Frequency Tables

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Answer Key & Explanations

Frequency Tables · Grade 8 · Worksheet 1

1. A school surveyed 200 students about their favorite after-school activity and whether they play a musical instrument. The results are shown in the two-way frequency table below: | | Sports | Arts & Crafts | Video Games | Total | |-----------------|--------|---------------|-------------|-------| | Plays Instrument| 35 | 28 | 17 | 80 | | No Instrument | 45 | 22 | 53 | 120 | | Total | 80 | 50 | 70 | 200 | What is the probability that a randomly selected student who plays a musical instrument prefers Arts & Crafts as their favorite activity? Express your answer as a simplified fraction. Answer: 7/20 Solution: We are asked for the probability that a randomly selected student who plays a musical instrument prefers Arts & Crafts. This means we are only looking at the "Plays Instrument" row, not the whole table. - Sports: 35 - Arts & Crafts: 28 - Video Games: 17 - Total who play an instrument: 80…
   Full step-by-step solution

   Step 1: Understand the question We are asked for the probability that a randomly selected student who plays a musical instrument prefers Arts & Crafts. This means we are only looking at the "Plays Instrument" row, not the whole table. Step 2: Identify the relevant numbers from the table From the row "Plays Instrument": - Sports: 35 - Arts & Crafts: 28 - Video Games: 17 - Total who play an instrument: 80 Step 3: Set up the conditional probability We want: Probability = (Number who play instrument and prefer Arts & Crafts) / (Total who play instrument) So: P = 28 / 80 Step 4: Simplify the fraction 28/80 Divide numerator and denominator by 4: 28 ÷ 4 = 7 80 ÷ 4 = 20 So 28/80 = 7/20 Step 5: Final answer The probability is 7/20.

2. (2x + 3)(x - 4) - (x + 2)(x - 5) = ? Answer: x² - 7x - 2 Solution: Expand (2x + 3)(x - 4) = 2x(x) + 2x(-4) + 3(x) + 3(-4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12 Expand (x + 2)(x - 5) = x(x) + x(-5) + 2(x) + 2(-5) = x² - 5x + 2x - 10 = x² - 3x - 10 = (2x² - 5x - 12) - (x² - 3x - 10) = 2x² - 5x - 12 - x² + 3x + 10 = (2x² - x²) + (-5x + 3x) + (-12 + 10) = x² - 2x -…
   Full step-by-step solution

   Step 1: Expand (2x + 3)(x - 4) = 2x(x) + 2x(-4) + 3(x) + 3(-4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12 Step 2: Expand (x + 2)(x - 5) = x(x) + x(-5) + 2(x) + 2(-5) = x² - 5x + 2x - 10 = x² - 3x - 10 Step 3: Subtract the second expression from the first = (2x² - 5x - 12) - (x² - 3x - 10) = 2x² - 5x - 12 - x² + 3x + 10 Step 4: Combine like terms = (2x² - x²) + (-5x + 3x) + (-12 + 10) = x² - 2x - 2 The answer is x² - 2x - 2.

3. A survey asked 120 students about their preference for two sports: soccer and basketball. 65 students like soccer, 55 students like basketball, and 25 students like both sports. Complete the two-way frequency table and find how many students like only soccer. Answer: 40 Solution: Set up the two-way frequency table with rows for 'Like Soccer' and 'Do Not Like Soccer', and columns for 'Like Basketball' and 'Do Not Like Basketball'. - Total students = 120 - Like soccer = 65 - Like basketball = 55 - Like both = 25 (this goes in the intersection of 'Like Soccer' and 'Like…
   Full step-by-step solution

   Step 1: Set up the two-way frequency table with rows for 'Like Soccer' and 'Do Not Like Soccer', and columns for 'Like Basketball' and 'Do Not Like Basketball'. Step 2: Place the given numbers: - Total students = 120 - Like soccer = 65 - Like basketball = 55 - Like both = 25 (this goes in the intersection of 'Like Soccer' and 'Like Basketball') Step 3: Calculate students who like soccer but not basketball: 65 (total like soccer) - 25 (like both) = 40 Step 4: Calculate students who like basketball but not soccer: 55 (total like basketball) - 25 (like both) = 30 Step 5: Calculate students who like neither sport: 120 (total) - 65 (like soccer) - 30 (like only basketball) = 25 Or: 120 - 55 (like basketball) - 40 (like only soccer) = 25 Step 6: The number of students who like only soccer is 40. The answer is 40.

