, ratios, proportions, and percentages
  • Scoring: Instant results with detailed explanations

Ready to Test Your Skills?

This simulation will help you prepare for the actual ACT math section.