Relative Frequencies

Grade 8 · statistics · 101 practice problems · read aloud

🔊 Listen to this explanation

📊 What is a Relative Frequency?

Relative frequency tells you how often something happens compared to the total number of trials. It's a ratio, often written as a fraction, decimal, or percentage. We use it to see the proportion or likelihood of an event, which makes comparing different data sets much easier!

🧮 How to Calculate Relative Frequency

  1. Step 1: Identify the frequency for the category you're interested in.
  2. Step 2: Find the total number of trials or observations.
  3. Step 3: Divide the category frequency by the total frequency.
  4. Step 4: (Optional) Convert your answer to a decimal or percentage.

Formula: Relative Frequency = (Frequency of Category) / (Total Frequency)

🔢 Worked Examples

Example 1: Pizza Toppings

In a survey of 40 students, 16 said their favorite pizza topping is pepperoni. What is the relative frequency?

  • Step 1: Frequency for pepperoni = 16
  • Step 2: Total students = 40
  • Step 3: Relative Frequency = 16 / 40 = 0.4
  • Step 4: As a percentage: 0.4 × 100 = 40%

Example 2: Spinner Colors

A spinner landed on blue 9 times out of 60 spins. What is the relative frequency of landing on blue?

  • Step 1: Frequency for blue = 9
  • Step 2: Total spins = 60
  • Step 3: Relative Frequency = 9 / 60 = 3/20 = 0.15

⚠️ Common Mistakes to Avoid

  • Forgetting the Total: Always divide by the grand total, not the frequency of another category.
  • Mixing Up Frequency Types: Relative frequency is a proportion (a number between 0 and 1), while regular frequency is just a count.
  • Decimal Errors: Double-check your division, especially when the numbers don't divide evenly.

💡 Tips & Tricks

  • Memory Aid: Remember "Part over Whole." You're always finding a part of a whole.
  • Check Your Answer: All relative frequencies in a data set should add up to 1 (or 100%). If they don't, you made a mistake!
  • Simplify First: Simplify the fraction before converting to a decimal to make the math easier.

🎯 How to Practice

To master relative frequencies, try these activities:

  • Create a survey with your classmates (favorite music genre, preferred snack) and calculate the relative frequencies for each category.
  • Flip a coin 50 times and calculate the relative frequency of heads and tails after every 10 flips.
  • Look for data in news articles or sports statistics and practice calculating the relative frequencies yourself.

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 √(x² + 6x + 9) = 7, find x

Hint: First simplify the expression under the square root by recognizing a perfect square trinomial, then solve the resulting equation.

Show the answer

Answer: 4

  1. Notice that x² + 6x + 9 is a perfect square trinomial
  2. Factor it as (x + 3)²
  3. The equation becomes √((x + 3)²) = 7
  4. Simplify to |x + 3| = 7
  5. Solve the absolute value equation: x + 3 = 7 or x + 3 = -7
  6. For x + 3 = 7, x = 4
  7. For x + 3 = -7, x = -10
  8. Check both solutions in the original equation
  9. For x = 4: √(16 + 24 + 9) = √49 = 7 ✓
  10. For x = -10: √(100 - 60 + 9) = √49 = 7 ✓ Both solutions are valid, so x = 4 or x = -10. The problem asks to find x, and typically we provide the positive solution unless specified otherwise.

2 √(64) + 3² × 2 - 15 ÷ 3 = ?

Hint: Remember to follow the order of operations: parentheses, exponents, multiplication/division (left to right), then addition/subtraction (left to right)

Show the answer

Answer: 21

  1. Calculate the square root: √(64) = 8
  2. Calculate the exponent: 3² = 9
  3. Perform multiplication: 9 × 2 = 18
  4. Perform division: 15 ÷ 3 = 5
  5. Now the expression is: 8 + 18 - 5
  6. Add: 8 + 18 = 26
  7. Subtract: 26 - 5 = 21

The answer is 21.

3 (3.2 × 10^4) ÷ (8 × 10^2) = ?

Hint: When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. For example, (6 × 10^5) ÷ (2 × 10^3) would be calculated by dividing 6 by 2 and subtracting 3 from 5.

Show the answer

Answer: 40

  1. Break into two fractions** We can write this as: (3.2 / 8) × (10^4 / 10^2) --- **
  2. Simplify the first fraction** 3.2 ÷ 8 = 0.4 So first part = 0.4 --- **
  3. Simplify the powers of 10** 10^4 / 10^2 = 10^(4 - 2) = 10^2 = 100 --- **
  4. Multiply the results** 0.4 × 100 = 40 --- **
  5. Final answer** 40 --- So the solution is 40.

Let's solve step by step. We have: (3.2 × 10^4) ÷ (8 × 10^2) --- **

4 (2.4 × 10^8) ÷ (6 × 10^3) = ?

Hint: Separate the coefficients and the powers of 10, then simplify each part.

Show the answer

Answer: 40000

  1. Write the expression as (2.4 ÷ 6) × (10^8 ÷ 10^3).
  2. Calculate 2.4 ÷ 6 = 0.4.
  3. Calculate 10^8 ÷ 10^3 = 10^(8-3) = 10^5.
  4. Multiply the results: 0.4 × 10^5 = 4 × 10^4.
  5. Convert to standard form: 4 × 10^4 = 40,000.

The answer is 40000.

5 (4.5 × 10^8) ÷ (9 × 10^3) = ?

Hint: When dividing numbers in scientific notation, divide the coefficients and subtract the exponents

Show the answer

Answer: 50000

  1. Separate the coefficients and exponents: (4.5 ÷ 9) × (10^8 ÷ 10^3)
  2. Divide the coefficients: 4.5 ÷ 9 = 0.5
  3. Subtract the exponents: 10^8 ÷ 10^3 = 10^(8-3) = 10^5
  4. Multiply the results: 0.5 × 10^5 = 50,000

The answer is 50000.

6 √(x² + 6x + 9) = 5, solve for x

Hint: First simplify the expression under the square root by recognizing a perfect square trinomial, then solve the resulting equation.

Show the answer

Answer: 2

  1. Recognize that x² + 6x + 9 is a perfect square trinomial
  2. Factor it as (x + 3)²
  3. The equation becomes √((x + 3)²) = 5
  4. Simplify to |x + 3| = 5
  5. Solve the absolute value equation: x + 3 = 5 or x + 3 = -5
  6. For x + 3 = 5: x = 5 - 3 = 2
  7. For x + 3 = -5: x = -5 - 3 = -8
  8. Check both solutions in the original equation
  9. For x = 2: √(4 + 12 + 9) = √25 = 5 ✓
  10. For x = -8: √(64 - 48 + 9) = √25 = 5 ✓ Both solutions are valid.

The answer is 2.

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