4. A survey of 150 students asked about their preferences for music genres. 80 students like pop music, 60 like rock music, and 35 like both genres. How many students like only pop music? Answer: 45 Solution: - Total students: 150 - Students who like pop: 80 - Students who like rock: 60 - Students who like both genres: 35 Students who like only pop = Total who like pop - Students who like both genres = 80 - 35 = 45 Pop only: 45 Rock only: 60 - 35 = 25 Both genres: 35 Neither genre: 150 - (45 + 25 +…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 150 - Students who like pop: 80 - Students who like rock: 60 - Students who like both genres: 35 Step 2: Calculate students who like only pop: Students who like only pop = Total who like pop - Students who like both genres = 80 - 35 = 45 Step 3: Verify with the two-way table: Pop only: 45 Rock only: 60 - 35 = 25 Both genres: 35 Neither genre: 150 - (45 + 25 + 35) = 45 Total: 45 + 25 + 35 + 45 = 150 ✓ The answer is 45.

5. A school surveyed its 8th grade students about their favorite type of movie and whether they prefer to watch movies at home or in theaters. The results are shown in the two-way frequency table below: | | Action | Comedy | Drama | Total | |---------------|--------|--------|-------|-------| | At Home | 24 | 32 | 16 | 72 | | In Theaters | 18 | 12 | 8 | 38 | | Total | 42 | 44 | 24 | 110 | What percentage of students who prefer watching movies at home chose Drama as their favorite movie type? Round your answer to the nearest whole percent. Answer: 22% Solution: *What percentage of students who prefer watching movies at home chose Drama as their favorite movie type?* We only look at the "At Home" row. Find how many of them chose Drama, then divide by the total number of students who prefer watching movies at home, and multiply by 100.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the question** We are asked: *What percentage of students who prefer watching movies at home chose Drama as their favorite movie type?* This means: We only look at the "At Home" row. Find how many of them chose Drama, then divide by the total number of students who prefer watching movies at home, and multiply by 100. --- **Step 2: Identify the numbers from the table** From the "At Home" row: - Drama (At Home) = 16 - Total At Home = 72 --- **Step 3: Set up the fraction** Fraction = (Students who prefer At Home and chose Drama) / (Total students who prefer At Home) Fraction = 16 / 72 --- **Step 4: Simplify the fraction** 16/72 = 8/36 = 4/18 = 2/9 --- **Step 5: Convert to a percentage** 2/9 as a decimal ≈ 0.222222... Multiply by 100: 0.222222 × 100 ≈ 22.2222% --- **Step 6: Round to nearest whole percent** 22.222% rounds to 22% --- **Final Answer:** 22%

6. A survey asked 150 students about their preference for two activities: reading and gaming. 80 students like reading, 70 students like gaming, and 30 students like both activities. Complete the two-way frequency table and find how many students like only reading. Answer: 50 Solution: - Total students: 150 - Students who like reading: 80 - Students who like gaming: 70 - Students who like both activities: 30 Students who like only reading = Total who like reading - Students who like both activities = 80 - 30 = 50 Only reading: 50 Only gaming: 70 - 30 = 40 Both activities: 30…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 150 - Students who like reading: 80 - Students who like gaming: 70 - Students who like both activities: 30 Step 2: Calculate students who like only reading: Students who like only reading = Total who like reading - Students who like both activities = 80 - 30 = 50 Step 3: Verify with the two-way table: Only reading: 50 Only gaming: 70 - 30 = 40 Both activities: 30 Neither activity: 150 - (50 + 40 + 30) = 30 Total: 50 + 40 + 30 + 30 = 150 ✓ The answer is 50.