Frequency Tables

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Frequency Tables · Grade 8 · Worksheet 1

1. Mason surveyed 132 students in the 8th grade about their preferred method of learning a new topic: by reading a textbook, watching a video tutorial, or doing a hands-on activity. He also asked each student whether they consider themselves a visual learner or not. The results are partially recorded in the two-way frequency table below. What percentage of students who prefer hands-on activities are NOT visual learners? Round your answer to the nearest whole percent. | | Reading | Video | Hands-On | Total | |--------------------|---------|-------|----------|-------| | Visual Learner | 17 | 32 | 12 | 61 | | Not a Visual Learner | 22 | 27 | ? | 71 | | Total | 39 | 59 | 34 | 132 | Answer: 65 Solution: Find the missing value for 'Not a Visual Learner' who prefer 'Hands-On'. The total for 'Hands-On' is 34, and 12 'Visual Learners' prefer 'Hands-On'. So, 34 - 12 = 22 'Not a Visual Learners' prefer 'Hands-On'.
   Full step-by-step solution

   Step 1: Find the missing value for 'Not a Visual Learner' who prefer 'Hands-On'. The total for 'Hands-On' is 34, and 12 'Visual Learners' prefer 'Hands-On'. So, 34 - 12 = 22 'Not a Visual Learners' prefer 'Hands-On'. Step 2: Identify the specific group for the percentage. The question asks about students who prefer 'Hands-On' activities. The total number of students who prefer 'Hands-On' is 34. Step 3: Identify the part of that group we are interested in. The question asks for the percentage of those students who are 'Not a Visual Learner'. From Step 1, that number is 22. Step 4: Calculate the percentage. Divide the part by the whole: 22 / 34 = 0.6470588... Step 5: Convert the decimal to a percentage by multiplying by 100: 0.6470588 * 100 = 64.70588... Step 6: Round to the nearest whole percent. The digit after the decimal point is 7, which is 5 or greater, so we round up. 64.70588... rounded to the nearest whole number is 65. The answer is 65.

2. A school surveyed 200 students about their favorite extracurricular activities. The results are shown in the two-way frequency table below: | | Sports | Music | Art | Total | |------------|--------|-------|-----|-------| | Boys | 45 | 20 | 15 | 80 | | Girls | 25 | 50 | 45 | 120 | | Total | 70 | 70 | 60 | 200 | What percentage of students who prefer music are girls? Round your answer to the nearest whole percent. Answer: 71% Solution: We are asked: "What percentage of students who prefer music are girls?" Understand the question. We want the percentage of students who prefer music that are girls. Look at the "Music" column: Boys = 20, Girls = 50, Total = 70.
   Full step-by-step solution

   We are asked: "What percentage of students who prefer music are girls?" Step 1: Understand the question. We want the percentage of students who prefer music that are girls. This means: among the students who prefer music, what fraction are girls? Then convert to percent. Step 2: From the table, find the total number of students who prefer music. Look at the "Music" column: Boys = 20, Girls = 50, Total = 70. So total students who prefer music = 70. Step 3: Among these 70 students, the number who are girls = 50. Step 4: Calculate the fraction: (Girls who prefer music) / (Total who prefer music) = 50 / 70. Step 5: Simplify 50/70 = 5/7. Step 6: Convert to a percentage: (5/7) × 100 = 500/7 ≈ 71.42857...% Step 7: Round to the nearest whole percent: 71.428...% rounds to 71%. Final answer: 71%

3. (3x² - 2x + 7) + (5x² + 4x - 3) = ? Answer: 8x² + 2x + 4 Solution: Write the expression: (3x² - 2x + 7) + (5x² + 4x - 3) Remove parentheses: 3x² - 2x + 7 + 5x² + 4x - 3 Combine x² terms: 3x² + 5x² = 8x² Combine x terms: -2x + 4x = 2x Combine constant terms: 7 - 3 = 4 Write the final expression: 8x² + 2x + 4 The answer is 8x² + 2x + 4.
   Full step-by-step solution

   Step 1: Write the expression: (3x² - 2x + 7) + (5x² + 4x - 3) Step 2: Remove parentheses: 3x² - 2x + 7 + 5x² + 4x - 3 Step 3: Combine x² terms: 3x² + 5x² = 8x² Step 4: Combine x terms: -2x + 4x = 2x Step 5: Combine constant terms: 7 - 3 = 4 Step 6: Write the final expression: 8x² + 2x + 4 The answer is 8x² + 2x + 4.

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. (3x - 2)(2x + 5) - (x + 3)² = ? Answer: 5x² + 5x - 19 Solution: Expand (3x - 2)(2x + 5) using FOIL 3x × 2x = 6x² 3x × 5 = 15x -2 × 2x = -4x -2 × 5 = -10 So (3x - 2)(2x + 5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10 Expand (x + 3)² using (a + b)² = a² + 2ab + b² x² + 2(x)(3) + 3² = x² + 6x + 9 (6x² + 11x - 10) - (x² + 6x + 9) = 6x² + 11x - 10 - x² - 6x - 9 6x² -…
   Full step-by-step solution

   Step 1: Expand (3x - 2)(2x + 5) using FOIL 3x × 2x = 6x² 3x × 5 = 15x -2 × 2x = -4x -2 × 5 = -10 So (3x - 2)(2x + 5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10 Step 2: Expand (x + 3)² using (a + b)² = a² + 2ab + b² x² + 2(x)(3) + 3² = x² + 6x + 9 Step 3: Subtract the second expression from the first (6x² + 11x - 10) - (x² + 6x + 9) = 6x² + 11x - 10 - x² - 6x - 9 Step 4: Combine like terms 6x² - x² = 5x² 11x - 6x = 5x -10 - 9 = -19 Step 5: Final result is 5x² + 5x - 19

6. A survey of 250 students asked about their preferences for music genres: Rock and Pop. 140 students like Rock, 120 like Pop, and 60 like both genres. How many students like only Pop? Answer: 60 Solution: - Total students: 250 - Students who like Rock: 140 - Students who like Pop: 120 - Students who like both genres: 60 Students who like only Pop = Total who like Pop - Students who like both genres = 120 - 60 = 60 Only Pop: 60 Only Rock: 140 - 60 = 80 Both genres: 60 Neither genre: 250 - (60 + 80…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 250 - Students who like Rock: 140 - Students who like Pop: 120 - Students who like both genres: 60 Step 2: Calculate students who like only Pop: Students who like only Pop = Total who like Pop - Students who like both genres = 120 - 60 = 60 Step 3: Verify with the two-way table: Only Pop: 60 Only Rock: 140 - 60 = 80 Both genres: 60 Neither genre: 250 - (60 + 80 + 60) = 50 Total: 60 + 80 + 60 + 50 = 250 ✓ The answer is 60.

7. (3x - 2)(2x + 5) - (x + 1)(x - 3) = ? Answer: 5x² + 12x - 7 Solution: Expand (3x - 2)(2x + 5) 3x × 2x = 6x² 3x × 5 = 15x -2 × 2x = -4x -2 × 5 = -10 So (3x - 2)(2x + 5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10 Expand (x + 1)(x - 3) x × x = x² x × (-3) = -3x 1 × x = x 1 × (-3) = -3 So (x + 1)(x - 3) = x² - 3x + x - 3 = x² - 2x - 3 (6x² + 11x - 10) - (x² - 2x - 3) =…
   Full step-by-step solution

   Step 1: Expand (3x - 2)(2x + 5) 3x × 2x = 6x² 3x × 5 = 15x -2 × 2x = -4x -2 × 5 = -10 So (3x - 2)(2x + 5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10 Step 2: Expand (x + 1)(x - 3) x × x = x² x × (-3) = -3x 1 × x = x 1 × (-3) = -3 So (x + 1)(x - 3) = x² - 3x + x - 3 = x² - 2x - 3 Step 3: Subtract the second expression from the first (6x² + 11x - 10) - (x² - 2x - 3) = 6x² + 11x - 10 - x² + 2x + 3 Step 4: Combine like terms 6x² - x² = 5x² 11x + 2x = 13x -10 + 3 = -7 Final answer: 5x² + 13x - 7